In this paper we show that area of any quadrilateral can be estimated from the four lengths sides. With the Triangle Inequality Theorem and a novel provided diagonal's formula, the boundaries of quadrilateral diagonals are found. Finally Bretschneider's formula can be applied to find a set of possible areas.

In this paper we will prove that if a compact $A$ in $R^n$ belongs to the unit ball in $R^n$, then $A$ has a slice of measure greater than a calculable constant times the measure of $A$. Our result is sharp.

Let $X$ be a complex projective manifold. Denote the singular cohomology with rational coefficients by $H^{i} (X;mathbb Q)$ where $i$ is the degree. For a Hodge class $uin H^{2p}(X;mathbb Q)$, we first prove it can be represented by the difference of two oriented singular cycles. Each one, when it is regarded as a current of integration, is of degree $(p, p)$ type and positive. Using the structures on positive currents, we prove that there exists an algebraic set $A$ of codimension $p$ such thatthe support of both currents is contained in $A$. Hence, begin{equation} uin kerbiggl ( H^{2p}(X;mathbb Q)to H^{2p}(X-A;mathbb Q)biggr) end{equation} where "ker" denotes the kernel of the restriction map. Then (0.1) implies that $u$ is represented by an algebraic cycle.

The present article provides a detailed mathematical treatise on the geometry of skateboard transition ramps. These usually form a circular segment shape and are used in skateboarding for transitioning from a horizontal plane to another angle of incline. Transitions play a crucial role in the flow of skateparks and are required, for example, for quarter pipes, mini ramps, vert ramps, or jump ramps. Skateboarding has been an Olympic sport since 2023, yet many publicly funded skateparks still do not meet the demands of the sport. On one hand, there is often a lack of willingness from authorities to engage with the athletes beforehand; on the other hand, the people involved in the planning process often lack the necessary experience and mathematical expertise to perfectly fulfill the users' needs. Thoughtful planning is particularly crucial for ramps with curved surfaces to ensure the flow of the skatepark. In addition to the mathematical analysis, source codes for programs are provided to make the calculations as convenient as possible for all users.

In this paper we continue the development of the circles of partition by introducing a certain geometry of the axes of complex circles of partition. We use this geometry to verify the condition in the squeeze principle in special cases with regards to the orientation of the axes of complex circles of partition.

In this paper, we explore the properties and optimization techniques related to polyhedral cones and energy numbers with a focus on the cone of positive nxn semidefinite matrices and efficient computation strategies for kernels. In Part (a), we examine the polyhedral nature of the cone of positive semidefinite matrices, S^n , establishing that it does not form a polyhedral cone for due to its infinite dimensional characteristics. In Part (b), we present an algorithm for efficiently computing the kernel function K(x, x prime) = (latex formatting in wysiwig)on-the-fly, leveraging a polyhedral description of the convex hull generated by the feature mappings phi and phi prime. By restructuring the problem and using gradient-based optimization techniques, our approach minimizes memory usage and computational overhead, thus enabling scalable computation. Through examples and visualizations, we demonstrate the practical applications and efficiency of the proposed algorithm in optimizing these kernel computations.

The generalized Hodge conjecture can be viewed as a prediction on the coniveau filtration at all levels $$degree-2 coniveau, $$ an integer associated to each cohomological subgroup in the filtration (see (1.3) below). At the level 0, the conjecture coincides with the millennium Hodge conjecture on Hodge classes, which remains outstanding. At the non negative levels, Voisin has worked out the detail of Grothendieck's vision: the conjecture at the level 1 is implied by the conjecture at the level 0. In this paper, we would like to introduce a method with a new interpretation of the coniveau filtration in general. In case of 3-folds, the new method shows that Voisin's proof for the level 1 could work in the same way without the millennium Hodge conjecture.

Using GA's capacities for rotating and reflecting vectors, we solve the classic 2-D version of the Snellious-Pothenot surveying problem. The method used here provides two solutions, which can be averaged to better estimate the location of the unknown point P. A link to a GeoGebra worksheet of the solutions is provided so that the reader may test the validity of the method.

We show how to use rotations of vectors in GA to solve the following problem: "The following are known about a triangle: The ratio of the lengths of two sides; the angle formed by those sides; and the length of that angle’s bisector. Find the length of the side opposite that angle."

Some theorems on the construction of links (defined herein) from Reeb chords are established and proved. We relate these concepts to physics by constructing an arbitrary spacetime from a background knot using Reidemeister moves.

Let $X$ be a differential manifold. Let $mathscr D'(X)$ be the space of currents, and $S^{infty}(X)$ the Abelian group freely generated by regular cells, each of which is a pair of a polyhedron $Uppi$ and a ifferential embedding of a neighborhood of ${Uppi}$ to $X$. In this paper, we define a variant that is a bilinear map begin{equation}begin{array}{ccc} S^{infty}(X)times S^{infty}(X) &ightarrow & mathscr D'(X)(c_1, c_2) &ightarrow & [c_1wedge c_2]end{array}end{equation} called the supportive intersection such thatpar 1) the support of $[c_1wedge c_2]$ is contained in the intersection of the supports of $c_1, c_2$; 2) if $c_1, c_2$ are closed, $[c_1wedge c_2]$ is also closed and its cohomology class is the cup-product of the cohomology classes of $c_1, c_2$.

We generalize the famous `Chromatic Number of the Plane'. For every finite metric space we define a similar question. We show that 15 colors suffice togenerate a coloring of the plane without monochromatic distances 1 or 2.

An analytical treatment of rotations in the Euclidean plane and 3-dimensional Euclidean space, using differential equations, is presented. Fundamental geometric results, such as the linear transformation for rotations, the invariance of the Euclidean norm, a proof of the Pythagorean theorem, and the existence of a period of rotations, are derived from a set of fundamental equations. Basic Euclidean geometry is also constructed from these equations.

In the local gluing one glues local neighborhoods around the critical point of the stable and unstable manifolds to gradient flow lines defined on a finite time interval [−T,T] for large T. If the Riemannian metric around the critical point is locally Euclidean, the local gluing map can be written down explicitly. In the non-Euclidean case the construction of the local gluing map requires an intricate version of the implicit function theorem.In this paper we explain a functional analytic approach how the local gluing map can be defined. For that we are working on infinite dimensional path spaces and also interpret stable and unstable manifolds as submanifolds of path spaces. The advantage of this approach is that similar functional analytical techniques can as well be generalized to infinite dimensional versions of Morse theory, for example Floer theory.A crucial ingredient is the Newton-Picard map. We work out an abstract version of it which does not involve troublesome quadratic estimates.

We propose some questions about Fukaya categories. Given a class of isomorphisms $0 sim tau$, where $tau$ represents the truth value of a particle, and $0$ is a $0$ object in a Fukaya category, what are its spectral homology theories? This is a variation on the works of P. Seidel and E. Riehl.

I define the notions of complex curvature and complex radius and prove that one of these complex numbers is exactly the inverse of the other.

One of the consequences of Fermat's last theorem is the existence of a countable infinite number of rational points on the unit circle, which allows in turn, to find the rational points on the unit sphere via the inverse stereographic projection of the homothecies of the rational points on the unit circle. We proceed to iterate this process and obtain the rational points on the unit $S^3$ via the inverse stereographic projection of the homothecies of the rational points on the previous unit $S^2$. One may continue this iteration/recursion process ad infinitum in order to find the rational points on unit hyper-spheres of arbitrary dimension $S^4, S^5, cdots, S^N$. As an example, it is shown how to obtain the rational points of the unit $ S^{24}$ that is associated with the Leech lattice. The physical applications of our construction follow and one finds a direct relation among the $N+1$ quantum states of a spin-N/2 particle and the rational points of a unit $S^N$ hyper-sphere embedded in a flat Euclidean $R^{N+1}$ space.

