A Brief Introduction to Geometry

2003

This is a guided tour through some selected topics in geometric analysis. We have chosen to illustrate many of the basic ideas as they apply to the theory of minimal surfaces. This is, in part, because minimal surfaces is, if not the oldest, then certainly one of the oldest areas of geometric analysis dating back to Euler's work in the

INTERNATIONAL JOURNAL OF RESEARCH IN EDUCATION METHODOLOGY, 2016

In this paper the origin of the concepts of geometry and the way and the reason for systematization is analyzed. A first approach to the analysis of the processes involved in the development of geometry determines its target and leads us to clarify what is meant by a mathematical object and, further, what is meant by geometric object, and leads into the structure essential geometry.

2004

Inverse and quotient of two vectors, 7.-Priority of algebraic operations, 8.-Geometric algebra of the vector plane, 9.-Exercises, 9. 2. A vector basis for the Euclidean plane Linear combination of two vectors, 10.-Basis and components, 10.-Orthonormal bases, 11.-Applications of formulae for products, 11.-Exercises, 12. 3. Complex numbers The subalgebra of complex numbers, 13.-Binomial, polar and trigonometric form of a complex number, 13.-Algebraic operations with complex numbers, 14.-Permutation of complex numbers and vectors, 17.-The complex plane, 18.-Complex analytic functions, 19.-Fundamental theorem of algebra, 24.-Exercises, 26. 4. Transformations of vectors Rotations, 27.-Axial symmetries, 28.-Inversions, 29.-Dilations, 30.-Exercises, 30 Second Part: Geometry of the Euclidean plane 5. Points and straight lines Translations, 31.-Coordinate systems, 31.-Barycentric coordinates, 33.-Distance between two points and area, 33.-Condition of collinearity of three points, 35.-Cartesian coordinates, 36.-Vectorial and parametric equations of a line, 36.-Algebraic equation and distance from a point to a line, 37.-Slope and intercept equations of a line, 40.-Polar equation of a line, 41.-Intersection of two lines and pencil of lines, 41.-Dual coordinates, 43.-Desargues's theorem, 48.-Exercises, 50. 6. Angles and elemental trigonometry Sum of the angles of a polygon, 53.-Definition of trigonometric functions and fundamental identities, 54.-Angle inscribed in a circle and double-angle identities, 55.-Addition of vectors and sum of trigonometric functions, 56.-Product of vectors and addition identities, 57.-Rotations and de Moivre's identity, 58.-Inverse trigonometric functions, 59.-Exercises, 60. 7. Similarities and simple ratio Direct similarity (similitude), 61.-Opposite similarity, 62.-Menelaus' theorem, 63.-Ceva's theorem, 64.-Homothety and simple ratio, 65.-Exercises, 67. 8. Properties of triangles Area of a triangle, 68.-Medians and centroid, 69.-Perpendicular bisectors and circumcentre, 70.-Angle bisectors and incentre, 72.-Altitudes and orthocentre, 73.-Euler's line, 76.-Fermat's theorem, 77.-Exercises, 78. XIII 9. Circles Algebraic and Cartesian equations, 80.-Intersections of a line with a circle, 80.-Power of a point with respect to a circle, 82.-Polar equation, 82.-Inversion with respect to a circle, 83.-The nine-point circle, 85.-Cyclic and circumscribed quadrilaterals, 87.-Angle between circles, 89.-Radical axis of two circles, 89.-Exercises, 91. 10. Cross ratios and related transformations Complex cross ratio, 92.-Harmonic characteristic and ranges, 94.-Homography (Möbius transformation), 96.-Projective cross ratio, 99.-Points at infinity and homogeneous coordinates, 102.-Perspectivity and projectivity, 103.-Projectivity as a tool for theorem demonstrations, 108.-Homology, 110.-Exercises, 115. 11. Conics Conic sections, 117.-Two foci and two directrices, 120.-Vectorial equation, 121.-Chasles' theorem, 122.-Tangent and perpendicular to a conic, 124.-Central equations for ellipse and hyperbola, 126.-Diameters and Apollonius' theorem, 128.-Conic passing through five points, 131.-Pencil of conics passing through four points, 133.-Conic equation in barycentric coordinates and dual conic, 133.