Appendix a Basic Mathematics for 3D Computer Graphics
Appendix A Basic Mathematics for 3D Computer Graphics A.1 Vector Operations (),, A vector v is a represented as v1 v2 v3 , which has a length and direction. The location of a vector is actually undefined. We can consider it is parallel to the line (),, (),, from origin to a 3D point v. If we use two points A1 A2 A3 and B1 B2 B3 to (),, represent a vector AB, then AB = B1 – A1 B2 – A2 B3 – A3 , which is again parallel (),, to the line from origin to B1 – A1 B2 – A2 B3 – A3 . We can consider a vector as a ray from a starting point to an end point. However, the two points really specify a length and a direction. This vector is equivalent to any other vectors with the same length and direction. A.1.1 The Length and Direction The length of v is a scalar value as follows: 2 2 2 v = v1 ++v2 v3 . (EQ 1) 378 Appendix A The direction of the vector, which can be represented with a unit vector with length equal to one, is: ⎛⎞v1 v2 v3 normalize()v = ⎜⎟--------,,-------- -------- . (EQ 2) ⎝⎠v1 v2 v3 That is, when we normalize a vector, we find its corresponding unit vector. If we consider the vector as a point, then the vector direction is from the origin to that point. A.1.2 Addition and Subtraction (),, (),, If we have two points A1 A2 A3 and B1 B2 B3 to represent two vectors A and B, then you can consider they are vectors from the origin to the points.
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