Index

5th postulate, 129 base of prism, 30 base of pyramid, 31 addition of vectors, 132 base of spherical frustum, 91 additivity of dot product, 114 base of spherical sector, 96 altitude of cone, 77 base of spherical segment, 91 altitude of cylinder, 76 bases of frustum, 32 altitude of frustum, 32 bilateral symmetry, 61 altitude of prism, 30 bilinearity, 137 altitude of pyramid, 31 box, 31 altitude of spherical segment, 91 angle between line and plane, 21 Cartesian coordinate system, 141 angle between lines, 20 Cartesian projection, 20 angle between planes, 18 Cauchy-Schwarz inequality, 138 angle between skew lines, 20 Cavalieri’s principle, 45 angle on hyperbolic plane, 150 center, 88 antiprism, 72, 74 center of homothety, 52 apothem, 32 center of mass, 119 Archimedean solids, 72 center of symmetry, 59 Archimedes’ axiom, 131 central symmetry, 59 area, 152 Ceva’s theorem, 122 area of sphere, 93 circle, 161 area of spherical frustum, 93 circumscribed prism, 77 area of spherical segment, 93 circumscribed pyramid, 78 associative, 109 circumscribed sphere, 103 associativity, 111, 120 collinear, 123 axial cross section, 86 common notion, 128 axiom, 128, 132 commutative, 109 axiom of completeness, 131 composition, 138, 154 axiom of dimension, 134 concurrent, 37 axioms of order, 130 cone, 77 axis of revolution, 75 congruent dihedral angles, 17 axis of symmetry, 61 congruent figures, 129, 138 conical frustum, 78 ball, 88 conical surface, 77 barycenter, 119 consistency, 144 base of cone, 77 convex polyhedral angle, 23 base of conical frustum, 78 convex polyhedron, 29 base of cylinder, 76 coordinate Euclidean space, 142

173 174 Index coordinates, 141 generatrix, 75–77 cube, 31 great circle, 89 cuboid, 31 greater dihedral angle, 17 curved surface, 78 cylinder, 76 half-plane, 16 cylindrical surface, 76 head of directed segment, 107 hemisphere, 89 definition, 128 hexahedron, 67 Desargues’ theorem, 127 higher order symmetry, 63 development, 81 Hilbert’s 3rd Problem, 48 diagonal of polyhedron, 29 Hilbert’s axioms, 130 diagonal plane, 31 homogeneity of dot product, 114 diameter, 88 homologous, 51 difference of vectors, 111 homothetic figures, 52 dihedral angle, 16 homothety, 52 dimension (of space), 133 homothety coefficient, 52 dimensions (of box), 31 hyperbola, 148 directed segment, 107 hyperbolic, 160 directrix, 76, 77 hyperbolic plane, 149 distance from point to plane, 12 hyperbolic rotation, 159 distributivity, 110, 111 hyperboloid, 149 dodecahedron, 67 hyperboloid model, 149 dome, 91 dot product, 112 icosahedron, 67 doubling the cube, 59 identity, 154 inner product, 137 edge of dihedral angle, 16 inscribed prism, 77 edge of half-plane, 16 inscribed pyramid, 78 edge of polyhedral angle, 23 inscribed sphere, 103 edge of polyhedron, 29 interior, 16 element, 131 inverse, 154 elliptic, 160 isometry, 138, 154 equal vectors, 108 equation of plane, 153 Klein model, 143 equivalent, 35, 39 Euclidean inner product, 137 lateral edge of prism, 30 Euclidean vector space, 137 lateral edge of pyramid, 31 Euler’s line, 118 lateral face of prism, 30 exterior, 16 lateral face of pyramid, 31 lateral surface, 91 face of dihedral angle, 16 lateral surface area, 35, 78 face of polyhedral angle, 23 lateral surface of cone, 77 face of polyhedron, 29 lateral surface of cylinder, 76 fixed point, 156 length, 151 foot of perpendicular, 11 lie between, 130 foot of slant, 11 light cone, 148 frustum of pyramid, 32 light-like vector, 148 line, 128, 134 generator, 75, 76 line of centers, 103 Index 175 line