Index of Definitions, Axioms, and Theorems
for Survey of Geometry
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Chapter One – Axiomatic Systems and Finite Geometry |
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| absolute consistency - 1.1.1, 1.1.2 | Hamming Distance - 1.4 |
| abstract model - 1.1.1 | independent - 1.1.1, 1.1.2 |
| axiom - 1.1.1 | isomorphic - 1.1.1, 1.1.2 |
| axiomatic system - 1.1.1 | model - 1.1.1, 1.1.2 |
| categorical - 1.1.1, 1.1.2 | abstract - 1.1.1 |
| Codeword Table - 1.4 | concrete - 1.1.1, 1.1.2 |
| common notion - 1.1.3 | nonisomorphic - 1.1.2 |
| complete - 1.1.1, 1.1.2 | parity matrix - 1.4 |
| concrete model - 1.1.1, 1.1.2 | postulate - 1.1.1, 1.1.3 |
| concurrent - 1.3 | postulate system - 1.1.1 |
| consistent - 1.1.1, 1.1.2 | primitive term - 1.1.1, 1.1.2 |
| absolute consistency - 1.1.1, 1.1.2 | principle of duality - 1.1.1, 1.1.2, 1.3 |
| relative consistency - 1.1.1, 1.1.2 | projective plane of order n - 1.3 |
| dual - 1.1.1, 1.1.2 | relation - 1.1.1, 1.1.2 |
| element - 1.1.1, 1.1.2 | relative consistent - 1.1.1, 1.1.2 |
| Fano's Geometry (axioms) - 1.2 | theorem - 1.1.1 |
| finite projective plane (axioms) - 1.3 | undefined term - 1.1.1, 1.1.2 |
| generator matrix - 1.4 | vacuously true - 1.1.2 |
| Hamming (7, 4) Code - 1.4 | word - 1.4 |
| Chapter Two – Euclidean and Non-Euclidean Geometry | |
| acute angle - 2.4.2 | linear pair - 2.5.1 |
| alternate interior angles - 2.6.1 | Klein model - 2.7.3 |
| parallel lines - 2.6.1 | Max-distance plane - 2.1.3 |
| angle - 2.4.2 | metric geometry - 2.1.1, Appendix B |
| bisector - 2.4.2 | midpoint - 2.3 |
| congruent - 2.4.2 | Missing Strip plane - 2.1.3 |
| interior of - 2.4.2 | Modified Riemann Sphere - 2.1.3, 2.7.3 |
| measure - Euclidean 2.1.3 Poincaré Half-plane 2.1.3 | neutral geometry - 2.1.1 |
| supplementary - 2.5.1 | noncollinear - 2.2 |
| Angle Addition Postulate - 2.4.1 | obtuse angle - 2.4.2 |
| Angle Construction Postulate - 2.4.1 | one-to-one and onto - 2.3 |
| Angle Measurement Postulate - 2.4.1 | parallel lines - |
| antipodal points - 2.1.3, 2.7.3, 2.7.3 | alternate interior angles - 2.6.1 |
| base - isosceles triangle 2.5.2 | existence - 2.6.1 |
| Saccheri quadrilateral 2.6.2 | perpendicular - 2.6.1 |
| base angles - isosceles triangle 2.5.2 | Pasch's Postulate - 2.4.1 |
| Saccheri quadrilateral 2.6.2 | perpendicular - 2.4.2 |
| betweenness of points - 2.3 (Hilbert's - Appendix A) | parallel - 2.6.1 |
| Birkhoff's axioms - Appendix B, 2.1.1 | Plane Separation Postulate - 2.4.1 |
| bisector of an angle - 2.4.2 | Playfair's Axiom - 2.1.2, 2.7.1 |
| Cartesian plane - 2.1.3 | polar points (poles) - 2.1.3, 2.7.3 |
| collinear - 2.2 | Poincaré Half-plane - 2.1.3 |
| congruent - angles 2.4.2 | point - betweenness of 2.3 (Hilbert's - Appendix A) |
| segments 2.3 | Cartesian (Euclidean, Taxicab, Max-dist.) - 2.1.3 |
| triangles 2.5.2 | Discrete - 2.1.3 |
| convex set - 2.4.1 | Missing Strip - 2.1.3 |
