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The Tangential Image

If we consider the unit tangent vector T(t) of a curve as a vector emanating from the origin, then as t varies this vector traces out a second curve, lying on the unit circle. We call this curve the tangential image or the tangential mapping. Investigating the tangential mapping of a closed smooth curve gives us a perspective on how the tangent behaves which is different from the one we get from seeing the tangent vector move along the original curve.

Because every point on the tangential image of a curve lies on the unit circle, it is difficult to see by just looking at it how it is traced out and whether it ever overlaps itself. For this reason, we will view the tangential image by gradually perturbing it, moving it from the unit circle. We can accomplish this by stretching the tangential image vector from 1 to (1+u) , where u is a small number, so that the curve we look at is changed from T(t) to (1+ut)T(t) .

The normal image of the curve X(t) is traced out by the unit vector, U(t). The same perturbation method can be employed in looking at the normal image.

Demonstration 7: Perturbed Tangential Image of Plane Curves
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The value of u in (1+ut)T(t) , that is, the value of the parameter that determines how much the tangential image is perturbed, is determined in the control panel. The best way to get a sense of how the tangent of a curve changes is to watch the tangential image traced out in real time. To do this, run the tapedeck in the control panel, which takes t from the left-hand to the right-hand endpoint of the domain. The tangent at the value of t determined by the tapedeck is displayed on the curve in the Curve with Tangent window.

Lab 1 shows that the singed curvature of a plane curves is related to the length of the tangential image. It is wise to keep in mind that when we perturb the tangential image in order to see it better, we also change its length. Watch the tangential images traced out on the unit circle, with u=0 , in order to get a sense of their lengths.

This demo shows the tangential image of a plane curve gradually perturbed from the unit circle.

    Investigate the behavior of the tangential image for the following family of curves (the default equation in the demo):

      Xc(t )=((c+cos( t))cos(t ),(c+cos(t))sin(t))
    with domain -p≤t≤p . For which c will the tangential image be one-to-one, so that each point of the circle is the image of exactly one point of the curve? For which values of c are there exactly two points on the curve that are sent by the tangential map to a given point on the circle? What happens for other values of c ?

    Consider the family of curves

      Xc(t )=(cos(t) ,cos(c)sin(t) +sin(c)sin(2t))
    with domain -p≤t≤p , obtained by projecting the space curve
      Y(t) =(cos(t),sin( t),sin(2t))
    into planes containing the x-axis. Observe that for certain values of c , there are points Q on the unit circle that are not the tangential image of any point on the curve. For which values of c does the curve Xc have singular points, and what does the tangential image look like when there is a singular point?

    What conjectures can be made about the number of times a point on the circle is covered positively and the number of times it is covered negatively? What happens when the point Q passes the image of a point of inflection of the original curve (or the point on the opposite side of the circle)?

    How many times will a given vector Q be equal to the normal of a point of the curve? How many times will Q be equal to the unit normal vector at a point where the geodesic curvature is positive, and how many times where it is negative? How will this change as we move Q around the circle? What happens when Q passes the image of an inflection point of the original curve?

The main thing here is the change in angle between the TU frame and the x-axis with respect to s. this section includes the development of "circular images."

Demonstration 8: The Famous Caterpillar Demo
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The caterpillar lives on a curve entered by the user. The user can change its length and location. Also displayed is the ratio of the angle swept out by the normal image of the caterpillar to the length of the caterpillar. This is to be compared with the curvature of the function as the length of the caterpillar gets small.

    What does the curve to at a point where the circular image reverses direction?

    What happens to the curvature function for positions of the caterpillar the where spread of the bristles is greatest?


Next: Distance Functions to Plane Curves