1.4: Smooth Curves and Curves with Singular Points

By this time, you may have noticed that some plane curves, such as the ellipse and the exponential spiral, appear to be smooth, while others, such as X(t) =(t4,t5) have cusps. In doing the reparametrization demo, you may also have seen some examples where there is no visible kink in the curve, but the car moving along the curve comes to a stop at some point. This section explores ways of defining these phenomena. A curve is said to be differentiable over a domain if its first derivative X'(t) is continuous over that domain. A curve is called regular at a point X(t) if X'(t) is not equal to the zero vector, and X(t) is called a singular point on the curve if X'(t) = 0. Thus X(t) is a singular point if X'(t)·X'(t)=0. A cusp is a special type of singular point. It is a sharp reversal of direction for a curve. In order for a curve to have a cusp at a point X(t0), the limit of the unit tangent vector T(t) as t → t+0 must be the negative of the limit of the unit tangent vector as t → t-0. There are two types of cusps: ordinary and rhamphoid. An ordinary cusp occurs at a point t where the two branches of the curve on each side of t0 are on opposite sides of the single tangent line to the curve at t0. A rhamphoid cusp occurs when the two branches lie on the same side of the tangent line to the curve at t0.


This demo displays a curve that has either an ordinary cusp or a rhamphoid cusp at t = 0. The equation for the curve is X(t) = (3+t)(tm,tn), and the values of m and n determine the nature of the cusp. For example, the values m = 2, n = 3 yield an ordinary cusp, and the values m = 2, n = 4 yield a rhamphoid cusp.



1. What relationship do the values of m and n have to the quadrant in which the curve lies?

2. Try to establish a rule about m and n that determines whether or not this curve has an ordinary cusp, a rhamphoid cusp or neither. In particular, look at high values of m and n.
A corner is a singular point at which the limit of the unit tangent vector approached from one direction exists but is not equal either to the limit of the unit tangent vector approached from the other direction, or to its negative. The graph (t, |t|) of the absolute value function has a corner at t = 0, and the derivative is not defined at that point. However, it is also possible to have a corner on a curve where the dervitave does exist at every point; an example is the graph (t3, |t3|) at the point t = 0. At such a corner point of a differentiable curve, the derivative must be zero. In any case, the unit tangent and unit normal vectors are not defined at a corner.



The default function traced out in this Demo is (t,|t|), which has a corner at t = 0. The derivative is therefore not defined at 0.



1. Curves whose representation involves absolute values will often have corners. For example, (t,|t2-c|), where the domain goes from -1 to 1, will have corners at t = 0.5 and t = -0.5 for a particular value of c. What is that value?
A removable singularity is a singular point at which the limit of the unit tangent vector approached from one direction is the same vector as the limit approached from the other direction. In this case, we may define T at the singular point to be this common limit:

T(t) = lim (t → t0) (1/s'(t))X'(t)

This type of singularity is called removable because the curve can be parametrized so that it does not have a singularity. An example of a removable singularity is the origin on the curve X(t) = (t3,t3). A singularity that is not removable is called essential
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