For a space of directed currents, geometric data may be accessible by means of a certain $frac{1}{n}$-type functor on a sheaf of germs. We investigate pointwise periodic homeomorphisms and their connections to foliations.

In recent years the As-Rigid-As-Possible with Smooth Rotations (SR-ARAP [5]) technique has gained popularity in applications where an isometric-type of surface mapping is needed. The advantage of SR-ARAP is that quality of deformation results is comparable to more costly volumetric techniques operating on tetrahedral meshes. The SR-ARAP relies on local/global optimisation approach to minimise the non-linear least squares energy. The power of this technique resides on the local step. The local step estimates the local rotation of a small surface region, or cell, with respect of its neighbouring cells, so a local change in one cell’s rotation affect the neighbouring cell’s rotations and vice-versa. The main drawback of this technique is that the local step requires a global convergence of rotation changes. Currently the local step is solved in an iterative fashion, where the number of iterations needed to reach convergence can be prohibitively large and so, in practice, only a fixed number of iterations is possible. This trade-off is, in some sense, defeating the goal of SR-ARAP. We propose a linear-time closed-form solution for estimating the codependent rotations of the local step by solving a sparse linear system of equations. Our method is more efficient than state-of-the-art since no iterations are needed and optimised sparse linear solvers can be leveraged to solve this step in linear time. It is also more accurate since this is a closed-form solution. We apply our method to generate interactive surface deformation, we also show how a multiresolution optimisation can be applied to achieve real-time animation of large surfaces.

This book focuses on the application of rational representations to plane geometry. Most plane geometry objects, such as circles, triangles, quadrilaterals, conic curves, and their composite figures, can be represented almost exclusively in terms of rational parameters, which makes the process of computation and proof straightforward.

Based on the remarkable property of the Darboux vector to be perpendicular to the normal, I define a new trihedron associated with curves in space and prove that this trihedron also satisfies Frenet's formulas. Unlike the previous paper, where I used the trigonometric form of Frenet's formulas for simplicity, in this paper I construct a proof based only on curvature and torsion, respectively, darbuzian and lancretian.

Bazându-mă pe proprietatea remarcabilă a vectorului lui Darboux de a fi perpendicular pe normală, definesc un nou triedru asociat curbelor din spațiu și demonstrez că și acest triedru satisface formulele lui Frenet. Spre deosebire de lucrarea anterioară, unde am folosit pentru simplitate forma trigonometrică a formulelor lui Frenet, în această lucrare construiesc o demonstrație bazată doar pe curbură și torsiune, respectiv, pe darbuzian și lancretian.<p>Based on the remarkable property of the Darboux vector to be perpendicular to the normal, I define a new trihedron associated with curves in space and prove that this trihedron also satisfies Frenet's formulas. Unlike the previous paper, where I used the trigonometric form of Frenet's formulas for simplicity, in this paper I construct a proof based only on curvature and torsion, respectively, darbuzian and lancretian.

I minimize the N charges electric potential on a sphere, the minimum potential optimize the distance between the charges and it is possible to obtain the polyhedrons from the N charge positions

In this paper it is solved the case n = 5 of the problem 1.345 of the Crux Mathematicorum journal, proposed by Paul Erdös and Esther Szekeres in1988. The problem was solved for n ≥ 6 by János Pach and the solution published by the Crux Mathematicorum journal, leaving the case n = 5open to the reader. In september 2021, user23571113 posed this problem at the post https://math.stackexchange.com/questions/4243661/prove-thatfor-one-vertex-of-a-convex-pentagon-the-sum-of-distances-to-the-othe/4519514#4519514,and it has finally been solved.

Clifford algebras provide the natural generalizations of complex, dual numbers and quaternions into the concept of non-commutative Clifford numbers.The paper demonstrates an algorithm for the computation of inverses of such numbers in a non-degenerate Clifford algebra of an arbitrary dimension.The algorithm is a variation of the Faddeev--LeVerrier--Souriau algorithm and is implemented in the open-source Computer Algebra System Maxima.Symbolic and numerical examples in different Clifford algebras are presented.

In this paper we continue the development of the circles of partition by introducing a certain geometry of the axes of complex circles of partition. We use this geometry to verify the condition in the squeeze principle in special cases with regards to the orientation of the axes of complex circles of partition.

In this paper, we prove the existence of the approximate $(sigma,tau)$-Hermitian Yang--Mills structure on the $(sigma,tau)$-semi-stable quiver bundle $mathcal{R}=(mathcal{E},phi)$ over compact Gauduchon manifolds. An interesting aspect of this work is that the argument on the weakly $L^{2}_1$-subbundles is different from ['{A}lvarez-C'{o}nsul and Garc'{i}a-Prada, Comm Math Phys, 2003] and [Hu--Huang, J Geom Anal, 2020].

The unit circle and the cuspidal cubic curve have been found to intersect at coordinates that can be defined by the Plastic constant, which is defined as the solution to the cubic function x^3 = x + 1. This report explores the connections between the algebraic properties of the Plastic constant and the geometric properties of the circle and this curve.

Critical points of a function subject to a constraint can be either detected by restricting the function to the constraint or by looking for critical points of the Lagrange multiplier functional. Although the critical points of the two functionals, namely the restriction and the Lagrange multiplier functional are in natural one-to-one correspondence this does not need to be true for their gradient flow lines. We consider a singular deformation of the metric and show by an adiabatic limit argument that close to the singularity we have a one-to-one correspondence between gradient flow lines connecting critical points of Morse index difference one. We present a general overview of the adiabatic limit technique in the article [FW22b].The proof of the correspondence is carried out in two parts. The current part I deals with linear methods leading to a singular version of the implicit function theorem. We also discuss possible infinite dimensional generalizations in Rabinowitz-Floer homology. In part II [FW22a] we apply non-linear methods and prove, in particular, a compactness result and uniform exponential decay independent of the deformation parameter.

In this second part to [FW22a] we finish the proof of the one-to-one correspondence of gradient flow lines of index difference one between the restricted functional and the Lagrange multiplier functional for deformation parameters of the metric close to the singular one. In particular, we prove that, although the metric becomes singular, we have uniform bounds for the Lagrange multiplier of finite energy solutions and all its derivatives. This uniform bound is the crucial ingredient for a compactness theorem for gradient flow lines of arbitrary deformation parameter. If the functionals are Morse we further prove uniform exponential decay. We finally show combined with the linear theory in part I that if the metric is Morse-Smale the adiabatic limit map is bijective. We present a general overview of the adiabatic limit technique in the article [FW22b].

Dark matter has two independent origins in the impedance model: Geometrically, extending two-component Dirac spinors to the full 3D Pauli algebra eight-component wavefunction permits calculating quantum impedance networks of wavefunction interactions. Impedance matching governs amplitude and phase of energy flow. While vacuum wavefunction is the same at all scales, flux quantization of wavefunction components yields different energies and physics as scale changes, with corresponding enormous impedance mismatches when moving far from Compton wavelengths, decoupling the dynamics. Topologically, extending wavefunctions to the full eight components introduces magnetic charge, pseudoscalar dual of scalar electric charge. Coupling to the photon is reciprocal of electric, inverting fundamental lengths - Rydberg, Bohr, classical, and Higgs - about the charge-free Compton wavelength $lambda=h/mc$. To radiate a photon, Bohr cannot be inside Compton, Rydberg inside Bohr,... Topological inversion renders magnetic charge `dark'.Dark energy mixes geometry and topology, translation and rotation gauge fields. Impedance matching to the Planck length event horizon exposes an identity between gravitation and mismatched electromagnetism. Fields of wavefunction components propagate away from confinement scale, are reflected back by vacuum wavefunction mismatches they excite. This attenuation of the `Hawking graviton' wavefunction results in exponentially increasing wavelengths, ultimately greater than radius of the observable universe. Graviton oscillation between translation and rotation gauge fields exchanges linear and angular momentum, is an invitation to modified Newtonian dynamics.

