Coordinate system transformations
Author
Hmm, I''m having trouble transforming points from the regular coordinate system to an arbitary one. It''s the rotations that are the problem, I''ll try and explain by an 2D example.
Regular coordinate system:
Vectors:
x(1 0)
y(0 1)
Diagram:
y
|
|___ x
New coordinate system (90 degree rotation):
x(0 -1)
y(1 0)
___ y
|
|
x
Let''s say a point in the original system is (2, 1), the point transformed to the new system should be at (1, -2)
Looking at my maths books I figure you would just do the matrix multiplication:
(0 -1)(2)
(1 0)(1) = (-2, 1) ...obviously the wrong answer
Where am I going wrong??
The point at (1, 0) becomes ( 0, 1)
The point at (0, 1) becomes ( -1, 0 )
so (2, 1) = 2 (1, 0) + (0, 1) becomes 2(0, 1) + (-1, 0) = (-1, 2) not (1, -2)
Besides, this is perfectly equivalent to the matrix multiplication...
ToohrVyk
The point at (0, 1) becomes ( -1, 0 )
so (2, 1) = 2 (1, 0) + (0, 1) becomes 2(0, 1) + (-1, 0) = (-1, 2) not (1, -2)
Besides, this is perfectly equivalent to the matrix multiplication...
ToohrVyk
Making it more clear :
ToohrVyk
Always be wary of typos. Even one is enough to turn your pubic hair public.
[edited by - ToohrVyk on July 3, 2003 8:19:18 AM]
[edited by - ToohrVyk on July 3, 2003 8:19:48 AM]
|1| |0| |0| |-1||0| becomes |1] and |1| becomes | 0|To get the matrix, one simply sticks toegether the images of the base :|0 -1||1 0|To multiply a vector by this matrix is to decompose the vector into its coordinates in the base, then calculate it's image along each axis, and finally sum it up toegether... so|0 -1| |2| |0| |-1| |-1||1 0| * |1| = 2 * |1| + 1 * | 0| = | 2| ToohrVyk
Always be wary of typos. Even one is enough to turn your pubic hair public.
[edited by - ToohrVyk on July 3, 2003 8:19:18 AM]
[edited by - ToohrVyk on July 3, 2003 8:19:48 AM]
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