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Normalized Parallel Curves

In the curvature section, we introduced the concept of parallel curves at a given distance d. At each stage we may scale the image so that it fits on the screen by multiplying the parallel curve Xd(t) =X(t) +dU(t) by v = 11+d. Thus we obtain

    Xv(t )=(1-v) X(t)+v U(t)
as v changes from 0 to 1 . This means that the scaled parallel curve is just a linear interpolation between the curve and its normal image. Stated somewhat picturesquely, "The normal circular image is the parallel curve at infinity." This becomes considerably more interesting in the case of hypersurfaces.

Normalized Parallel Curve Demo

The Parallel at Distance v/(1-v) window shows the curve along with the parallel at its actual distance. When the distance gets large, you will have to rescale the display (by pressing the "s" key with the mouse cursor in the window) in order to see the parallel curve; as you do this, you can see that the original curve diminishes to a point in relative size. The Normalized Parallel Curve window shows the curve, along with the parallel curve scaled according to the function Xv(t) . The size of the original curve is not being changed in any way. You can set v between 0 and 1 using the slider in the Parallels window.

The purpose of this demo is to help visualize the interpolation between the parallel curves and the normal image of a curve.


Next: Reconstruction from Curvature