To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we show how to calculate the angle through which two unit vectors must be rotated in order to be parallel to each other. Among the ideas that we use are a transformation of the usual GA formula for rotations, and the use of GA products to eliminated variables in simultaneous equations. We will show the benefits of (1) examining an interactive GeoGebra construction before attempting a solution, and (2) considering a range of implications of the given information.

In this paper we study the global electrostatic energy behaviour of mutually repelling charged electrons on the surface of a unit-radius sphere. Using the method of compression, we show that the total electrostatic energy $U_k(N)$ of $N$ mutually repelling particles on a sphere of unit radius in $mathbb{R}^k$ satisfies the lower boundbegin{align} U_k(N)gg_{epsilon}frac{N^{2}}{sqrt{k}}.onumberend{align}.

In this paper we derive the possibly simplest integral representations for the Riemann zeta function and its generalizations (the Lerch function, $\Phi(e^m,-k,b)$, the Hurwitz zeta, $\zeta(-k,b)$, and the polylogarithm, $\mathrm{Li}_{-k}(e^m)$), valid in the whole complex plane relative to all parameters, except for singularities. We also present the relations between each of these functions and their partial sums. It allows one to figure, for example, the Taylor series expansion of $H_{-k}(n)$ about $n=0$ (when $-k$ is a positive integer, we obtain a finite Taylor series, which is nothing but the Faulhaber formula). With these relations, one can also obtain the simplest integral representation of the derivatives of the zeta function at zero. The method used requires evaluating the limit of $\Phi\left(e^{2\pi\ii\,x},-2k+1,n+1\right)+\pi\ii\,x\,\Phi\left(e^{2\pi\ii\,x},-2k,n+1\right)/k$ when $x$ goes to $0$, which in itself already constitutes an interesting problem.

Using the method of compression we show that the number of points that can be placed in a plane figure with mutual distances at least $d>0$ satisfies the lower bound \begin{align} \gg_2 d^{\epsilon}\nonumber \end{align}for some small $\epsilon>0$.

Using the method of compression, we show that volume $Vol(K)$ of a ball $K$ in $\mathbb{R}^n$ with a single lattice point in it's interior as center of mass satisfies the lower bound \begin{align} Vol(K)\gg \frac{n^n}{\sqrt{n}}\nonumber \end{align}thereby disproving the Ehrhart volume conjecture, which claims that the upper bound must hold \begin{align} Vol(K) \leq \frac{(n+1)^n}{n!}\nonumber \end{align}for all convex bodies with the required property.

Using the method of compression we obtain a lower bound for the average number of $d^r$-unit distances that can be formed from a set of $n$ points in the euclidean space $\mathbb{R}^k$. By letting $\mathcal{D}_{n,d^r}$ denotes the number of $d^r$-unit distances~($r>1$~fixed) that can be formed from a set of $n$ points in $\mathbb{R}^k$, then we obtain the lower bound \begin{align} \sum \limits_{1\leq d\leq t}\mathcal{D}_{n,d^r}\gg n\sqrt[2r]{k}\log t.\nonumber \end{align}for a fixed $t>1$.

Using the method of compression, we show that the number of integral points on the boundary of a $k$-dimensional sphere of radius $r$ satisfies the lower bound \begin{align} \mathcal{N}_{r,k} \gg r^{k-1}\sqrt{k}.\nonumber \end{align}

Using the method of compression we show that the number of integral points in the region bounded by the $2r\times 2r \times \cdots \times 2r~(k~times)$ grid containing the sphere of radius $r$ and a sphere of radius $r$ satisfies the lower bound \begin{align} \mathcal{N}_{r,k} \gg r^{k-\delta}\times \frac{1}{\sqrt{k}}\nonumber \end{align}for some small $\delta>0$.

Using the method of compression we show that the number of integral points in a $k$ dimensional sphere of radius $r>0$ is \begin{align} N_k(r)\gg \sqrt{k} \times r^{k-1+o(1)}.\nonumber \end{align}

Using the method of compression we show that the number of integral points in the region bounded by the $2r\times 2r$ grid containing the circle of radius $r$ and a circle of radius $r$ satisfies the lower bound \begin{align} \mathcal{N}_r \gg r^{2-\delta}\nonumber \end{align}for some small $\delta>0$.

In this note, we would like to refer simply to the great history of Euclidean geometry and as a result we would like to state the great and essential development of Euclidean geometry by the new discovery of division by zero and division by zero calculus. We will be able to see the important and great new world of Euclidean geometry by Hiroshi Okumura.

We introduce compact subsets in the plane and in R3, which we call Polyorthogon and Polycuboid, respectively. We consider a usual chessboard. We display it by equal bricks or mirrored bricks.

In this paper, we will give a pleasant representation of the orthogonality of two lines by means of the division by zero calculus. For two lines with gradients $m$ and $ M$, they are orthogonal if $ m M = - 1. $ Our common sense will be so stated. However, note that for the typical case of $x,y$ axes, the statement is not valid. Even for the high school students, the new result may be pleasant with surprising new results and ideas.

In this paper, using the method of compression, we prove a stronger upper bound for the Erd\H{o}s unit distance problem in the plane by showing that \begin{align} \# \bigg\{||\vec{x_j}-\vec{x_t}||:\vec{x}_t, \vec{x_j}\in \mathbb{E}\subset \mathbb{R}^2,~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j \leq n\bigg\}\ll_2 n^{1+o(1)}.\nonumber \end{align}

In this second paper about the geodesic lines of the torus, we calculate in detail the integrals giving the length $s=s(\fii)$ and the longitude $\lm=\lm(\fii)$ of a point on the geodesic lines of the torus.

We consider special semicircles, whose endpoints lie on a circle, for a generalized arbelos called the arbelos with overhang considered in [4] with division by zero.

We consider Enomoto's problem involving a chain of circles touching two parallel lines and three circles with collinear centers. Generalizing the problem, we unexpectedly get a generalization of a property of the power of a point with respect to a circle.

We demonstrate several results in plane geometry derived from division by zero and division by zero calculus. The results show that the two new concepts open an entirely new world of mathematics.

In this paper we show that the number of points that can be placed in the grid $ntimes ntimes cdots times n~(d~times)=n^d$ for all $din mathbb{N}$ with $dgeq 2$ such that no three points are collinear satisfies the lower boundbegin{align}gg n^{d-1}sqrt[2d]{d}.onumberend{align}This pretty much extends the result of the no-three-in-line problem to all dimension $dgeq 3$.

Let $\mathcal{R}\subset \mathbb{R}^n$ be an infinite set of collinear points and $\mathcal{S}\subset \mathcal{R}$ be an arbitrary and finite set with $\mathcal{S}\subset \mathbb{N}^n$. Then the number of points with mutual integer distances on the shortest line containing points in $\mathcal{S}$ satisfies the lower bound \begin{align} \gg_n \sqrt{n}|\mathcal{S}\bigcap \mathcal{B}_{\frac{1}{2}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]}[\vec{x}]|\sum \limits_{\substack{k\leq \mathrm{max}_{\vec{x}\in \mathcal{S}\cap \mathcal{B}_{\frac{1}{2}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]}[\vec{x}]}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]\\k\in \mathbb{N}\\k>1}}\frac{1}{k},\nonumber \end{align}where $\mathcal{G}\circ \mathbb{V}_1[\vec{x}]$ is the compression gap of the compression induced on $\vec{x}$. This proves that there are infinitely many collinear points with mutual integer distances on any line in $\mathbb{R}^n$ and generalizes the well-known Erd\H{o}s-Anning Theorem in the plane $\mathbb{R}^2$.

We present two Geometric-Algebra (GA) solutions to a vector-rotation problem posed by Professor Miroslav Josipovic. We follow the sort of solution process that might be useful to students. First, we review concepts from GA and classical geometry that may prove useful. Then, we formulate and carry-out two solution strategies. After testing the resulting solutions, we propose an extension to the original problem.

