2.2: 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.



The Curve window displays the graph of a curve X(t), which you can define in the control panel. You can also define u, the parameter that determines how much the tagential image is perturbed. To get a sense of how the tangent of a curve changes, scroll through the t0 tapedeck and watch the tangential image traced out in the Perturbed Tangential Image window.



1. Investigate the behavior of the tangential image for the following family of curves: X(t) = (c + cos(t))*(cos(t), sin(t)). 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?

2. Consider the family of curves X(t) = (cos(t), cos(c)*sin(t) + sin(c)*sin(2*t)). 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 have singular points, and what does the tangential image look like when there is a singular point?

3. 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)?

4. 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?
In section 2.1, we showed that the geodesic curvature of a plane curve is related to the motion of its tangent vector on the unit circle. In particular, we derived the equation κg(s) = θ'(s) using an arclength parametrization. Integrating both sides of this equation gives us

∫κg(s)ds = ∫θ'(s)ds.

Since the tangential image of the curve lies on the unit circle (r = 1), the right-hand side is equal to the length of the tangential image. Meanwhile, we define the left-hand side to be the total geodesic curvature of the curve. Note that this is the total signed geodesic curvature, so, for example, it is completely possible for a curve whose tangential image covers the entire unit circle to have zero total geodesic curvature.


This demonstration draws a parameterized curve X(t) and colors it by the sign of its geodesic curvature. Blue is used if κg < 0; white is used if κg = 0; and red is used if κg > 0. The tangential image is also shown with the same color coding. For the default curve, which is a figure eight, the total geodesic curvature is zero since the red and blue arcs on the tangential image cancel out exactly. As you animate the variable t0 forward in time in the control panel notice that we can also tell the sign of the geodesic curvature by the motion of the tangent vector. The geodesic curvature is positive when the tangent vector moves counterclockwise around the unit circle and negative when the tangent vector moves clockwise.


This demonstration is known as the Caterpillar Demo. The caterpillar lives on a curve X(t) entered by the user. There are two tapedecks in the control panel: the parameter t0 controls the location of the caterpillar along the curve, while the parameter h controls the length of the caterpillar. Along the length of the caterpillar there are orange spikes which are all normal to the curve. These normal vectors sweep out an angle Δθ on the unit circle. In another window, we see a graph of the ratio of the angle swept out by the normal image of the caterpillar to the length of the caterpillar. That is, we graph (t, Δθ(t)/Δs(t)), where Δs(t) is the length of the caterpillar at t. Since θ'(s) is the same for both the tangent vector T(t) and the normal vector U(t), the graph of this ratio should resemble (t, |κg(t)|) as the length of the caterpillar becomes small (i.e. as h → 0).



1. What does the curve do at a point where the circular image reverses direction?

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