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Pérez Águila, Ricardo
- Capítulo 3. 4D
Geometric Transformation-
4D Orthogonal Polytopes
-- Licenciatura en Ingeniería
en Sistemas Computacionales. - Departamento de Ingeniería en
Sistemas Computacionales. - Escuela de Ingeniería, - Universidad de las Américas
Puebla.


Introducción



Chapter 3 4D Geometric Transformations 3.1 3D Geometric Transformations as Extension of 2D Geometric Transformations [Hearn, 95] considers 3D geometric transformations (translation, rotation, scaling, etc.) as extensions of the 2D geometric transformations for these same operations with the consideration of the Z coordinate. Translating in the 2D space implies a displacement of a polygon in direction of X and Y-axis, in other words, we apply a translation over a polygon to change its position.
A 2D point is converted when the translation distances tx and ty are added to the original coordinate (x,y) to move it to the new position (x',y'): x'  x  t x y'  y  t y Or using homogeneous coordinates and the matrix representation:   x' y ' 1   x 1 y 1   0  t  x  0 1 ty 0 0 1  Based in the previous idea, translating in 3D space implies a displacement of an object in direction of the X, Y and Z-axis.
We translate a 3D object when it is moved in 25 each one of the three directions of the coordinates.
We translate a point (x,y,z) to the position (x',y',z') adding the distances tx, ty and tz: x'  x  t x y'  y  t y z'  z  t z or  1 0 0 1 0 0 0 0 0  t  x 0 ty 1 tz 0 1   x' y ' z ' 1   x y z 1    Scaling in 2D space implies a change of size (and in some cases of shape and position) of an object through two factors each one with relation with X and Y coordinates. A 2D point is converted when it is multiplied by the scaling factors SX and SY to produce the transformed coordinates (x',y').
The scaling factor SX scales objects in the direction parallel to X axis, while the scaling factor SY scales objects in the direction parallel to Y axis.
We have then: x'  S X  x y '  SY  y or   x' y ' 1   x  SX y 1   0  0  0 SY 0 0 0 1  Again, it is possible to extend the previous 2D concept and to conclude that scaling in 3D space implies a change of size of a polyhedron by three factors each one wit...






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