In the first part, we deal with the question of which points we have to connect togenerate a non self-intersecting polygon. Afterwards, we introduce a polyhole, whichis a generalization of a polygon. Roughly speaking a polyhole is a big polygon, wherewe cut out a finite number of small polygons.In the second part, we present two ‘centers’, which we call center and barycenter.In the case that both centers coincide, we call these polygons nice. We show thatif a polygon has two symmetry axes, it is nice. We yield examples of polygons witha single symmetry axis which are nice and which are not nice. We show that allsymmetry axes intersect at one point.In the third part, we introduce the Spieker center and the Point center for polygons.We define beautiful polygons and perfect polygons.

We prove the Intersecting-Chords Theorem as a corollary to a relation-ship, derived via Geometric Algebra, about the product of the lengths of two segments of a single chord. We derive a similar theorem about the product of the lengths of a secant a chord.

Euclid proved (Elements, Book III, Propositions 20 and 21) proved that an angle inscribed in a circle is half as big as the central angle that subtends the same arc. We present a high-school level version of Hestenes' GA-based proof ([1]) of that same theorem. We conclude with comments on the need for learners of GA to learn classical geometry as well.

In this paper we introduce and develop the notion of simple close curve magnetization. We provide an application to Bellman's lost in the forest problem assuming special geometric conditions between the hiker and the boundary of the forest.

In this paper, we will discuss Euclidean geometry from the viewpoint of the division by zero calculus with typical examples. Where is the point at infinity? It seems that the point is vague in Euclidean geometry in a sense. Certainly we can see the point at infinity with the classical Riemann sphere. However, by the division by zero and division by zero calculus, we found that the Riemann sphere is not suitable, but D\"aumler's horn torus model is suitable that shows the coincidence of the zero point and the point at infinity. Therefore, Euclidean geometry is extended globally to the point at infinity. This will give a great revolution of Euclidean geometry. The impacts are wide and therefore, we will show their essence with several typical examples.

This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid’s Postulate 1), and Axiom 8 (an extended version of Euclid’s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid's First Postulate, Euclid's Second Postulate, Hilbert's Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid's Postulate 4, Posidonius-Geminus' Axiom, Proclus' Axiom, Cataldi's Axiom, Tacquet's Axiom 11, Khayyam's Axiom, Playfair's Axiom, and an extended version of Euclid's Fifth Postulate.

For a parametric equation of circles touching two externally touching circles, we consider its Laurent expansion around one of the singular points. Then we can find an equation of a notable circle and the equations of the external common tangents of the two circles from the coefficient of the Laurent expansion. However it is a mystery why we can find such things.

We generalize problems in Wasan geometry which nvolve no folded figures but are related to Haga's fold in origamics. Using the tangent circles appeared in those problems with division by zero, we give a parametric representation of the generalized Haga's fold given in the first part of these two-part papers.

We provide a formula that expresses the number of (n − 2)-gaps of a generic digital n-object. Such a formula has the advantage to involve only a few simple intrinsic parameters of the object and it is obtained by using a combinatorial technique based on incidence structure and on the notion of free cells. This approach seems suitable as a model for an automatic computation, and also allow us to find some expressions for the maximum number of i-cells that bound or are bounded by a fixed j-cell.

After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenient, we identify a common theme through which those types of calculations can be used to find the intersections of spheres with lines, planes, and other spheres.

Using rows 2 through 4 of a unimodular 8X8 rotation matrix, the vertices of E8 4_21, 2_41, and 1_42 are projected to 3D and then gathered & tallied into groups by the norm of their projected locations. The resulting Platonic and Archimedean solid 3D structures are then used to study E8's relationship to other research areas, such as sphere packings in Grassmannian spaces, using E8 Eisenstein Theta Series in recent proofs for optimal 8D and 24D sphere packings, nested lattices, and quantum basis critical parity proofs of the Bell-Kochen-Specker (BKS) theorem.

We generalize a problem in Wasan geometry involving a triangle and its incircle and get simple relationships between the three chains arising from three lines.

For a family of connections in a vector fiber bundle over a riemannian manifold, a Yang-Mills flow is defined with help of the riemannian curvature of the connections.

We consider a problem in Wasan geometry involving a golden arbelos and give a characterization of the golden arbelos involving an Archimedean circle. We also construct a self-similar circle configuration using the figure of the problem.

As a high-school-level example of solving a problem via Geometric (Clifford) Algebra, we show how to calculate the distance and direction between two points on Earth, given the locations' latitudes and longitudes. We validate the results by comparing them to those obtained from online calculators. This example invites a discussion of the benefits of teaching spherical trigonometry (the usual way of solving such problems) at the high-school level versus teaching how to use Geometric Algebra for the same purpose.

This problem first appeared in the American Mathematical Monthly in 1965, proposed by Sir Alexander Oppenheim. As a matter of curiosity, the American Mathematical Monthly is the most widely read mathematics journal in the world. On the other hand, Oppenheim was a brilliant mathematician, and for the excellence of his work in mathematics, obtained the title of “ Sir ”, given by the English to English citizens who stand out in the national and international scenario.Oppenheim is better known in the academic world for his contribution to the field of Number Theory, known as the Oppenheim Conjecture.

This problem first appeared in the American Mathematical Monthly in 1965, proposed by Sir Alexander Oppenheim. As a matter of curiosity, the American Mathematical Monthly is the most widely read mathematics journal in the world. On the other hand, Oppenheim was a brilliant mathematician, and for the excellence of his work in mathematics, obtained the title of “ Sir ”, given by the English to English citizens who stand out in the national and international scenario.Oppenheim is better known in the academic world for his contribution to the field of Number Theory, known as the Oppenheim Conjecture.

Translation of the article of Emanuels Grinbergs, ОБ ОДНОЙ ГЕОМЕТРИЧЕСКОЙ ВАРИАЦИОННОЙ ЗАДАЧЕ that is published in LVU Zinātniskie darbi, 1958.g., sējums XX, izlaidums 3, 153.-164., in Russian https://dspace.lu.lv/dspace/handle/7/46617.

As a high-school-level application of Geometric Algebra (GA), we show how to solve a simple vector-triangle problem. Our method highlights the use of outer products and inverses of bivectors.

As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to derive an equation for the line of intersection between two given planes. The solution method that we use emphasizes GA's capabilities for expressing and manipulating projections and rotations of vectors.

This paper explains in algebraic detail how two-dimensional conics can be defined by the outer products of conformal geometric algebra (CGA) points in higher dimensions. These multivector expressions code all types of conics in arbitrary scale, location and orientation. Conformal geometric algebra of two-dimensional Euclidean geometry is fully embedded as an algebraic subset. With small model preserving modifications, it is possible to consistently define in conic CGA versors for rotation, translation and scaling, similar to Hrdina et al. (Adv. Appl Cliff. Algs. Vol. 28:66, pp. 1–21, https://doi.org/10.1007/s00006-018-0879-2,2018), but simpler, especially for translations.

This work explains how to extend standard conformal geometric algebra of the Euclidean plane in a novel way to describe cubic curves in the Euclidean plane from nine contact points or from the ten coefficients of their implicit equations. As algebraic framework serves the Clifford algebra Cl(9,7) over the real sixteen dimensional vector space R^{9,7}. These cubic curves can be intersected using the outer product based meet operation of geometric algebra. An analogous approach is explained for the description and operation with cubic surfaces in three Euclidean dimensions, using as framework Cl(19,16). Keywords: Clifford algebra, conformal geometric algebra, cubic curves, cubic surfaces, intersections

Expanding the root form of a polynomial for large numbers of roots can be complicated. Such polynomials can be used to prove the irrationality of powers of pi, so a technique for arriving at expanded forms is needed. We show here how roots of polynomials can generate regular polygons whose vertices considered as roots form known expanded polynomials. The product of these polynomials can be simple enough to yield the desired expanded form.

