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Name: Bernard P.
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: 4/29/2004


Question:
I have a circular disk 10" in diameter laying in a flat plane.

I am viewing it from above, what I see is a circular disc.

I draw two lines on the disk, 90 degrees apart, from the center of the disk to the outside edge. "looks like the cross hairs of a rifle scope"

I identify the lines as an "X" and "Y" axis.

Using a Standard Unit Circle to describe position, I rotate the disk ten (10) degrees on the "X" axis, from the 90 degree location on the disk, up, toward me. I no longer recognize the object as a circular disk.

Using a Standard Unit Circle to describe position, I rotate the disk ten (10) degrees on the "Y" axis, from the "0" degree location on the disk, up, toward me. I have further distorted the original shape.

My question is; how would this configuration be described.

Example:
Using the edge of the disk, furthest from the center of the disk and myself, as a tool, I press and drag the disc across a slab of clay to produce a concave configuration in the clay. How is this depression described? Is it a concave elliptical arc segment developed by the compound angle of the disk?



Replies:
Let the diameter of the circle be 'a'. You are on the z-axis looking at the circle in the x/y plane where 'x' is the horizontal axis and 'y' is the vertical axis. As you "tip" the circle about the 'x' axis by an angle (theta) the major axis of the ellipse along the 'x' axis does not change. It remains 'a'. However, the projection of the circle onto your line of sight, that is back onto the 'y' axis is: a*cos(theta) = 'b' the minor axis of the projected ellipse. You can convince yourself that this is correct by the limiting cases. If 'theta' = 0 [no tipping], cos(0) = 1 and you view the circle with diameter 'a' in both the 'x' and 'y' directions. But if 'theta' = 90 degrees [ pi/2 radians ], you will be viewing the tipped circle "edge-on" and the minor axis 'b' = 0.

I do not have a clear enough grasp of your example, so as vengeful mathematicians do, to the agonized cries of students, the proof is left to the "student". AAHRRG!!!!

Vince Calder


Bernard,

What you see is an ellipse. It would be effectively the same as scaling down one axis. The axis parallel to the axis of rotation remains constant. The axis perpendicular to the rotation will change. If the entire object begins in the horizontal plane, all points appear to move toward the axis of rotation. The distance scales down by cosine of the rotation angle. If the object is 3-dimensional, not beginning in the horizontal plane, the initial height will matter. Imagine looking at the disk from the side, observing the rotation around the axis. This will give you the information you need to take accommodate initial heights.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College



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