Approximate Rectangle's Rotational Axis Given Corner Vectors?

IllusionSector

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Hi, freeMATHhelp. First time poster here.
I'm trying to approximate the rotation pivot coordinates of a rectangle based on corner vertex movement in 2d space. In the figure below are all the color coded elements, where the actual rotational axis (the cross shown in green) is known. However, I'd like to be able to deduce it as if it were unknown based on the change of vertex positions between two instances in time.

Rotational Axis of a Rectangle from Corner Vectors.jpg
Is that even possible?
Hope this makes sense.

Thank you in advance.
 
Hi, freeMATHhelp. First time poster here.
I'm trying to approximate the rotation pivot coordinates of a rectangle based on corner vertex movement in 2d space. In the figure below are all the color coded elements, where the actual rotational axis (the cross shown in green) is known. However, I'd like to be able to deduce it as if it were unknown based on the change of vertex positions between two instances in time.

Is that even possible?
Hope this makes sense.

Thank you in advance.

Geometrically, this is easy. I'll let you translate this into vector calculations.

For any given point, the old and new locations must lie on a circle whose center is the center (axis) of rotation. So the perpendicular bisector of the segment you have drawn as a vector will pass through the center. Do this for two points, and the intersection of the two lines will give you the center. If you do it for all four points, in principle the lines should all meet at one point; in practice, rounding or other errors may cause the intersection points to differ slightly; averaging the points of intersection might give you a more accurate result, or you might drop any for which the lines being intersected are to close to the same slope.

FMH430223.jpg
 
Thank you!

Thank you very much for that elegant solution, Dr.Peterson.
My next step with be to translate this to code.
In case you happen to be curious as to the purpose of this, it has to to do with optimizing motion blur for animations.

Thank you again.
 
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