Treatise Of Plane Geometry Through Geometric Algebra
Ramon González Calvet
ISBN: 84-699-3197-0
First Catalan edition: June 1996
First English edition: June 2000 to June 2001
The geometric algebra, initially discovered by Hermann Grassmann (1809-1877) was reformulated by William Kingdon Clifford (1845-1879) through the synthesis of the Grassmann’s extension theory and the quaternions of Sir William Rowan Hamilton (1805-1865). In this way the bases of the geometric algebra were established in the XIX century. Notwithstanding, due to the premature death of Clifford, the vector analysis − a remake of the quaternions by Josiah Willard Gibbs (1839-1903) and Oliver Heaviside (1850-1925) − became, after a long controversy, the geometric language of the XX century; the same vector analysis whose beauty attracted the attention of the author in a course on electromagnetism and led him - being still undergraduate - to read the Hamilton’s Elements of Quaternions. Maxwell himself already applied the quaternions to the electromagnetic field. However the equations are not written so nicely as with vector analysis. In 1986 Ramon contacted Josep Manel Parra i Serra, teacher of theoretical physics at the Universitat de Barcelona, who acquainted him with the Clifford algebra. In the framework of the summer courses on geometric algebra which they have taught for graduates and teachers since 1994, the plan of writing some books on this subject appeared in a very natural manner, the first sample being the Tractat de geometria plana mitjançant l’àlgebra geomètrica (1996) now out of print. The good reception of the readers has encouraged the author to write the Treatise of plane geometry through geometric algebra (a very enlarged translation of the Tractat) and publish it at the Internet site http://campus.uab.es/~PC00018, writing it not only for mathematics students but also for any person interested in geometry. The plane geometry is a basic and easy step to enter into the Clifford-Grassmann geometric algebra, which will become the geometric language of the XXI century.
Dr. Ramon González Calvet (1964) is high school teacher of mathematics since 1987, fellow of the Societat Catalana de Matemàtiques and also of the Societat Catalana de Gnomònica.
CONTENTS
First Part: The vector plane and the complex numbers
1. The vectors and their operations. Vector addition, 1.
- Product of a vector and a real number, 2.
- Product of two vectors, 2.
- Product of three vectors: associative property, 5.
- Product of four vectors, 7.
- Inverse and quotient of two vectors, 7.
- Hierarchy of algebraic operations, 8.
- Geometric algebra of the vector plane, 8.
- Exercises, 9.
2. A base of vectors for the plane. Linear combination of two vectors, 10.
- Base and components, 10.
- Orthonormal bases, 11.
- Applications of the formulae for the products, 11.
- Exercises, 12.
3. The complex numbers. Subalgebra of the 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.
- The fundamental theorem of algebra, 24.
- Exercises, 26.
4. Transformations of vectors. Rotations, 27.
- Reflections, 28.
- Inversions, 29.
- Dilatations, 30.
- Exercises, 30
Second Part: The 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 alignment 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, 40.
- Intersection of two lines and pencil of lines, 41.
- Dual coordinates, 43.
- The Desargues theorem, 47.
- 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 single ratio. Direct similarity, 61.
- Opposite similarity, 62.
- The theorem of Menelaus, 63.
- The theorem of Ceva, 64.
- Homothety and single ratio, 65.
- Exercises, 67.
8. Properties of the 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.
- The Fermat's theorem, 77.
- Exercises, 78.
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.
- The homography (Möbius transformation), 96.
- Projective cross ratio, 99.
- The points at the infinity and homogeneous coordinates, 102.
- Perspectivity and projectivity, 103.
- The projectivity as a tool for theorems demonstration, 108.
- The homology, 110.
- Exercises, 115.
11. Conics Conic sections, 117.
- Two foci and two directrices, 120.
- Vectorial equation, 121.
- The Chasles' theorem, 122.
- Tangent and perpendicular to a conic, 124.
- Central equations for the ellipse and hyperbola, 126.
- Diameters and Apollonius' theorem, 128.
- Conic passing through five points, 131.
- Conic equations in barycentric coordinates and tangential conic, 132.
- Polarities, 134.
- Reduction of the conic matrix to a diagonal form, 136.
- Using a base of points on the conic, 137.
- Exercises, 137.
Third Part: Pseudo-Euclidean geometry
12. Matrix representation and hyperbolic numbers. Rotations and the representation of complex numbers, 139.
- The subalgebra of the 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 the 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 on 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 tridimensional spaces 14.
Spherical geometry in the Euclidean space. The geometric algebra of the Euclidean space, 170.
- Spherical trigonometry, 172.
- The dual spherical 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.
- The central or gnomonic projection, 177.
- Stereographic projection, 180.
- Orthographic projection, 181.
- Spherical coordinates and cylindrical equidistant (Plate Carré) projection, 182.
- Mercator's projection, 183.
- Peter's projection, 184.
- Conic projections, 184.
- Exercises, 185.
15. Hyperboloidal geometry in the pseudo-Euclidean space (Lobachevsky's geometry). The geometric algebra of the pseudo-Euclidean space, 188.
- The hyperboloid of two sheets, 190.
- The central projection (Beltrami disk), 191.
- Hyperboloidal (Lobachevskian) trigonometry, 196.
- Stereographic projection (Poincaré disk), 198.
- Azimuthal equivalent projection, 200.
- Weierstrass coordinates and cylindrical equidistant projection, 201.
- Cylindrical conformal projection, 202.
- Cylindrical equivalent projection, 203.
- Conic projections, 203.
- On the congruence of geodesic triangles, 205.
- Comment about the names of the non-Euclidean geometry, 205.
- Exercises, 205.
16. Solutions of the proposed exercises.
- 1. The vectors and their operations, 207.
- 2. A base of vectors for the plane, 208.
- 3. The complex numbers, 209.
- 4. Transformations of vectors, 213.
- 5. Points and straight lines, 214.
- 6. Angles and elemental trigonometry, 223.
- 7. Similarities and single ratio, 226.
- 8. Properties of the triangles, 228.
- 9. Circles, 236.
- 10. Cross ratios and related transformations, 240.
- 11. Conics, 245.
- 12. Matrix representation and hyperbolic numbers, 250.
- 13. The hyperbolic or pseudo-Euclidean plane, 251.
- 14. Spherical geometry in the Euclidean space, 254.
- 15. Hyperboloidal geometry in the pseudo-Euclidean space (Lobachevsky's geometry), 260.
Bibliography, 266.
Index, 270.
Chronology, 275.
http://ifile.it/awfvgm6/index.html_d%3D4XMCZ52M
