Index of Definitions, Axioms, and Theorems for Survey of Geometry
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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 - 4.7.14.7.24.7.3 line conic - 4.7.14.7.2
        definition - 4.7.1 Pascal's Mystic Hexagon Theorem - 4.7.14.7.2
        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 - 4.2.14.34.4 perspectivity - 4.2.14.6.1
diagonal point - 4.2.1, 4.7.2 point conic - 4.7.14.7.24.7.3
        of a quadrangle - 4.2.1 projectivity - 4.2.14.6.1
        of a hexagon - 4.7.2         alternate construction of - 4.6.4
diagonal line - 4.34.7.2         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