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While at UNC I had access to more powerful equipment. We did not have a Cray supercomputer with a couple of powerful processors, but we did have a massively parallel machine appropriately named the MASPAR with 8196 processors, individually not very powerful, but a natural architecture for 2D image processing. Still, there was a strong attraction to compute numerically on a personal computer, to see the results graphically and immediately, if not sooner. At the time I had upgraded my Intel 8086 machine to an Intel 80486 machine with a floating point coprocessor.
4t / v'1+ 4t 2 • The acceleration is a = (0, 2). The unit-length tangent and normal vectors are T = (1,2t) N , = v'1 + 4t 2 Finally, the curvature is K (-2t, 1) v'1 + 4t 2 = 2/(1 + 4t 2 )3/2. Planar Motion in Polar Coordinates The particle motion may also be represented using polar coordinates. The choice of Cartesian form versus polar form depends on your application. 10) Since R= r/lrl is unit length, we may write it as R= (cos e, sin e), where 8 depends on t. A unit-length vector perpendicular to R is P = (- sin cos 8) and is obtained by a 7T /2 counterclockwise rotation of R in the plane.
The curvature function that must be associated with the second choice is the negative of the first, -K (s), so that the product of curvature and normal still produces dTI ds. If the curve were planar, we could resort to the two-dimensional construction and select a normal that is a counterclockwise rotation from the tangent within that plane. This reasoning does not apply to a nonplanar curve. 2 Rigid Body Kinematics 21 of normal should be made in an attempt to maintain a continuous function N(s).