Discuss: Code copyright and legal issues
you perform a translation transformation on the object.
this would also set the direction of the rotation axis
then you allow the axis to not go through the origin. So you are moving the axis, or moving the coordinate system, or moving the object.
In the end you've got an arbitrary translation and an arbitrary rotation.
this would also set the direction of the rotation axis
then you allow the axis to not go through the origin. So you are moving the axis, or moving the coordinate system, or moving the object.
In the end you've got an arbitrary translation and an arbitrary rotation.
| Let the starting object be o1 and the result of the motion be object o2. Translate o2 to an object o2' so that o1 and o2' have a common point. This means that to transform o1 to o2' you need a fixed point rotation (usual rotation). That is given by a 3x3 orthonormal matrix. Every real 3x3 matrix has a real eigenvector (why?), which gives you the direction of the axis of rotation. If you want to transform o1 to o2', the axis of rotation would pass through the fixed point selected by you. If you translate this axis around, so it doesn't pass through the point selected by you, this makes o1 transform to objects that are translates of o2 (and o2'). You need a small computation to find exactly where to put the axis (with already determined direction) so that you transform o1 to an object o1' so that the line connecting the fixed point on o1' and o2' is parallel to the axis of rotation. Now with a single rotation you've placed the two objects directly beneath one another relative to the direction of the rotation. All you are left to do is translate the objects along that line parallel to the axis of rotation. |
Going back to my previous example, let's say I move a step forward. Now I have a vertical axis that passes through my head in common with my target placement. Now, this is a problem. No matter where I move this axis of rotation, applying the rotation to my initial placement will leave me in a horizontal plane common with my target placement, moving neither up nor down. But a translation that's parallel to the rotation axis will require me to be outside that plane.
The missing step here is proving that for any rotation that transforms A into B there's another non-parallel rotation that also transforms A into B, which I don't believe is true. Maybe if you're embedded in R^4.
Programmers. When you want to know something, they teach you something else, then an all out maths war takes place.
Programmers. They nitpick at what you say, substituting their reality for yours.
Programmers. Yer all a bunch a nutters!
Don't mind me, I'm not talking. You are talking to yourselves, in your own mind.
Programmers. They nitpick at what you say, substituting their reality for yours.
Programmers. Yer all a bunch a nutters!
Don't mind me, I'm not talking. You are talking to yourselves, in your own mind.
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