The Axioms of Descriptive GeometryUniversity Press, 1907 - 74 pagina's |
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A. N. WHITEHEAD a₁ axes B₂l BC and B'C belong centre class of points cohere collinear concurrent concurrent lines congruence group consider convex region coordinates coplanar Dedekind property defined Definition Descriptive Geometry Descriptive Space dt dt dt dy dt dz dt dx dt dy dz Euclidean axiom exist family of spheres Geom given half-rays harmonic conjugate Hence cf homogeneous coordinates IDEAL POINTS imaginary infinite plane infinitesimal rotation round infinitesimal transformation intersect the segment lies metrical geometry N₁ neighbourhood normal reference tetrahedron origin P₁ Peano's axioms Proj Projective Geometry projective group projective line projective space projective transformation proper projective plane proper projective points proposition quadric relation relatum respect round the axis segment AC segment OA semi-rays sheaf similar equations straight line surface of revolution theorem three lines trihedrons vertex x₁X y₁ Y₁Y аф
Populaire passages
Pagina 5 - If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.
Pagina 5 - ABC consists of all points collinear with any two points of the sides of the triangle. AXIOM IX. If there exist three points not lying in the same line, there exists a plane ABC such that there is a point D not lying in the plane ABC. DEF. 7. If A, B, C, and D are four points not lying in the same plane, they form a tetrahedron ABCD whose faces are the interiors of the triangles ABC, BCD, CD A, DAB (if the triangles exist) whose vertices are the four points, A, B, C, and D, and whose edges are the...
Pagina 4 - V. // three distinct points, A, B, and C do not lie on the same line and D and E are two points in the orders { BCD } and { CEA } , then a point F exists in the order { AFB } and such that D, E and F lie on the same line.
Pagina 4 - ... orders ABX, AXB, XAB. The points X in the order AXB constitute the 'segment' AB. A and B are the ' end-points ' of the segment, but are not included in it. VI. If points C and D (C^D) lie on the line AB, then A lies on the line CD. VII. If there exist three distinct points, there exist three points A, B, C not in any of the orders ABC, BCA, or CAB. Definition 2. Three distinct points not lying on the same line are the 'vertices
Pagina 5 - ABCD whose faces are the interiors of the triangles ABC, BCD, CD A, DAB (if the triangles exist) whose vertices are the four points, A, B, C, and D, and whose edges are the segments AB, BC, CD, DA, AC, BD. The points of faces, edges, and vertices constitute the surface of the tetrahedron. DEF. 8. If A, B, C, D are the vertices of a tetrahedron, the space ABCD consists of all points collinear with any two points of the faces of the tetrahedron. AXIOM X. If there exist four points neither lying in...
Pagina 4 - If A and B are any two distinct points, there exists a point C such that A, B, C are in the order ABC. Definition 1.
Pagina 4 - B) consists of A and B and all points X in one of the possible orders ABX, AXB, XAB. The points X in the order AXB constitute the segment AB.