| Abstract: |
There are several kinds of projection in three-dimensional space, in which we state eleven kinds of them.The analytical solution of some problems in Ell specially in E3 and E4 is introduced in this work such as : incidence, parallelism, intersection, orthogonality, aXIS subspace and distance between two parallel or skew subspaces.In four-dimensional space the reference system is taken as four mutually perpendicular hyperplanes II, I2, )’3 and I4 . These hyperplanes intersect in six mutually perpendicular planes Te I, Te2, Te3, Te4, Tes and Te6. These planes intersect in four mutually perpendicular axes x, y, z and t, which intersect in the origin of the system O.In four-dimensional space any point A can be represented by its four coordinates (tA, XA, YA, ZA.) relative to the four hyperplanes of the system of reference where I tA I = d (A , I4) , I XA 1= d (A , L3) ,I YA I = d (A, L2) and I ZA I = d (A, II), x, y, z and t E] - 00, 00[.In the Mongean system in £4 we consider Tel: {z, t} as a fixed plane (plane of the paper of drawing) and rotate the other planes about it. So we rotate the two planes Te2 and Te3 about the t-axis till they coincide with Tel , where the positive direction of x and y-axes coincide with the negative direction of z-axis. Also we rotate the two planes Tes and Te6 about the z-axis till they coincide with Te I where the positive direction of x and y-axes coincide with the negative direction of t-axis .The solid in E”~ is called a polychoron , which is bounded by 3-D faces called cells.There are sixteen regular polychora , six of which are convex and ten are stellated.The central projection in E4 is simply defined in two steps:1- The four dimensional space E4 is firstly projected from a special center of projection (0) on a three dimensional space of projection 2.::(xyz) .2- The obtained projection will be projected again inside L, from a center of projection 0 on a plane of projection n(yz) which is embedded inThe concept of the Axonometric projection in E4 is that the system of reference is firstly projected onto a selected hyperplane 2.::, which is called the Axonometric hyperplane of projection, then the projection of the system of reference on 2.:: is projected onto a plane XYZ which isL, where theorem 6-1 and theorem 6-2 are provedmathematically.The indexed projection in E’~ is a normal projection on a plane, which we considered as n4 = L I n L4 .A computer proeramwritten by matlab language IS used to classify, represent and transform the surfaces E3 in .A computer pro~ramdesigned by visual basic language to represent the main geometrical elements and also to solve some fundamental problems in E3.8-2 Recommendation for future work:In the future work we can study the following:1- The representation of surfaces in E3 by the methods of computer graphics and programming.2- The representation of hypersurfaces lil the central, indexed and axonometric projections with hyperplane sections.3- The several kinds of projections in E4 which are not mentioned in this work like double central projection, steriographic projection and cyclographic projection. Also obliqu projections and central axonometric projections in E4 will be studied4- The representation of surfaces in E~ by the methods of computergraphics is to be discussed.5- Discussing the representation of surfaces in E5, E6, ---and Ell.6- The metric problems in the Four-Dimensional Central projection. 7- The metric problems in the Four-Dimensional Indexed projection.8- The representation of the sphere in the four-Dimensional space using Pelz method.9- The visibility in the representation of the hyper solids.
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