Computer-aided Design (CAD) models of thin-walled parts such as sheet metal or plastics are often reduced dimensionally to their corresponding midsurfaces for quicker and fairly accurate results of Computer-aided Engineering (CAE) analysis. Generation of the midsurface is still a time-consuming and mostly, a manual task due to lack of robust and automated techniques. Midsurface failures manifest in the form of gaps, overlaps, not-lying-halfway, etc., which can take hours or even days to correct. Most of the existing techniques work on the complex final shape of the model forcing the usage of hard-coded heuristic rules, developed on a case-by-case basis. The research presented here proposes to address these problems by leveraging feature-parameters, made available by the modern feature-based CAD applications, and by effectively leveraging them for sub-processes such as simplification, abstraction and decomposition. In the proposed system, at first, features which are not part of the gross shape are removed from the input sheet metal feature-based CAD model. Features of the gross-shape model are then transformed into their corresponding generic feature equivalents, each having a profile and a guide curve. The abstracted model is then decomposed into non-overlapping cellular bodies. The cells are classified into midsurface-patch generating cells, called ‘solid cells’ and patch-connecting cells, called ‘interface cells’. In solid cells, midsurface patches are generated either by offset or by sweeping the midcurve generated from the owner-feature’s profile. Interface cells join all the midsurface patches incident upon them. Output midsurface is then validated for correctness. At the end, real-life parts are used to demonstrate the efficacy of the approach.

It is explained why the geometry of space-time, first found by Rainich, is generally valid. The equations of this geometry, the known Einstein-Maxwell equations, are discussed, and results are listed. We shall see how these tensor equations can be solved. As well, neutrosophics is more supported than dialectics. We shall find even more categories than described in neutrosophics.

In this paper, we will introduce the division by zero calculus in triangles and trigonometric functions as the first stage in order to see the elementary properties.

In this note, we give an application of the Method of the Repère Mobile to the Ellipsoid of Reference in Geodesy using a symplectic approach.

This work explains how three dimensional quadrics can be defined by the outer products of conformal geometric algebra points in higher dimensions. These multivector expressions code all types of quadrics in arbitrary scale, location and orientation. Furthermore, a newly modified (compared to Breuils et al, 2018, https://doi.org/10.1007/s00006-018-0851-1.) approach now allows not only the use of the standard intersection operations, but also of versor operators (scaling, rotation, translation). The new algebraic form of the theory will be explained in detail.

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to solve one of the beautiful sangaku problems from 19th-Century Japan. Among the GA operations that prove useful is the rotation of vectors via the unit bivector, i.

By similarity with the Seiberg-Witten equations, we propose two differential equations, depending of a spinor and a vector field, instead of a connection. Good moduli spaces are espected as a consequence of commutativity.

We define here a generalization of the well-know Levi-Civita connection. We choose an automorphism and define a connection with help of a (non-symmetric) bilinear form.

We propose here a generalization of the Clifford algebra by mean of two endomorphisms. We deduce a generalized Lichnerowicz formula for the space of modified spinors.

As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two bivectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors dened for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to denefine certain surface measures on surfaces of convex bodies in a novel and simple way.

In this paper, by the method of heat flow and the method of exhaustion, we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundle over a class of non-compact Gauduchon manifold.

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction ``north" as traced on the ground at the gnomon's location. To validate our equations, we convert those angles to Solar azimuths and altitudes for comparison with simulations made by the program Stellarium. As further validation, we analyze our equations algebraically to predict (for example) the precise timings and locations of sunrises, sunsets, and Solar zeniths on the solstices and equinoxes. We emphasize that the accuracy of the results is only to be expected, given the high accuracy of the heliocentric model itself, and that the relevance of this work is the efficiency with which that model can be implemented via GA for teaching at the introductory level. On that point, comments and debate are encouraged and welcome.

A doubling of the cube is attempted as a problem equivalent to the doubling of a horn torus. Both doublings are attained through the circle of Apollonius.

Following the definition of the flow of Ricci and with help of the Dirac operator, we construct a flow of hermitian metrics for the spinors fiber bundle.

We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve the problem by deriving expressions for the sine and cosine of the angle between projections of two vectors upon a plane. Geometric Algebra enables us to do so without deriving expressions for the projections themselves.

In this paper, we give a gradient estimate of positive solution to the equation $$\Delta u=-\lambda^2u, \ \ \lambda\geq 0$$ on a complete non-compact Finsler manifold. Then we obtain the corresponding Liouville-type theorem and Harnack inequality for the solution. Moreover, on a complete non-compact Finsler manifold we also prove a Liouville-type theorem for a $C^2$-nonegative function $f$ satisfying $$\Delta f\geq cf^d, c>0, d>1, $$ which improves a result obtained by Yin and He.

In this article, we will discuss a localization formulas of equlvariant cohomology about two Killing vector fields on the set of zero points ${\rm{Zero}}(X_{M}-\sqrt{-1}Y_{M})=\{x\in M \mid |Y_{M}(x)|=|X_{M}(x)|=0 \}.$ As application, we use it to get formulas about characteristic numbers and to get a Duistermaat-Heckman type formula on symplectic manifold.

We extend the reslut about Poincar\'e-Hopf type formula for the difference of the Chern character numbers (cf.[3]) to the non-isolated singularities, and establish a Poincar\'e-Hopf type formula for a pair of vector field with the function $h^{T_{\mathbb{C}}M}(\cdot,\cdot)$ has non-isolated zero points over a closed, oriented smooth manifold of dimension $2n$.

Published times of the Earth's perihelions do not refer to the perihelions of the orbit that the Earth would follow if unaffected by other bodies such as the Moon. To estimate the timing of that ``unperturbed" perihelion, we fit an unperturbed Kepler orbit to the timings of the year 2017's equinoxes and solstices. We find that the unperturbed 2017 perihelion, defined in that way, would occur 12.93 days after the December 2016 solstice. Using that result, calculated times of the year 2017's solstices and equinoxes differ from published values by less than five minutes. That degree of accuracy is sufficient for the intended use of the result.

We show how to calculate the projection of a vector, from an arbitrary direction, upon a given plane whose orientation is characterized by its normal vector, and by a bivector to which the plane is parallel. The resulting solutions are tested by means of an interactive GeoGebra construction.

We show how to express the representations of single, composite, and ``rotated" rotations in GA terms that allow rotations to be calculated conveniently via spreadsheets. Worked examples include rotation of a single vector by a bivector angle; rotation of a vector about an axis; composite rotation of a vector; rotation of a bivector; and the ``rotation of a rotation". Spreadsheets for doing the calculations are made available via live links.

We show how to express the representation of a composite rotation in terms that allow the rotation of a vector to be calculated conveniently via a spreadsheet that uses formulas developed, previously, for a single rotation. The work presented here (which includes a sample calculation) also shows how to determine the bivector angle that produces, in a single operation, the same rotation that is effected by the composite of two rotations.

We show how to transform a "rotate a vector around a given axis" problem into one that may be solved via GA, which rotates objects with respect to bivectors. A sample problem is worked to show how to calculate the result of such a rotation conveniently via an Excel spreadsheet, to which a link is provided.

Let $V$ be an asymptotically cylindrical K\"{a}hler manifold with asymptotic cross-section $\mathfrak{D}$. Let $E_\mathfrak{D}$ be a stable Higgs bundle over $\mathfrak{D}$, and $E$ a Higgs bundle over $V$ which is asymptotic to $E_\mathfrak{D}$. In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant Hermitian projectively Hermitian Yang-Mills metric on $E$.

