Chapter One – Axiomatic Systems and Finite Geometry |
|
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 (one-to-one correspondence) - 2.3 |
Angle Measurement Postulate - 2.4.1 | parallel lines - 2.6.1 |
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 of a segment - 2.3 | 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 | rectangle - 2.6.2 |
Missing Strip - 2.1.3 | remote interior angles - 2.6.1 |
Poincaré Half-plane - 2.1.3 | Riemann Sphere - 2.1.3, 2.7.3 |
Taxicab - 2.1.3 | Modified - 2.1.3, 2.7.3 |
edge of a half-plane - 2.4.1 | right angle - 2.4.2 |
elliptic geometry - 2.1.2, 2.7.3, 2.8 | ruler (coordinate system) - 2.3 |
double elliptic - 2.7.3 | axioms - 2.3 (Birkhoff's - Appendix B) |
single elliptic - 2.7.3 | rulers (standard) - Euclidean - 2.1.3 |
Elliptic Parallel Postulate - 2.7.3 | Max-distance 2.1.3 |
endpoint - ray 2.3 | Missing Strip - 2.1.3 |
segment 2.3 | Poincaré Half-plane - 2.1.3 |
equilateral triangle - 2.5.2 | Taxicab - 2.1.3 |
equivalence relation - 2.3 | 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 of a triangle - 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 - 2.3 |
vertex angle - 2.5.2 | Triangle Inequality - (link) |
line - Cartesian (Euclidean, Taxicab, Max-distance) 2.1.3 | vertex angle of an isosceles triangle - 2.5.2 |
Missing Strip - 2.1.3 | vertex of an angle - 2.4.2 |
Poincaré Half-plane - 2.1.3 | vertical angles - 2.5.1 |
Vertical Angle Theorem - 2.5.1 | |
Chapter Three – Transformational Geometry |
|
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 |
congruent - 3.3.1, 3.3.3 | isometry - 3.3.3 |
dilation - 3.6.1 | reflection - 3.5.2 |
affine (matrix) - 3.6.2 | rotation -3.4.2 |
direct isometry - 3.3.3 | similarity - 3.6.2 |
rotation - 3.4.2 | transformation - 3.2.3 |
translation - 3.4.2 | translation - 3.4.2 |
distance (analytic model) - 3.2.2 | one-to-one and onto - 3.2.1 |
glide reflection - 3.5.1 | point (analytic model) - 3.2.2 |
group - 3.2.1 | reflection - 3.5.1 |
affine direct isometry - 3.3.3 | affine (matrix) - 3.5.2 |
isometry - 3.3.1 (affine) - 3.3.3 | indirect isometry - 3.5.2 |
rotation - 3.4.1 (affine) - 3.4.2 | isometry - 3.5.1 |
translation - 3.4.1 (affine) - 3.4.2 | rotation - 3.4.1 |
incident (analytic model) - 3.2.2 | affine (matrix) - 3.4.2 |
indirect isometry - 3.3.3 | direct isometry - 3.4.2 |
reflection - 3.5.2 | isometry - 3.4.1 |
invariant - 3.3.1 | similar - 3.6.1 |
angle measure - 3.3.1, 3.6.1 | similarity transformation - 3.6.1 |
betweenness - 3.3.1, 3.6.1 | affine (matrix) - 3.6.2 |
collinearity - 3.3.1, 3.3.2, 3.6.1 | symmetry - 3.4.1 |
glide reflection - 3.5.1 | transformation - 3.2.1 |
reflection - 3.5.1, 3.5.2 | affine - 3.2.3 |
rotation - 3.4.1, 3.4.2 | of a plane - 3.2.1 |
translation - 3.4.1, 3.4.2 (line) 3.4.1 | translation - 3.4.1 |
isometry - 3.3.1 | affine (matrix) - 3.4.2 |
affine - 3.3.3 | direct isometry - 3.4.2 |
direct - 3.3.3 | isometry - 3.4.1 |
glide reflection - 3.5.1 | triangle - 2.3 |
indirect - 3.3.3 | Triangle Inequality - (link) |
reflection - 3.5.1 | |
rotation - 3.4.1 | |
translation - 3.4.1 | |
Chapter Four – Projective Geometry |
|
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