-Polarities, 135.-Reduction of the conic matrix to diagonal form, 136.-Exercises, 137. Third part: Pseudo-Euclidean geometry 12. Matrix representation and hyperbolic numbers Rotations and the representation of complex numbers, 139.-The subalgebra of hyperbolic numbers, 140.-Hyperbolic trigonometry, 141.-Hyperbolic exponential and logarithm, 143.-Polar form, powers and roots of hyperbolic numbers, 144.-Hyperbolic analytic functions, 147.-Analyticity and square of convergence of power series, 150.-About the isomorphism of Clifford algebras, 152.-Exercises, 153. 13. The hyperbolic or pseudo-Euclidean plane Hyperbolic vectors, 154.-Inner and outer products of hyperbolic vectors, 155.-Angles between hyperbolic vectors, 156.-Congruence of segments and angles, 158.-Isometries, 158.-Theorems about angles, 160.-Distance between points, 160.-Area in the hyperbolic plane, 161.-Diameters of the hyperbola and Apollonius' theorem, 163.-The law of sines and cosines, 164.-Hyperbolic similarity, 167.-Power of a point with respect to a hyperbola with constant radius, 168.-Exercises, 169. Fourth part: Plane projections of three-dimensional spaces 14. Spherical geometry in the Euclidean space The geometric algebra of the Euclidean space, 170.-Spherical trigonometry, 172.-The dual spherical triangle of a given triangle, 175.-Right spherical triangles and Napier's rule, 176.-Area of a spherical triangle, 176.-Properties of the projections of the spherical surface, 177.-Central or gnomonic projection, 177.-Stereographic projection, 180.-Orthographic projection, 181.-Lambert's azimuthal equivalent projection, 182.-Spherical coordinates and cylindrical equidistant (plate carré) projection, 183.-Mercator XIV projection, 184.-Cylindrical equivalent projection, 184.-Conic projections, 185.-Exercises, 186. 15. Hyperboloidal geometry in the pseudo-Euclidean space The geometric algebra of the pseudo-Euclidean space, 189.-The hyperboloid of two sheets (Lobachevskian surface), 191.-Central projection (Beltrami disk), 192.-Lobachevskian trigonometry, 197.-Stereographic projection (Poincaré disk), 199.-Azimuthal equivalent projection, 201.-Weierstrass coordinates and cylindrical equidistant projection, 202.-Cylindrical conformal projection, 203.-Cylindrical equivalent projection, 204.-Conic projections, 204.-About the congruence of geodesic triangles, 206.-The hyperboloid of one sheet, 206.-Central projection and arc length on the one-sheeted hyperboloid, 207.-Cylindrical projections, 208.-Cylindrical central projection, 209.-Cylindrical equidistant projection, 210.-Cylindrical equivalent projection, 210.-Cylindrical conformal projection, 210.-Area of a triangle on the onesheeted hyperboloid, 211.-Trigonometry of right triangles, 214.-Hyperboloidal trigonometry, 215.-Dual triangles, 218.-Summary, 221.-Comment about the names of the non-Euclidean geometry, 222.-Exercises, 222. 16. Solutions to the proposed exercises 1. Euclidean vectors and their operations, 224.-2. A vector basis for the Euclidean plane, 225.-3. Complex numbers, 227.-4. Transformations of vectors, 230.-5. Points and straight lines, 231.-6. Angles and elemental trigonometry, 241.-7. Similarities and simple ratio, 244.-8. Properties of triangles, 246.-9. Circles, 257.-10. Cross ratios and related transformations, 262.-11. Conics, 266.-12. Matrix representation and hyperbolic numbers, 273.-13. The hyperbolic or pseudo-Euclidean plane, 275.-14. Spherical geometry in the Euclidean space, 278.-15. Hyperboloidal geometry in the pseudo-Euclidean space, 284. Bibliography, 293. Internet bibliography, 296. Index, 298. Chronology of the geometric algebra, 305.

In this paper the origin of the concepts of geometry and the way and the reason for systematization is analyzed. A first approach to the analysis of the processes involved in the development of geometry determines its target and leads us to clarify what is meant by a mathematical object and, further, what is meant by geometric object, and leads into the structure essential geometry.