When a meromorphic vector field is given on the projective plane, a complete holomorphic limit cycle, because it is a closed singular submanifold of projective space, is defined by algebraic equations. Also the meromorphic vector field is an algebraic object. Poincare had asked, is there just an algebraic calculation leading from the vector field to the defining equations of the solution, without the mysterious intermediary of the dynamical system. The answer is yes, that there is nothing more mysterious or wonderful that happens when a complete holomorphic limit cycle is formed than could have been defined using algebra.

The derivation of multilinear forms used to compute the moments of sets bounded by subdivision surfaces requires solving a number of systems of linear equations. As the support of the subdivision mask or the degree of the moment grows, the corresponding linear system becomes intractable to construct, let alone to solve by Gaussian elimination. In the paper, we argue that the power iteration and the geometric series are feasible methods to approximate the multilinear forms. The tensor iterations investigated in this work are shown to converge at favorable rates, achieve arbitrary numerical accuracy, and have a small memory footprint. In particular, our approach makes it possible to compute the volume, centroid, and inertia of spatial domains bounded by Catmull-Clark and Loop subdivision surfaces.

To the collections of problems solved via Geometric Algebra (GA) in References 1-13, this document adds a solution, using only dot products, to the Problem of Apollonius. The solution is provided for completeness and for contrast with the GA solutions presented in Reference 3.

Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this document presents, explains, and discusses common solution strategies. Included are a list of formulas and a guide to nomenclature.

This document adds to the collection of solved problems presented in References [1]-[6]. The solutions presented herein are not as efficient as those in [6], but they give additional insight into ways in which GA can be used to solve this problem. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the CLP limiting case of the Problem of Apollonius in three ways, some of which identify the the solution circles' points of tangency with the given circle, and others of which identify the solution circles' points of tangency with the given line. For comparison, the solutions that were developed in [1] are presented in an Appendix.

The new solutions presented herein for the CLP and CCP limiting cases of the Problem of Apollonius are much shorter and more easily understood than those provided by the same author in References 1 and 2. These improvements result from (1) a better selection of angle relationships as a starting point for the solution process; and (2) better use of GA identities to avoid forming troublesome combinations of terms within the resulting equations.

In this article, we will discuss a new operator $d_{C}$ on $W(\mathfrak{g})\otimes\Omega^{*}(M)$ and to construct a new Cartan model for equivariant cohomology. We use the new Cartan model to construct the corresponding BRST model and Weil model, and discuss the relations between them.

This document adds to the collection of solved problems presented in References 1-4. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the LLP limiting case of the Problem of Apollonius in three ways.

This document is intended to be a convenient collection of explanations and techniques given elsewhere in the course of solving tangency problems via Geometric Algebra.

NOTE: A new Appendix presents alternative solutions. The famous "Problem of Apollonius", in plane geometry, is to construct all of the circles that are tangent, simultaneously, to three given circles. In one variant of that problem, one of the circles has innite radius (i.e., it's a line). The Wikipedia article that's current as of this writing has an extensive description of the problem's history, and of methods that have been used to solve it. As described in that article, one of the methods reduces the "two circles and a line" variant to the so-called "Circle-Line-Point" (CLP) special case: Given a circle C, a line L, and a point P, construct the circles that are tangent to C and L, and pass through P. This document has been prepared for two very different audiences: for my fellow students of GA, and for experts who are preparing materials for us, and need to know which GA concepts we understand and apply readily, and which ones we do not.

Note: The Appendix to this new version gives an alternate--and much simpler--solution that does not use reflections. The beautiful Problem of Apollonius from classical geometry ("Construct all of the circles that are tangent, simultaneously, to three given coplanar circles") does not appear to have been solved previously by vector methods. It is solved here via Geometric Algebra (GA, also known as Clifford Algebra) to show students how they can make use of GA's capabilities for expressing and manipulating rotations and reflections. As Viète did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students ``see" geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equality of multivectors to obtain solvable equations; and (3) recognizing complex algebraic expressions that reduce to simple rotations of multivectors. As an aid to students, the author has prepared a dynamic-geometry construction to accompany this article.

Written as somewhat of a "Schaums Outline" on the subject, which is especially useful in robotics and mechatronics. Geometric Algebra (GA) was invented in the 1800s, but was largely ignored until it was revived and expanded beginning in the 1960s. It promises to become a "universal mathematical language" for many scientific and mathematical disciplines. This document begins with a review of the geometry of angles and circles, then treats rotations in plane geometry before showing how to formulate problems in GA terms, then solve the resulting equations. The six problems treated in the document, most of which are solved in more than one way, include the special cases that Viete used to solve the general Problem of Apollonius.

This paper gives an overview of two different, but closely related, double conformal geometric algebras. The first is the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), and the second is the G(4,8) Double Conformal Space-Time Algebra (DCSTA). DCSTA is a straightforward extension of DCGA. The double conformal geometric algebras that are presented in this paper have a large set of operations that are valid on general quadric surface entities. These operations include rotation, translation, isotropic dilation, spacetime boost, anisotropic dilation, differentiation, reflection in standard entities, projection onto standard entities, and intersection with standard entities. However, the quadric surface entities and other "non-standard entities" cannot be intersected with each other.

Squaring the Circle is a famous geometry problem going all the way back to the ancient Greeks. It is the great quest of constructing a square with the same area as a circle using a compass and straightedge in a finite number of steps. Since it was proven that π was a transcendental number in 1882, the task of Squaring the Circle has been considered impossible. Here, we will show it is possible to Square the Circle in space-time. It is not possible to Square the Circle in Euclidean space alone, but it is fully possible in space-time, and after all we live in a world with not only space, but also time. By drawing the circle from one reference frame and drawing the square from another reference frame, we can indeed Square the Circle. By taking into account space-time rather than just space the Impossible is possible! However, it is not enough simply to understand math in order to Square the Circle, one must understand some “basic” space-time physics as well. As a bonus we have added a solution to the impossibility of Doubling the Cube. As a double bonus we also have also Boxed the Sphere! As one will see, one can claim we simply have bent the rules and moved a problem from one place to another. One of the main essences of this paper is that we can move challenging space problems out from space and into time, and vice versa.

There are many method for nding whether a point is inside a polygon or not. The congregation of all points inside a polygon can be referred point congregation of polygon. Assume on a plane there are N points. Assume the polygon have M vertexes. There are O(NM) calculations to create the point congregation of polygon. Assume N>>M, we oer a parallel calculation method which is suitable for GPU programming. Our method consider a polygon is consist of many fan regions. The fan region can be positive and negative. We wold like to extended this method to 3 D problem where a polyhedron instead of polygon should be drawn using cones.

This paper introduces the differential operators in the G(8,2) Geometric Algebra, called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). The differential operators are three x, y, and z-direction bivector-valued differential elements and either the commutator product or the anti-commutator product for multiplication into a geometric entity that represents the function to be differentiated. The general form of a function is limited to a Darboux cyclide implicit surface function. Using the commutator product, entities representing 1st, 2nd, or 3rd order partial derivatives in x, y, and z can be produced. Using the anti-commutator product, entities representing the anti-derivation can be produced from 2-vector quadric surface and 4-vector conic section entities. An operator called the pseudo-integral is defined and has the property of raising the x, y, or z degree of a function represented by an entity, but it does not produce a true integral. The paper concludes by offering some basic relations to limited forms of vector calculus and differential equations that are limited to using Darboux cyclide implicit surface functions. An example is given of entity analysis for extracting the parameters of an ellipsoid entity using the differential operators.

This note of differential geometry concerns the formulas of Elie Cartan about the differntial forms on a surface. We calculate these formulas for an ellipsoïd of revolution used in geodesy.

The G(8,2) Geometric Algebra, also called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), has entities that represent conic sections. DCGA also has entities that represent planar sections of Darboux cyclides, which are called cyclidic sections in this paper. This paper presents these entities and many operations on them. Operations include reflection, projection, rejection, and intersection with respect to spheres and planes. Other operations include rotation, translation, and dilation. Possible applications are introduced that include orthographic and perspective projections of conic sections onto view planes, which may be of interest in computer graphics or other computational geometry subjects.

