Lecture 3 23

(x, y) =

(t2, t3)

(x, y) =

(t3, t3)

Figure 1.17. Two curves with a vanishing tangent vector at

t = 0.

where t lies in the interval [a, b], whose endpoints a and b may be ±∞.

In this representation, the circle discussed above would be written as

(x, y, z) = (cos t, sin t, 0)

with 0 ≤ t ≤ 2π. If we replace the equation z = 0 with z = t, we

obtain not a circle, but a helix; it takes a little more imagination to

picture this as the intersection of two surfaces. We could also multiply

the expressions for x and y by t to describe a spiral on the cone, whose

implicit representation is again not immediate.

Exercise 1.12. Find two equations whose common solution set is

the helix.

If we expect our curve to be smooth, we must impose certain

conditions on the coordinate functions fi. The first condition is that

each fi be continuously differentiable; this will guarantee the existence

of a continuously varying tangent vector at every point along the

curve. However, if we do not impose the further requirement that this

tangent vector be non-vanishing, that is, that (f1)2 +(f2)2 +(f3)2 = 0

holds everywhere on the curve, then the curve may still fail to be

smooth.

As a simple but important example of what may happen when

this condition is violated, consider the curve (x, y) =

(t2, t3).

The

tangent vector (2t,

3t2)

vanishes at t = 0, which appears as a cusp at

the origin in Figure 1.17. So in this case, even though f1 and f2 are

perfectly smooth functions, the curve itself is not smooth.