Frame in Three-dimension Space
First, we construct the rotation matrix as following :
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Condition |
Equation |
Rotation
Matrix |
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Rotate
around x-axis counterclockwise with degree of θ. |
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Rotate
around y-axis counterclockwise with degree of θ. |
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Rotate
around z-axis counterclockwise with degree of θ. |
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If we rotate a vector around x-axes with x degree, around y-axes
with y degree, and around z-axes with z degree, we can get a rotated
matrix M from multiply each rotation matrixes :
M
=
Considering three basic vectors in 3-dimensional space: v1 = (1, 0, 0) v2 = (0, 1, 0) v3 = (0, 0, 1) Multiplying each vector with rotation matrix M, we have : M v1 = ( cos(z)cos(y), sin(z)cos(y), -sin(y) ) M v2 = ( -sin(z)cos(x)-cos(z)sin(y)sin(x), cos(z)cos(x)-sin(z)sin(y)sin(x), -cos(y)sin(x) ) M v3 = ( -sin(z)sin(x)+cos(z)sin(y)cos(x), cos(z)sin(x)+sin(z)sin(y)cos(x), cos(y)cos(x) ) If we want to model a 3-dimensional figure in computer screen, we just need to project these rotated vectors onto x-z plane since y-direction towards us. Then new frame could be constructed from above vectors having their y-component to disappear. Whatever constructing them by calculating or by vector performing, a new frame thus formed. In my constructions, in order to simplify some complicated condition I restricted each rotated angles from zero to half of pi.
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Examples of 3-dimensional figure : |