| coordinate system (ruler) - 2.3 | Modified Riemann Sphere 2.1.3, 2.7.3 |
| Crossbar Theorem - 2.5.1, 2.5.2 | Poincaré Half-plane - 2.1.3 |
| distance - 2.1.3 (Birkhoff - Appendix B) | Riemann Sphere - 2.1.3, 2.7.3 |
| axioms - 2.3 | Poincaré Disk - 2.1.3, 2.7.2, 2.7.3 |
| Discrete - 2.1.3 | Pons Asinorum - 2.5.2 |
| Euclidean - 2.1.3 | ray - 2.3 |
| Max-Distance - 2.1.3 | remote interior angles - 2.6.1 |
| Missing Strip - 2.1.3 | Riemann Sphere - 2.1.3, 2.7.3 |
| Poincaré Half-plane - 2.1.3 | Modified - 2.1.3, 2.7.3 |
| Taxicab - 2.1.3 | right angle - 2.4.2 |
| edge of a half-plane - 2.4.1 | ruler (coordinate system) - 2.3 |
| elliptic geometry - 2.1.2, 2.7.3, 2.8 | axioms - 2.3 (Birkhoff's - Appendix B) |
| double elliptic - 2.7.3 | rulers (standard) - Euclidean - 2.1.3 |
| single elliptic - 2.7.3 | Max-distance 2.1.3 |
| Elliptic Parallel Postulate - 2.7.3 | Missing Strip - 2.1.3 |
| endpoint - ray 2.3 | Poincaré Half-plane - 2.1.3 |
| segment 2.3 | Taxicab - 2.1.3 |
| equilateral triangle - 2.5.2 | Ruler Postulate - 2.3 |
| Euclid's Elements - 2.1.1 | Ruler Placement Postulate - 2.3 |
| Euclid's Fifth Postulate - 2.1.2, 2.6.2, 2.7.1 | Saccheri quadrilateral - 2.1.2, 2.6.2 |
| Euclidean geometry - 2.7.1, 2.8 | SAS Postulate - 2.5.2 |
| Euclidean Parallel Postulate - 2.7.1 | segment (line) - 2.3 |
| Euclidean plane (model) - 2.1.3 | congruent - 2.3 |
| exterior angle of a triangle - 2.6.1 | side of an angle - 2.4.2 |
| Exterior Angle Theorem - 2.6.1 | of a half-plane - 2.4.1 |
| half-plane - 2.4.1 | SMSG axioms - Appendix C, 2.1.1 |
| Hilbert's axioms - Appendix A, 2.1.1 | summit of a Saccheri quadrilateral - 2.6.2 angles - 2.6.2 |
| hyperbolic geometry - 2.7.2, 2.8 | Supplement Postulate - 2.5.1 |
| Hyperbolic Parallel Postulate - 2.7.2 | supplementary angles - 2.5.1 |
| incidence axioms - 2.2 (Hilbert's Appendix A) | synthetic geometry - 2.1.1, Appendix A |
| interior of an angle - 2.4.2 | Taxicab plane - 2.1.3 |
| isosceles triangle - 2.5.2 | transversal - 2.6.1 |
| base angles - 2.5.2 pons asinorum 2.5.2 | Triangle Inequality - (link) |
| vertex angle - 2.5.2 | vertex angle of an isosceles triangle - 2.5.2 |
| line - Cartesian (Euclidean, Taxicab, Max-distance) 2.1.3 | vertex of an angle - 2.4.2 |
| Missing Strip - 2.1.3 | vertical angles - 2.5.1 |
| Poincaré Half-plane - 2.1.3 | Vertical Angle Theorem - 2.5.1 |
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Chapter Three – Transformational Geometry |
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| affine transformation - 3.2.3 | line (analytic model) - 3.2.2 |
| angle between two lines (analytic model) - 3.2.2 | homogeneous coordinates - 3.2.2 |
| collinear (collinearity) - | measure of angle between - 3.2.2 |
| affine transformation - 3.3.2 | image of - 3.3.2 |
| analytic model - 3.2.2 | mapping - 3.2.1 |
| invariant - 3.3.1, 3.3.2, 3.6.1 | matrix - |
| isometry - 3.3.1 | dilation - 3.6.2 |
| similarity transformation - 3.6.1 | direct isometry - 3.3.3 |
