Parameterized Surfaces and Surface Area

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) −→ 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
on
the 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
Jf(s, t) =


f1s f1t
f2s f2t
f3s f3t

(1)
Date: June 24, 2015.

Parameterized Surfaces and Surface Area Poramate (Tom) Pranayanuntana
Figure
2. The surface parameterized
by r = [s, 1 − t2
, t3
− t]T
, where
−1 ≤ s ≤ 1 and −1.2 ≤ t ≤ 1.2,
is not simple.
Figure 3. Astroidal sphere
parameterized by r =
[sin3
s cos3
t, sin3
s sin3
t, cos3
s]T
,
where 0 ≤ s ≤ π and
0 ≤ t < 2π, is not smooth.
has continuous entries and the normal vector nS = rs × rt = fs × ft never zero. A surface S
is simple if it has a parameterization that is given by a one-to-one function. A surface S is
said to be smooth if it has a one-to-one smooth parameterization.
The requirement that Jf(s, t) be continuous is to ensure a continuously varying normal nS to
the surface, and the nonvanishing cross product is to assure that the normal never becomes
the zero vector. These together hold the intuitive idea that a smooth surface is one without
cusps or creases. The deﬁnition of a simple surface is designed to take out self-intersections
such as that shown in Figure 2.
In Figure 3, for instance, is the surface parameterized by
r =


sin3
s cos3
t
sin3
s sin3
t
cos3
s

(2)
where 0 ≤ s ≤ π and 0 ≤ t < 2π. It is not smooth because, for example, nS = rs ×rt vanishes
when s = π/2 and t = 0; this is the sharp point on the surface at the point (1, 0, 0). A smooth
surface without self-intersections is sometimes called a manifold. Roughly speaking, a
manifold 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 at
every point of the surface in a continuous way. The unit normal vector in the direction of the
orientation is denoted by ˆnS. For a closed surface (that is, the boundary of a solid region),
we usually choose the outward orientation.
June 24, 2015 Page 2 of 4

The Area Vector Later on, when we talk about flux of a vector field, the flux through a
flat surface depends both on the area of the surface and its orientation. Thus, it is useful to
represent 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 the
area of the surface, and whose direction is the direction of the orientation vector ˆnS; that is
AS = AS ˆnS.
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 partition
T 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 small
rectangular 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 of
the 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

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
∂s
on 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
∂t
on 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 the
direction of the unit normal orientation vector ˆnS. If the vector rs ×rt points in the opposite
direction to ˆnS, we reverse the order of the cross-product. Replacing ∆AS, ∆s, and ∆t by
dAS, ds, and dt, we write
dAS =
∂r
∂s
ds ×
∂r
∂t
dt =
∂r
∂s
×
∂r
∂t
dsdt.
Area of a Parameterized Surfaces The area ∆AS of a small parameter rectangle, which
is 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 the
following 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
dAS =
T
rs × rt dsdt
dAT
. (3)

Parameterized Surfaces and Surface Area

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