5.2: Velocity and Arc Length of Parameter Curves

The definition of the partial derivative of a vector function is very similar to the definition of a partial derivative of a function of two variables. The u-partial derivative of the position vector is defined by

    Xu(u 0,v0)=lim[h->0][X(u0+h,v0) -X(u0,v 0)]/h
if this limit exists. The v-partial derivative Xv(u 0,v0) is defined analogously. If these two vectors exist and are linearly independent, they define a plane called the tangent plane of the surface at X(u0,v 0). We avoid pathological situations by assuming that our functions have an appropriate number of derivatives.

If the two partial derivative vectors are linearly independent at each point of a domain, then the parametrization is said to be regular; if the two partial derivative vectors are linearly dependent at a point of a domain, then the parametrization is said to be singular at the point.

In certain cases a singularity occurs as a result of the parametrization, while in others, it reflects a geometric property of the surface itself. We have already seen this phenomenon for curves: the velocity vector of the plane curve X(t) =(t3,t3) is X'( t)=(3t2,3t2) , so the curve has a singularity when t=0 . However we may reparametrize the curve as X(s) =(s,s) , and with this parametrization, the velocity vector is non-zero at each point.

Consider the parametrization of the plane given by

    X(u,v) =(vcos(u),vsin( u),0)
defined over the domain with u between 0 and 2p , and v non-negative. Then the partial derivatives are
    Xu(u ,v)=(-vsin(u) ,vcos(u),0)
and
    Xv(u ,v)=(cos(u) ,sin(u),0)
Since
    Xu(u ,0)=(0,0,0)
for any u , the two partial derivative vectors are automatically dependent at (u,0) , so the origin is a singularity for this parametrization. On the other hand, it is easy to give a parametrization of the plane which has no singularities, for example
    X(u,v) =(u,v,0)
On the other hand,
    X(u,v) =(vcos(u),vsin( u),v)
over the same domain describes a cone with a singularity that can not be removed by reparametrization.

Demonstration 3: Partial Derivatives Demo
Java not enabled.

A point widget marked by an x in light blue enables you to choose a point in the domain. As we move the point around in the domain, we see the u- and v- partial derivative vectors moving along the surface depending on whethter or not u-curve or v-curve are on. If both of these checkboxes are turned on, we display the tangent plane by drawing the parallelogram determined by the partial derivative vectors at the point. To facilitate the viewing of the tangent plane the surface will not solidify, only the plane will solidify if you press the space bar.

In this demonstration, when we choose a point in the domain of a surface, in addition to the two parameter curves, the partial derivative vectors are also displayed.

    With this demo, quickly look at some smooth surfaces and observe how the tangent plane behaves. Then, look a surfaces which you know to have singularities. For example, try the Cone , which is entered as [u,v] --> [v cos(u),v sin(u),v] . What happens at the poles of a sphere? Why? Analyze also the tangential surface of a circular helix. This latter has what is called a cuspidal edge . One parametrization is [u,v] --> [cos(u)-v*sin(u),sin(u)+v*cos(u),0.5*u+v].

Arc Length of Parameter Curves

The velocity vector of a u-parameter curve is the u-partial derivative vector at that point, so we can integrate the length of that vector to calculate the length of the u-parameter curve with v=vo. We obtain

    s(b)-s(a) =[Integral(a,b)][sqrt]X u(u,v0) ·Xu( u,v0)du

The expression

    Xu(u ,v)·Xu(u,v)
occurs so frequently in the differential geometry of surfaces that it has a symbol g11(u,v) . The length of the u-parameter curve is then given by
    s(b)-s(a) =[integral(a,b)][sqrt]g11(u,v0) du

Similarly we define the symbol

    g22(u,v)= Xv(u,v) ·Xv( u,v)
and this can be used to compute the length of v-parameter curves. For example, if u=u0 , then the length of this v- parameter curve is
    s(d)-s(c) =[integral(c,d)][sqrt]g22(u0,v) dv

    Find some examples of surfaces for which at certain points the length functions of the parameter curves have an inflection point. When will this happen?

Next: Velocity and Arc Length of General Curves on Surfaces