Subject:
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Re: 3D geometry question
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Newsgroups:
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lugnet.cad
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Date:
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Wed, 13 Oct 1999 12:54:41 GMT
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Viewed:
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834 times
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Fredrik Glöckner:
> Say I have a vector v with length normalized to 1. For
> simplicity, picture this vector as starting in the origin.
Vectors don't start anywhere. A vector is just a direction
and a length.
> Now, I need a rotation matrix M with the following
> properties:
>
> o If I rotate a part with one edge going from (0,0,0) to
> (0,-1,0) (i.e., pointing upwards), this edge should
> coincide with the vector v after the transformation
Lets call the vector along the edge e.
> o M should be a proper transformation, i.e., no scaling or
> shearing.
There are infinitely many solutions to the problem. When you
have one solution, any rotation around v is also a solution.
We have to solve
v = M e
with respect to M, with the additional constraint
det(M) = 1
Hmmm? 9 variables and 4 equations. - That can't be sufficient
constraints to ensure M is a rotation matrix.
Lets try something else...
Any direction in three-dimensional space can be constructed
by one rotation around the x axis followed one rotation
around the y axis:
/ 1 0 0 \ / cos(b) 0 sin(b) \
M = | 0 cos(a) sin(a) | | 0 1 0 |
\ 0 -sin(a) cos(a) / \ -sin(b) 0 cos(b) /
/ cos(b) 0 sin(b) \
= | -sin(a)sin(b) cos(a) sin(a)cos(b) |
\ -cos(a)sin(b) -sin(a) cos(a)cos(b) /
Now you have a rotation matrix parametrised by
the two rotation angles (a and b). The equation
v = M * e
can then be rewritten as three equations:
v1 = cos(b) e1 + sin(b) e3
v2 = -sin(a) sin(b) e1 + cos(a) e2 + sin(a) cos(b) e3
v3 = -cos(a) sin(b) e1 - sin(a) e2 + cos(a) cos(b) e3
(which it should be possible to solve)
Play well,
Jacob
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Message has 1 Reply:
 | | Re: 3D geometry question
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| (...) Sure. But to picture the vector as starting in the origin makes it easier to understand what I want in the paragraph below: (...) In fact, for my application, any of these solutions would suffice. (...) I think we also need some constraints on (...) (25 years ago, 13-Oct-99, to lugnet.cad)
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Message is in Reply To:
| | 3D geometry question
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| Hi, I have this problem I hope someone may solve. Say I have a vector v with length normalized to 1. For simplicity, picture this vector as starting in the origin. Now, I need a rotation matrix M with the following properties: o If I rotate a part (...) (25 years ago, 13-Oct-99, to lugnet.cad)
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