Parametric Surfaces, Part 3

Parametric Representations of Surfaces

Part 3: The Fundamental Vector Product and Surface Area

In Part 2, we saw that changes in coordinates for surfaces give rise to a scale factor for area that can be computed as the Jacobian of the coordinate transformation. Here we extend this idea parametric representations of surfaces, which we now think of as coordinate transformations from the parameter plane into space. We will calculate the change-in-area factor in exactly the same way, as an area of a parallelogram, which will again turn out to be the magnitude of a cross product of tangent vectors. However, since the image of the parameter rectangle need not be in the xy- plane, the cross product is not limited to a k component only -- or, indeed, even to a constant direction.

As an example, consider the parameterization of the torus from Part 1:

x(u,v) = (a + b cos v) cos u,
y(u,v) = (a + b cos v) sin u,
z(u,v) = b sin v,

0 < u < 2, 0 < v < 2.

In the following figure we show a portion of the parameter plane and a portion of the torus. On the left we have highlighted a typical rectangle R in the parameter plane, and on the right the corresponding portion of the surface, labeled S. As in Part 2, the change-in-area factor will be the ratio of the area of S to the area of R.

The next figure shows a close-up of the typical parameter rectangle R and its image S on the parameterized surface (both yellow), along with the tangent vectors at one corner (blue and red), the parallelogram determined by those vectors (green), and the cross product of those tangent vectors (black).

This time (in contrast to Part 2) the tangent vectors are given by

t_u = x_u(u,v) i + y_u(u,v) j + z_u(u,v) k,
t_v = x_v(u,v) i + y_v(u,v) j + z_v(u,v) k.

As in Part 2, these vectors are scaled by the grid spacings du and dv, respectively, to get the tangent vectors to the surface that approximate the curved images of the parameter segments. We calculate the approximate area of S as

dA = |t_u x t_v| du dv.

Since du dv is the area of R, the length of the cross product is again the local change-in-area factor. To find the total surface area determined by a region of the parameter plane, we integrate dA over that region. The cross product of the tangent vectors t_u and t_v is called the fundamental vector product of the parameterization, and its length is again called the Jacobian, a shorter but less descriptive name than "local change-in-area factor."

For the parameterization of the torus given above, calculate the fundamental vector product.

Find the length of the fundamental vector product, i.e., the local change-in-area factor. Simplify as much as you can. Your answer should be independent of one of the two parameters -- explain geometrically why that should be the case.

Calculate the surface area of the torus with "big radius" a and "small radius" b. Explain why it is little surprise that the answer turns out to be the product of the circumferences of the two circles.

Find a formula for the surface area of the graph of a function z = f(x,y) for a < x < b, c < y < d. [Hint: Use u = x and v = y as the parameters.]

Use your formula in step 4 to find the area of the saddle surface f(x,y) = x² - y² for -1 < x < 1, -1 < y < 1.

There is nothing in the formula in step 4 that requires the parameters to be Cartesian variables. Apply your formula to find the surface area of a cone of height h and base radius a, using the polar coordinate formula az = hr. Relate your answer geometrically to the height, radius of the base, and slant height. Give a geometric argument to show that your answer is correct. [Hint: If you slice the cone along one side, it flattens out to a piece of a circle.]

In Part 1, step 5, you constructed a parameterization of the sphere of radius 2, using spherical coordinates. Modify that parameterization to describe a sphere of any radius a, and calculate both the fundamental vector product and the Jacobian. Then derive the well-known formula for surface area of a sphere.