This paper introduces the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general Darboux cyclide surfaces in Euclidean 3D space. The general Darboux cyclide is a quartic surface. Darboux cyclides include circular tori and all quadrics, and also all surfaces formed by their inversions in spheres. Dupin cyclide surfaces can be formed as inversions in spheres of circular toroid, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversion spheres centered on other surfaces. All DCGA entities can be conformally transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. All entities can be inversed in general spheres and reflected in general planes. Entities representing the intersections of surfaces can be created by wedge products. All entities can be intersected with spheres, planes, lines, and circles. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying implicit surface equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.

A Java program is presented which extracts star positions from the Hipparcos main catalogue and places them into a sphere collection rendered in a X3D, VRML97 or HTML file. The main options to the executable are a cut-off distance to some center of the scene (the sun by default) and labeling some or all of the spheres with common names, Henry Draper numbers or Hipparcos ID's.

Recently the general orthogonal planes split with respect to any two pure unit quaternions $f,g \in \mathbb{H}$, $f^2=g^2=-1$, including the case $f=g$, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) [E.Hitzer, S.J. Sangwine, The orthogonal 2D planes split of quaternions and steerable quaternion Fourier Transforms, in E. Hitzer, S.J. Sangwine (eds.), "Quaternion and Clifford Fourier Transforms and Wavelets", TIM \textbf{27}, Birkhauser, Basel, 2013, 15--39.]. Applications include color image processing, where the orthogonal planes split with $f=g=$ the grayline, naturally splits a pure quaternionic three-dimensional color signal into luminance and chrominance components. Yet it is found independently in the quaternion geometry of rotations [L. Meister, H. Schaeben, A concise quaternon geometry of rotations, MMAS 2005; \textbf{28}: 101--126], that the pure quaternion units $f,g$ and the analysis planes, which they define, play a key role in the spherical geometry of rotations, and the geometrical interpretation of integrals related to the spherical Radon transform of probability density functions of unit quaternions, as relevant for texture analysis in crystallography. In our contribution we further investigate these connections.

This paper will present various techniques for visualizing a split real even $E_8$ representation in 2 and 3 dimensions using an $E_8$ to $H_4$ folding matrix. This matrix is shown to be useful in providing direct relationships between $E_8$ and the lower dimensional Dynkin and Coxeter-Dynkin geometries contained within it, geometries that are visualized in the form of real and virtual 3 dimensional objects.

Se presenta el estudio de propiedades geométricas de un cuadrilátero inscrito en una circunferencia, en un plano de Minkowski. Se estudian las relaciones entre los cuatro triángulos formados por los vértices del cuadrilátero, sus antitriángulos y puntos de simetría, sus baricentros y otros puntos asociados con dichos triángulos, respectivamente. Se introduce la noción de anticuadrilátero y se extiende la noción de circunferencia de Feuerbach de un cuadriláteros, inscritos en una circunferencia, a planos de Minkowski en general. --- The study of geometric properties of a inscribed quadrangle in a circle, in a Minkowski plane is presented. We study the relations between the four triangles formed by the vertices of the quadrangle, its anti-triangles and points of symmetry, its barycenters and other points associated with such triangles, respectively. The notion of anti-quadrangle is introduced and extends the notion of Feuerbach circle for quadrangles, inscribed in a circle, to Minkowski planes in general.

Usando la noción de C-ortocentro se extienden, a planos de Minkowski en general, nociones de la geometría clásica relacionadas con un triángulo, como por ejemplo: puntos de Euler, triángulo de Euler, puntos de Poncelet. Se muestran propiedades de estas nociones y sus relaciones con la circunferencia de Feuerbach. Se estudian sistemas C-ortocéntricos formados por puntos presentes en dichas nociones y se establecen relaciones con la ortogonalidad isósceles y cordal. Además, se prueba que la imagen homotética de un sistema C-ortocéntrico es un sistema C-ortocéntrico. --- Using the notion of C-orthocenter, notions of the classic euclidean geometry related with a triangle, as for example: Euler points; Euler’s triangle; and Poncelet’s points, are extended to Minkowski planes in general. Properties of these notions and their relations with the Feuerbach’s circle, are shown. C-orthocentric systems formed by points in the above notions are studied and relations with the isosceles and chordal orthogonality, are established. In addition, there is proved that the homothetic image of a C-orthocentric system is a C-orthocentric system.

Mediante el estudio de ciertas propiedades geométricas de los sistemas C-ortocéntricos, relacionadas co las nociones de ortogonalidad (Birkhoff, isósceles, cordal), bisectriz (Busemann, Glogovskij) y línea soporte a una circunferencia, se muestran nueve caracterizaciones de euclidianidad para planos de Minkowski arbitrarios. Tres de estas generalizan caracterizaciones dadas para planos de Minkowski estrictamente convexos en [8, 9], y las otras seis son nuevos aportes sobre el tema. -- By studying certain geometric properties of C-orthocentric systems related to the notions of orthogonality (Birkhoff, isosceles, chordal), angular bisectors (Busemann, Glogovskij) and support line to a circumference, shows nine characterizations of the Euclidean plane for arbitrary Minkowski planes. Three of these generalized characterizations given for strictly convex Minkowski planes in [8, 9], and the other six are new contributions on subject.

In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete case examples, we mention that iteratively generated sequences like Fibonacci's numbers and (the complex moduli of) Mandelbrot's numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin's holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli's iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.

In this work, we deploy Santilli's iso-dual iso-topic lifting and Inopin's holographic ring (IHR) topology as a platform to introduce and assemble a tesseract from two inter-locking, iso-morphic, iso-dual cubes in Euclidean triplex space. For this, we prove that such an "iso-dual tesseract" can be constructed by following a procedure of simple, flexible, topologically-preserving instructions. Moreover, these novel results are significant because the tesseract's state and structure are directly inferred from the one initial cube (rather than two distinct cubes), which identifies a new iso-geometrical inter-connection between Santilli's exterior and interior dynamical systems.

Descriptions of 1-dimensional projective space in terms of the cross ratio, in one-dimensional geometry as a projective line, in two-dimensional geometry as a circle, and in three-dimensional geometry as a regulus. A characterization of projective 3-space is given in terms of polarity. This paper differs from the original version by addition of a section showing that the circle is distinguished from other meridians by its compactness and the existence of exponential functions.

Most matter in nature and technology is composed of crystals and crystal grains. A full understanding of the inherent symmetry is vital. A new interactive software tool is demonstrated, that visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space group visualizer (SGV) is a script for the open source visual CLUCalc, which fully supports geometric algebra computation. In our presentation we will first give some insights into the geometric algebra description of space groups. The symmetry generation data are stored in an XML file, which is read by a special CLUScript in order to generate the visualization. Then we will use the Space Group Visualizer to demonstrate space group selection and give a short interactive computer graphics presentation on how reflections combine to generate all 230 three-dimensional space groups.

In this article, we will discuss the smooth $(X_{M}+\sqrt{-1}Y_{M})$-invariant forms on M and to establish a localization formulas. As an application, we get a localization formulas for characteristic numbers.

Inspired by the recent sums of the squares law obtained by Kovacs-Fang-Sadler-Irwin we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions after using Clifford algebraic methods.

A method for dealing with the product of step discontinuous and delta functions is proposed. A standard method for applying the above defined product of distributions to polyhedron vertices is analysed and the method is applied to a special case where the well known angle defect formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus. The angle defect formula is the discrete version of the curvature for vertices of polyhedra. Among other things, this paper is basically the formal proof of the above statement.

This paper presents the correspondences of the eccentric mathematics of cardinal and integral functions and centric mathematics, or ordinary mathematics. Centric functions will also be presented in the introductory section, because they are, although widely used in undulatory physics, little known.

