Parametric Representation of Surfaces

Section 9.1 Parametric Representation of Surfaces

As noted in the introduction to the present chapter, a surface,

S,

is a deformed plane domain,

D \subset R^{2},

lying in space. Hence to describe

S

mathematically we need to say which point in

D

corresponds to what point on

S,

just as a map of the world (represented by

D

) shows the geographic locations in the real world (the surface

S

). This correspondence defines a (vector valued) one-to-one function

g

from

D

R^{3}

whose image is

S .

The function

g : D \to R^{3}

is called a parametrisation of

S .

Some surfaces cannot be represented by one single parametrisation, or it is not convenient to do so. (To represent the world we often use a collection of maps, an atlas, and not a single map of the whole world.) As with parametrisations of a curve, a parametrisation of a surface is never unique! We call a representation by means of a parametrisation a parametric representation of a surface.

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The above class of surfaces is too big. We only want to work with smooth (or at least piecewise smooth) surfaces. To define exactly what we mean by a smooth surface let us assume that

g : D \to R^{3}

is the parametrisation of (part of) a surface. Suppose that

a = (a_{1}, a_{2}) \in D,

so that

g (a)

is the corresponding point on

S .

Then the map

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x_{1} \to g (x_{1}, a_{2})

is the parametrisation of a curve on

S

through

g (a) .

Similarly, the map

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x_{2} \to g (a_{1}, x_{2})

is the parametrisation of another curve on

S

through

g (a) .

In order for

S

to be smooth we assume that these curves are smooth. More precisely this means that the above are regular parametrisations, so in particular we require that

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v_{i} := \frac{\partial}{\partial x_{i}} g (a) \neq 0

for

i = 1, 2 .

For

S

to be smooth we require more. If we are very unlucky, the image of

D

is only a curve, and not a surface. In that case the vectors

v_{1}

and

v_{2}

are parallel. To prevent this we require in addition to the above that

v_{1}

and

v_{2}

not be parallel. Both of the above requirements are satisfied if and only if the area of the parallelogram spanned by

v_{1}

and

v_{2}

is nonzero. In Theorem 1.16 we derived a formula to compute the area of a parallelogram. We have to form the matrix with columns

v_{1}

and

v_{2},

which in our present case is the Jacobian matrix

J_{g} (a)

g

a .

According to Theorem 1.16 the surface area spanned by

v_{1}

and

v_{2}

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\sqrt{det ((J_{g} (y))^{T} J_{g} (y))} .

Because of its significance in the theory of surfaces, or more generally in differential geometry, the the square root has a name.

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Definition 9.1. Jacobian of a parametrisation.

Suppose that

g : R^{k} \to R^{N}

is a differentiable map. Then the expression

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\sqrt{det ((J_{g} (y))^{T} J_{g} (y))}

is called the Jacobian of the parametrisation

g

y .

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Definition 9.2. Regular parametrisation.

Suppose that

g : D \to R^{3}

is the parametrisation of (part of) a surface. We say that

g

is a regular parametrisation of (part of)

S,

if the Jacobian of

g

is nowhere zero. Finally,

S

is called a smooth surface, if it can be fully described by one (or several) regular parametrisation. If several parametrisations are required then their images on

S

must not overlap by more than just a line.

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We also sometimes look at piecewise smooth surfaces which are finite unions of smooth surfaces.

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Example 9.3. Jacobian of a sphere.

Suppose that

S

is the sphere of radius

R

centred at the origin in

R^{3} .

We can describe every point on that sphere by indicating two angles, one from the vertical, the other from the horizontal. (Called latitude and longitude on a map of the world). According to the considerations in Subsection 6.4.1 a parametrisation is given by

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