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Introduction. The ability to visualize or depict the 3D object is the basis for understanding the shape of the object

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The ability to visualize or depict the 3D object is the basis for understanding the shape of the object. In addition, in many cases this is the capacity to rotate, move and build kinds of projections of the object. All this is easily illustrated by the example of our acquaintance with relatively complex unfamiliar object. To understand its form, we immediately begin to rotate an object, move away at arm's length, move up and down, back and forth, etc. To do the same with a computer, we must extend our previous two-dimensional analysis to three dimensions. Based on experience, we immediately introduce homogeneous coordinates. Thus, a 3D point [ x y z ] is 4D vector:

(2.1)

where [ T ] is the matrix of a transformation. As before, the transformation from the homogeneous coordinates to ordinary coordinates is given by formula:

(2.2)

Generalized transformation matrix 4×4 for three-dimensional homogeneous coordinates can be represented as follows:

(2.3)

Transformation matrix 4×4 can be divided into four parts. The upper-left (3×3)-submatrix defines a linear transformation in the form of scaling, translation, reflection, and rotation. Lower left (1×3)-submatrix defines the movement, and the upper right (3×1)-submatrix - perspective transformation. Last lower right (1×1)-submatrix sets the general scaling. The total conversion obtained after the application of this (4×4) matrix to the homogeneous vector and calculations ordinary coordinates, is called the bilinear transformation. In general, this transformation provides a combination of shear, local scaling, rotation, reflection, movement, perspective transformation and general scaling.


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