Parametric Surfaces

Surfaces can also be defined parametrically. In particular, suppose each component of a vector-valued function is a function of two variables u and v:

r(u,v) = á f(u,v) ,g(u,v) ,h(u,v) ñ

(1)

Then the graph of r( u,v) over some region S in the uv-plane is a surface in R³, and r(u,v) is called a parameterization of that surface. Equivalently, x = f(u,v) , y = g( u,v) , and z = h( u,v) for (u,v) in S defines a surface, and if each point on the surface corresponds to only one point in S, then the variables u and v are the coordinates of the surface with respect to the mapping r(u,v).

Given a parametric surface (1), we often desire to transform it into a level surface representation of the form U(x,y,z) = k. To do so, we often use the trigonometric identities, such as the Pythagorean identities

cos2(t) + sin2(t) = 1 1 + tan2(t) =  sec2( t) 1 - 2sin2(t)  = cos(2t) cosh2(t) - sinh2(t) = 1 1 + cot2(t) =  csc2( t) 2cos2(t) - 1 = cos(2t)

Other identities that may occur include 2sin(t) cos(t) = sin(2t) and e^t e^-t = 1.       

EXAMPLE 2    Find a level surface representation of the surface parameterized by

r( u,v) = á cos( u) cosh(v) ,sin( u) cosh( v) ,sinh( v) ñ

Solution: Since x = cos( u) cosh( v) , y = sin( u) cosh( v) , and z = sinh( v), the identity cos²( u) +sin²( u) = 1 leads to

x2+y2 = cos2( u) cosh2( v) +sin2( u) cosh2( v) = cosh2( v) [ cos2( u) +sin2( u) ] = cosh2( v)

As a result, the identity cosh²( v) -sinh²(v) = 1 leads to

x2 + y2 - z2 = cosh2(v) - sinh2(v) = 1

Thus, r( u,v) = á cos( u) cosh( v) ,sin( u) cosh( v) ,sinh(v) ñ is a parameterization of the level surface

x2 + y2 - z2  =  1

which we recognize as a hyperboloid in one sheet.

Graphic of Directional Derivative

       

A sphere of radius R centered at the origin is often parameterized in terms of longitude q and latitude j, which results in the parameterization

r( q,j) = á Rcos( j) cos( q) ,Rcos( j) sin(q) ,Rsin( j) ñ

(2)

where q in [ 0,2p] and j is in [-p/2,p/2] .

EXAMPLE 3    Show that (2) is a parameterization of the sphere of radius R centered at the origin.       

Solution: Since x = Rcos( j) cos( q) , y = Rcos( j) sin( q) and z = Rsin( j) , the identity cos²( q) +sin²( q) = 1 leads to

x2+y2 = R2cos2( j) cos2( q) +R2cos2( j) sin2( q) = R2cos2( j) [ cos2( q) +sin2( q) ] = R2cos2( j)

Moreover, z² = R²sin²( j) implies that

x2+y2+z2 = R2cos2( j) +R2sin2(j) = R2

which is the equation of the sphere of radius R centered at the origin.

However, cartographers and mathematicians have long used parameterizations of the sphere other than (2). For example, in 1599, the mapmaker Gerard Mercator constructed a projection of the earth's surface in which a straight line on a map corresponds to a fixed compass bearing on the earth's surface. To do so, he imagined that the sphere was inside of a cylinder with radius R.

Q is the Mercator Projection of P onto the cylinder

Click and drag P to see more projections

This led to the Mercator parameterization of the sphere:

r( q,m) = á R sech( m) cos( q) ,R sech( m) sin( q) ,Rtanh( m) ñ

where the hyperbolic secant and tangent functions are defined

sech( m) =  1 cosh( m) ,       tanh( m) =  sinh( m) cosh( m)

We will examine the Mercator parameterization more closely in the exercises.

Important Note: Parameterizations of the sphere illustrate a couple of important details about parameterizations of surfaces. First, points on a level surface U(x,y,z) = k may correspond to more than one (u,v) pair under r(u,v)-- for example, points on a sphere with longitude q = 0 are the same as points with longitude q = 2p. Second, not all points on a level surface U(x,y,z) = k necessarily correspond to a (u,v) pair under r(u,v) -- for example, the north and south poles do not correspond to any points on the cylinder under the Mercator projection.

Important Notation: Given the note above, say that r(u,v) over some region S in the uv-plane is a coordinate patch on a level surface U(x,y,z) = k if r(S) is a region on the surface and there is only one (u,v) for each point in r(S) (i.e., r is 1-1 on S). In this case, a pair (u,v) is said to be the coordinates of a point r(u,v) on the surface.