Previous: Osculating Circles
The Tangential ImageIf 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 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): Consider the family of curves
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
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? |