| concurrent (analytic model) - 3.2.2 | indirect isometry - 3.3.3 |
| dilation - 3.6.1 | isometry - 3.3.3 |
| affine (matrix) - 3.6.2 | reflection - 3.5.2 |
| direct isometry - 3.3.3 | rotation -3.4.2 |
| rotation - 3.4.2 | similarity - 3.6.2 |
| translation - 3.4.2 | transformation - 3.2.3 |
| distance (analytic model) - 3.2.2 | translation - 3.4.2 |
| glide reflection - 3.5.1 | one-to-one and onto - 3.2.1 |
| group - 3.2.1 | point (analytic model) - 3.2.2 |
| affine direct isometry - 3.3.3 | reflection - 3.5.1 |
| isometry - 3.3.1 (affine) - 3.3.3 | affine (matrix) - 3.5.2 |
| rotation - 3.4.1 (affine) - 3.4.2 | indirect isometry - 3.5.2 |
| translation - 3.4.1 (affine) - 3.4.2 | isometry - 3.5.1 |
| incident (analytic model) - 3.2.2 | rotation - 3.4.1 |
| indirect isometry - 3.3.3 | affine (matrix) - 3.4.2 |
| reflection - 3.5.2 | direct isometry - 3.4.2 |
| invariant - 3.3.1 | isometry - 3.4.1 |
| angle measure - 3.3.1, 3.6.1 | similar - 3.6.1 |
| betweenness - 3.3.1, 3.6.1 | similarity transformation - 3.6.1 |
| collinearity - 3.3.1, 3.3.2, 3.6.1 | affine (matrix) - 3.6.2 |
| glide reflection - 3.5.1 | symmetry - 3.4.1 |
| reflection - 3.5.1, 3.5.2 | transformation - 3.2.1 |
| rotation - 3.4.1, 3.4.2 | affine - 3.2.3 |
| translation - 3.4.1, 3.4.2 (line) 3.4.1 | of a plane - 3.2.1 |
| isometry - 3.3.1 | translation - 3.4.1 |
| affine - 3.3.3 | affine (matrix) - 3.4.2 |
| direct - 3.3.3 | direct isometry - 3.4.2 |
| glide reflection - 3.5.1 | isometry - 3.4.1 |
| indirect - 3.3.3 | Triangle Inequality - (link) |
| reflection - 3.5.1 | |
| rotation - 3.4.1 | |
| translation - 3.4.1 | |
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Chapter Four – Projective Geometry |
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| axioms for projective geometry - 4.2.1 | harmonic conjugate - 4.5.1 |
| Brianchon's Theorem - 4.7.2 | harmonic sets and music - 4.5.2 |
| collinear - 4.2.1 | hexagon (simple) - 4.7.2 |
| concurrent - 4.2.1 | homology (axis/center of) - 4.6.4 |
| conic - 4.7.1, 4.7.2, 4.7.3 | line conic - 4.7.1, 4.7.2 |
| definition - 4.7.1 | Pascal's Mystic Hexagon Theorem - 4.7.1, 4.7.2 |
| hexagon - 4.7.2 | pencil of points - 4.2.1 |
| tangent - 4.7.3 | perspective from a point - 4.2.1 |
| cross joins - 4.6.4 | from a line - 4.2.1 |
| Desargues' Theorem - 4.2.1, 4.3, 4.4 | perspectivity - 4.2.1, 4.6.1 |
| diagonal point - 4.2.1, 4.7.2 | point conic - 4.7.1, 4.7.2, 4.7.3 |
| of a quadrangle - 4.2.1 | projectivity - 4.2.1, 4.6.1 |
| of a hexagon - 4.7.2 | alternate construction of - 4.6.4 |
| diagonal line - 4.3, 4.7.2 | construction of - 4.6.2 |
| of a quadrilateral - 4.3 | harmonic sets - 4.6.3 |
| of a hexagon - 4.7.2 | quadrangle (complete) - 4.2.1 |
| duality in projective geometry - 4.3 | quadrilateral (complete) - 4.3 |
| elementary correspondence - 4.6.1 | Theorem of Pappus - 4.6.4 |
| Fundamental Theorem of Projective Geometry - 4.6.2 | triangle - 4.2.1 |
| harmonic sets - 4.5.1 | |
| projectivity - 4.6.3 | |
© Copyright 2006 - Timothy Peil