This is an introduction into Geometry and geometric analysis, taught in the fall term 1995 at Caltech. It introduces geometry on manifolds, tensor analysis, pseudo Riemannian geometry. General relativity is used as a guiding example in the last part. Exercises, midterm and final with solutions as well as 4 appendices listing some results and definitions in real analysis, geometry, measure theory and differential equations are located at the end of the text. The material contains hardly anything which can not be found in the union of the textbooks listed in the bibliography. In retrospect the material appears a bit too condensed for a 9 week undergraduate course. 8) Can you hear the shape of a drum? On any Riemannian manifold M , there exists a notion of a gradient and a notion of divergence. As in the Euclidean space, one can define the Laplacian ∆f = div gradf. If M is compact, then ∆ has eigenvalues 0 ≤ λ 1 ≤ λ 2. .. which converge to infinity. An interesting problem is to find the set of isospectral manifolds to a given manifold. An other problem is to find the set of metrics on M , which maximize det(∆), where det is a regularized determinant. The question "Can you hear ..." is due to M. Kac and is formulated for manifolds with boundaries, where the Laplace-Beltrami operator is taken with Dirichlet boundary conditions. There are now known two-dimensional isospectral domains in the plane, but it is for example not known, whether there exist smooth isospectral domains or convex isospectral domains in the plane. Chapter 1 Manifolds 1.1 Definition of manifolds Definition. A locally Euclidean space M of dimension n is a Hausdorff topological space, for which each point x ∈ M has a neighborhood U , which is homeomorphic to an open subset φ(U) of R n. The pair (U, φ) is called a coordinate system or a chart. Definition. A C k atlas on a locally Euclidean space M is a collection F = {U i , φ i } i∈I of charts such that i∈I U i = M , φ ij = φ i • φ −1 j is in C k (φ i (U j ∩ U i), R n). An atlas is maximal, if it has the property that if (U, φ) is a chart such that φ • φ −1 i and φ i • φ −1 are C k for all i ∈ I, then (U, φ) ∈ F. Definition. Two atlases F and G are called equivalent if F ∪ G is an atlas. Given an atlas A, the union of all atlases equivalent to A is a maximal and called the differentiable structure generated by A. Definition. A n-dimensional C k-differentiable manifold is a pair (M, F), where M is a n-dimensional second countable locally Euclidean space and where F is a differentiable structure on M. A manifold is called smooth if it is a C k-differentiable manifold for all k > 0. Index C k-map, 10 Alexander horned sphere, 4 Algebra of Lie, 57 Alternation mapping,

Seven papers presented at a research conference on space and geometry are contained in this monograph. The first paper gives an historical sketch of the development of geometry and discusses several considerations for selecting geometric content for the elementary school. Two papers deal with Piaget's research into the child's development of space and geometry concepts, and another paper suggests directions for further research on space from the Piagetian perspective. A fifth paper reviews the van Hiele levels of development in geometry and discusses the new Soviet geometry curriculum, another paper reviews cross-cultural research on perception, and the final paper examines some research issues concerning children's concepts of transformation geometry. (DT)

Oxford Scholarship Online, 2018

In the Anglophone world, the philosophical treatment of geometry has fallen on hard times. This chapter argues that philosophy can come to a better understanding of mathematics by providing an account of modern geometry, including its development of new forms of space, both for mathematical physics and for arithmetic. It returns to the discussions of Weyl and Cassirer on geometry whose concerns are very much relevant today. A way of encompassing a great part of modern geometry via homotopy toposes is discussed, along with the `cohesive’ variant of their internal language, known as `homotopy type theory’. With these tools in place, we can now start to see what an adequate philosophical account of current geometry might look like.

Theorem 24. Isometries of the plane are classified as follows: -Direct isometries: * Rotations of angle θ, including the identity (θ = 2πk, k ∈ Z) * Translations by a fixed vector v (again including the identity, v = 0) * Central symmetries, -Indirect isometries: * reflection symmetries, * glide reflections * Rotations, including the identity * Translations by a fixed vector * Central symmetries, -Indirect (or odd) isometries: * reflections symmetries, * glide reflections • In other geometries, e.g. hyperbolic geometries, the problem becomes more complicated: