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.
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