PARAMETERIZED SURFACES AND SURFACE AREA

PORAMATE (TOM) PRANAYANUNTANA

Definition. Given a function ~f : T ⊂st-plane︷︸︸︷R2 −→ S ⊂

xyz-space︷︸︸︷R3 , we define the surface param-

eterized by ~r = ~f to be the set of points S =

~r(s, t) ∈ R3︸︷︷︸xyz-space

∣∣∣∣(s, t) ∈ T ⊂ R2︸︷︷︸st-plane

.

~f =

f1

f2

f3

−−−−−−−−−−−→

(s, t) 7−→ ~r = ~f(s, t)

Figure 1. The parameterization sends each point (s, t) in the parameter re-

gion, T , to a point (x, y, z) or position vector ~r = [f1(s, t), f2(s, t), f3(s, t)]T onthe surface, S.

That is, S is the image of T under ~f . The equation ~r = ~f(s, t) is a parameterization of S.

We say that the parameterization by ~f =

f1

f2

f3

is smooth if the Jacobian matrix

J ~f(s, t) =

f1s f1t

f2s f2t

f3s f3t

(1)

Date: June 24, 2015.

Parameterized Surfaces and Surface Area Poramate (Tom) Pranayanuntana

Figure2. The surface parameterizedby ~r = [s, 1− t2, t3 − t]T , where−1 ≤ s ≤ 1 and −1.2 ≤ t ≤ 1.2,is not simple.

Figure 3. Astroidal sphereparameterized by ~r =[sin3 s cos3 t, sin3 s sin3 t, cos3 s]T ,where 0 ≤ s ≤ π and0 ≤ t < 2π, is not smooth.

has continuous entries and the normal vector ~nS = ~rs × ~rt = ~fs × ~ft never zero. A surface Sis simple if it has a parameterization that is given by a one-to-one function. A surface S issaid to be smooth if it has a one-to-one smooth parameterization.

The requirement that J ~f(s, t) be continuous is to ensure a continuously varying normal ~nS tothe surface, and the nonvanishing cross product is to assure that the normal never becomesthe zero vector. These together hold the intuitive idea that a smooth surface is one withoutcusps or creases. The definition of a simple surface is designed to take out self-intersectionssuch as that shown in Figure 2.

In Figure 3, for instance, is the surface parameterized by

~r =

sin3 s cos3 tsin3 s sin3 t

cos3 s

(2)

where 0 ≤ s ≤ π and 0 ≤ t < 2π. It is not smooth because, for example, ~nS = ~rs×~rt vanisheswhen s = π/2 and t = 0; this is the sharp point on the surface at the point (1, 0, 0). A smoothsurface without self-intersections is sometimes called a manifold. Roughly speaking, amanifold then should, in the vicinity of each point not on its boundary, resemble a plane.

Surface Area

Orientation of a Surface At each point on a smooth surface there are two unit normals,one in each direction. Choosing an orientation means picking one of these normals atevery point of the surface in a continuous way. The unit normal vector in the direction of theorientation is denoted by n̂S. For a closed surface (that is, the boundary of a solid region),we usually choose the outward orientation.

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Parameterized Surfaces and Surface Area Poramate (Tom) Pranayanuntana

The Area Vector Later on, when we talk about flux of a vector field, the flux through aflat surface depends both on the area of the surface and its orientation. Thus, it is useful torepresent its area by a vector called the area vector, denoted ~AS.

Definition. The area vector for a flat oriented surface S is a vector whose magnitude is thearea of the surface, and whose direction is the direction of the orientation vector n̂S; that is

~AS = ASn̂S.

To obtain a reasonable definition of the area of a non-flat surface S lying in R3 that is

parameterized by ~r = ~f(s, t), where (s, t) ∈ T ⊂ R2, we reason as follows. If we partitionT into many small rectangles, then S is partitioned into many pieces, each of which is the

image under ~f of one of these small rectangles. See Figure 4.

Figure 4. Parameter rectangle on the surface S corresponding to a smallrectangular region in the parameter region, T .

If ~f is differentiable on T , then on each of these small rectangles, ~f has a good linear

approximation, so the image of a small rectangle under ~f closely resembles the image ofthe same rectangle under the linear approximation ~L. We consider a parameter rectangle(a patch) on the surface S corresponding to a rectangular region with sides ∆s and ∆t in

the parameter region, T . If ∆s and ∆t are small, the area vector, ∆ ~AS, of the patch is

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Parameterized Surfaces and Surface Area Poramate (Tom) Pranayanuntana

approximately the area vector of the parallelogram defined by the vectors

~r(s+ ∆s, t)− ~r(s, t)︸ ︷︷ ︸secant vector displaced from one point

to another point on surface S : ~r = ~f

corresponding to moving from (s, t)

to (s + ∆s, t) on parameter region T

≈ ∂~r

∂s∆s︸ ︷︷ ︸

tangent vector∂~r

∂son tangent plane:

~r = ~L, multiplied by the run ∆s

, and

~r(s, t+ ∆t)− ~r(s, t)︸ ︷︷ ︸secant vector displaced from one point

to another point on surface S : ~r = ~f

corresponding to moving from (s, t)

to (s, t + ∆t) on parameter region T

≈ ∂~r

∂t∆t︸ ︷︷ ︸

tangent vector∂~r

∂ton tangent plane:

~r = ~L, multiplied by the run ∆t

.

Thus

∆ ~AS ≈∂~r

∂s∆s× ∂~r

∂t∆t =

(∂~r

∂s× ∂~r

∂t

)∆s∆t.

From the reasoning above, we assume that the vector ~rs × ~rt is never zero and points in thedirection of the unit normal orientation vector n̂S. If the vector ~rs×~rt points in the oppositedirection to n̂S, we reverse the order of the cross-product. Replacing ∆ ~AS, ∆s, and ∆t byd ~AS, ds, and dt, we write

d ~AS =∂~r

∂sds× ∂~r

∂tdt =

(∂~r

∂s× ∂~r

∂t

)dsdt.

Area of a Parameterized Surfaces The area ∆AS of a small parameter rectangle, whichis approximately flat, is the magnitude of its area vector ∆ ~AS. Therefore,

Area of S =∑

∆AS =∑∥∥∥∆ ~AS

∥∥∥ ≈∑ ‖~rs × ~rt‖∆s∆t.

Taking the limit as the area of the parameter rectangles tends to zero, we are led to thefollowing expression for the area of S.

The Area of a Parameterized Surface The area of a surface S which is

parameterized by ~r = ~r(s, t) = ~f(s, t), where (s, t) varies in a parameter region T ,is given by

AS =

∫S:~r(s,t),(s,t)∈T

dAS =

∫S:~r(s,t),(s,t)∈T

∥∥∥d ~AS

∥∥∥ =

∫T

‖~rs × ~rt‖ dsdt︸︷︷︸dAT

. (3)

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