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Describing a Geometric Construct
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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Update: June 2012
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