A tendering is a negotiating process for a contract through by a tenderer issuing an invitation, bidders submitting bidding documents and the tenderer accepting a bidding by sending out a notification of award. As a useful way of purchasing, there are many norms and rulers for it in the purchasing guides of the World Bank, the Asian Development Bank,..., also in contract conditions of various consultant associations. In China, there is a law and regulation system for tendering and bidding. However, few works on the mathematical model of a tendering and its evaluation can be found in publication. The main purpose of this paper is to construct a Smarandache multi-space model for a tendering, establish an evaluation system for bidding based on those ideas in the references [7] and [8] and analyze its solution by applying the decision approach for multiple objectives and value engineering. Open problems for pseudo-multi-spaces are also presented in the final section.

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers (part IV)

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers (part III)

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n &t; 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multirings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers.

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multirings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers.

A Smarandache multi-space is a union of n spaces A1,A2,...,An with some additional conditions holding. Combining Smarandache multispaces with classical metric spaces, the conception of multi-metric space is introduced. Some characteristics of a multi-metric space are obtained and Banach�s fixed-point theorem is generalized in this paper.

A Smarandache multi-space is a union of n spaces A1,A2,...,An with some additional conditions holding. Combining Smarandache multispaces with linear vector spaces in classical linear algebra, the conception of multi-vector spaces is introduced. Some characteristics of a multi-vector space are obtained in this paper.

A Smarandache multi-space is a union of n spaces A1,A2,...,An with some additional conditions holding. Combining Smarandache multispaces with rings in classical ring theory, the conception of multi-ring spaces is introduced. Some characteristics of a multi-ring space are obtained in this paper

A Smarandache multi-space is a union of n spaces A1,A2, ... ,An with some additional conditions holding. Combining classical of a group with Smarandache multi-spaces, the conception of a multi-group space is introduced in this paper, which is a generalization of the classical algebraic structures, such as the group, filed, body,..., etc.. Similar to groups, some characteristics of a multi-group space are obtained in this paper.

For an integer m > 1, a combinatorial manifold fM is defined to be a geometrical object fM such that for(...) there is a local chart (see paper) where Bnij is an nij -ball for integers 1 < j < s(p) < m. Integral theory on these smoothly combinatorial manifolds are introduced. Some classical results, such as those of Stokes� theorem and Gauss� theorem are generalized to smoothly combinatorial manifolds in this paper.

For an integer m ≥ 1, a combinatorial manifold fM is defined to be a geometrical object fM such that for (...), there is a local chart (see paper) where Bnij is an nij -ball for integers 1 ≤ j ≤ s(p) ≤ m. Topological and differential structures such as those of d-pathwise connected, homotopy classes, fundamental d-groups in topology and tangent vector fields, tensor fields, connections, Minkowski norms in differential geometry on these finitely combinatorial manifolds are introduced. Some classical results are generalized to finitely combinatorial manifolds. Euler-Poincare characteristic is discussed and geometrical inclusions in Smarandache geometries for various geometries are also presented by the geometrical theory on finitely combinatorial manifolds in this paper.

A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969), i.e., an axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways and a Smarandache n-manifold is a n-manifold that support a Smarandache geometry. Iseri provided a construction for Smarandache 2-manifolds by equilateral triangular disks on a plane and a more general way for Smarandache 2-manifolds on surfaces, called map geometries was presented by the author in [9]-[10] and [12]. However, few observations for cases of n ≥ 3 are found on the journals. As a kind of Smarandache geometries, a general way for constructing dimensional n pseudo-manifolds are presented for any integer n ≥ 2 in this paper. Connection and principal fiber bundles are also defined on these manifolds. Following these constructions, nearly all existent geometries, such as those of Euclid geometry, Lobachevshy-Bolyai geometry, Riemann geometry, Weyl geometry, Kähler geometry and Finsler geometry, ...,etc., are their sub-geometries.

Combinatorics is a powerful tool for dealing with relations among objectives mushroomed in the past century. However, an more important work for mathematician is to apply combinatorics to other mathematics and other sciences not merely to find combinatorial behavior for objectives. Recently, such research works appeared on journals for mathematics and theoretical physics on cosmos. The main purpose of this paper is to survey these thinking and ideas for mathematics and cosmological physics, such as those of multi-spaces, map geometries and combinatorial cosmoses, also the combinatorial conjecture for mathematics proposed by myself in 2005. Some open problems are included for the 21th mathematics by a combinatorial speculation.

Parallel lines are very important objects in Euclid plane geometry and its behaviors can be gotten by one�s intuition. But in a planar map geometry, a kind of the Smarandache geometries, the situation is complex since it may contains elliptic or hyperbolic points. This paper concentrates on the behaviors of parallel bundles in planar map geometries, a generalization of parallel lines in plane geometry and obtains characteristics for parallel bundles.

On a geometrical view, the conception of map geometries is introduced, which is a nice model of the Smarandache geometries, also new kind of and more general intrinsic geometry of surfaces. Some open problems related combinatorial maps with the Riemann geometry and Smarandache geometries are presented.

One is reminded in this paper of the often overlooked fact that the geometric straight line, or GSL, of Euclidean geometry is not necessarily identical with its usual Cartesian coordinatisation given by the real numbers in <b>R</b>. Indeed, the GSL is an abstract idea, while the Cartesian, or for that matter, any other specific coordinatisation of it is but one of the possible mathematical models chosen upon certain reasons. And as is known, there are a a variety of mathematical models of GSL, among them given by nonstandard analysis, reduced power algebras, the topological long line, or the surreal numbers, among others. As shown in this paper, the GSL can allow coordinatisations which are arbitrarily more rich locally and also more large globally, being given by corresponding linearly ordered sets of no matter how large cardinal. Thus one can obtain in relatively simple ways structures which are more rich locally and large globally than in nonstandard analysis, or in various reduced power algebras. Furthermore, vector space structures can be defined in such coordinatisations. Consequently, one can define an extension of the usual Differential Calculus. This fact can have a major importance in physics, since such locally more rich and globally more large coordinatisations of the GSL do allow new physical insights, just as the introduction of various microscopes and telescopes have done. Among others, it and general can reassess special relativity with respect to its independence of the mathematical models used for the GSL. Also, it can allow the more appropriate modelling of certain physical phenomena. One of the long vexing issue of so called �infinities in physics� can obtain a clarifying reconsideration. It indeed all comes down to looking at the GSL with suitably constructed microscopes and telescopes, and apply the resulted new modelling possibilities in theoretical physics. One may as well consider that in string theory, for instance, where several dimensions are supposed to be compact to the extent of not being observable on classical scales, their mathematical modelling may benefit from the presence of infinitesimals in the mathematical models of the GSL presented here. However, beyond all such particular considerations, and not unlikely also above them, is the following one : theories of physics should be not only background independent, but quite likely, should also be independent of the specific mathematical models used when representing geometry, numbers, and in particular, the GSL. One of the consequences of considering the essential difference between the GSL and its various mathematical models is that what appears to be the definitive answer is given to the intriguing question raised by Penrose : �Why is it that physics never uses spaces with a cardinal larger than that of the continuum ?�.

In this study, we present a proof of the Menelaus theorem for quadrilaterals in hyperbolic geometry, and a proof for the transversal theorem for triangles.

Given a vector space V of dimension n and a natural number k < n, the grassmannian G_k(V) is defined as the set of all subspaces W ⊂ V such that dim(W) = k. In the case of V = Rⁿ, G_k(V) is the set of k-fl ats in Rⁿ and is called real grassmannian [1]. Recently the study of these manifolds has found applicability in several areas of mathematics, especially in Modern Differential Geometry and Algebraic Geometry. This work will build two differential structures on the real grassmannian, one of which is obtained as a quotient space of a Lie group [1], [3], [2], [7]