perpendicular to plane, 10 plane of symmetry, 60 linear angle, 16 plane perpendicular to line, 10 linear combination, 133 plane surface, 128 linearly dependent, 133 Platonic solids, 67 linearly independent, 133 point, 128, 134 lune, 145 polyhedral angle, 23 polyhedron, 29 mass, 119 postulate, 128 material point, 119 prism, 29, 30 measure of polyhedral angle, 105 product of scalar and vector, 110 measure of solid angle, 105 projection, 20 Menelaus’ theorem, 124 projectionoffigure,20 meridian, 76 projectionofslant,12 meridional, 76 projective geometry, 126 Minkowski inner product, 147 projective plane, 146 Minkowski space, 147, 148 proportional vectors, 110 multiplication, 110 proposition, 129 multiplication by scalars, 132 pseudo-mass, 119 pyramid, 31 net, 81, 82 pyramidal frustum, 32 net of cone, 83 net of conical frustum, 83 quadrangular prism, 30 net of cylinder, 82 quadrangular pyramid, 32 non-Euclidean geometries, 129 oblique, 11 radius, 88 oblique cylinder, 76 radius-vector, 111, 134 oblique prism, 30 rectangular parallelepiped, 31 octahedron, 67 reflection, 155, 157 opposite vector, 132 regrouping, 120 opposite vectors, 111 regular polyhedral angle, 66 order of symmetry, 63 regular polyhedron, 66 origin, 111, 134 regular prism, 30 orthogonal projection, 20 regular pyramid, 32 orthogonal vectors, 148 regular pyramidal frustum, 32 regular tetrahedron, 67 Pappus’ theorem, 127 relation, 130 parabolic, 160 represent the same vector, 108 parallel line and plane, 4 Riemann surface, 160 parallel planes, 5 right circular cone, 77 parallel postulate, 129 right circular cylinder, 76 parallelepiped, 31 right cylinder, 76 pentahedralangle,23 right dihedral angle, 17 perpendicular, 10 right prism, 30 perpendicular cross section, 35 rotation, 155 perpendicular lines, 20 perpendicular planes, 19 scalar, 132 plane, 134 scalar product, 112 plane angle, 23 scissors-congruent, 48 176 Index set, 131 three perpendiculars, 13 set of real numbers, 131 time-like vector, 148 signed projection, 113 total surface area, 79–81 similar, 56 translation, 54, 134, 155 similar cones, 84 triangle inequality, 140 similar cylinders, 84 triangular prism, 30 similar figures, 56 triangular pyramid, 32 similar polyhedra, 51 trihedral angle, 23 skew lines, 4 slant, 11 undefinable notions, 130 slant to plane, 11 unit dihedral angle, 18 smaller dihedral angle, 17 unit vector, 113, 141 solid angle, 102 solid geometry, 1 vector, 108, 132 space-like vector, 148 vector space, 132 sphere, 88 vertex, 77 spherical frustum, 91 vertex of polyhedral angle, 23 spherical geometry, 144 vertex of polyhedron, 29 spherical lune, 145 vertex of pyramid, 31 spherical model, 146 vertex of solid angle, 102 spherical sector, 96 vertical dihedral angles, 17 spherical segment, 91 vertical polyhedral angles, 26 spherical surface, 88 volume, 39 stereometry, 1 volume of ball, 99 straight line, 128 volume of cone, 83 subset, 131 volume of conical frustum, 83 subspace, 133 volume of cylinder, 83 sum of vectors, 108 volume of prism, 44 supplementary dihedral angles, 17 volume of spherical sector, 98 surface, 128 weighted sum, 120 surface area, 35 surface of revolution, 75 zero vector, 110 symmetric figures, 139 zone, 91 symmetric polyhedral angles, 26 symmetricity, 137 symmetricity of dot product, 114 symmetry about line, 61 symmetry about plane, 60 tail of directed segment, 107 tangent balls, 103 tangent cones, 86 tangent line, 91 tangent plane, 86, 90 tangent spheres, 103 test for perpendicular planes, 19 tetrahedral angle, 23 tetrahedron, 32