05 parametric surfaces - handout

Parametric Surfaces and Surfaces of Revolution

Math 55 - Elementray Analysis III

Institute of MathematicsUniversity of the Philippines

Diliman

Math 55 Parametric Surfaces 1/ 15

Recall

A curve in R3 is given by a vector function

R(t) = f(t)+ g(t)+ h(t)k

or a set of parametric equations

x = f(t), y = g(t), z = h(t).


Parametric Surfaces

A surface in R3 can be described by a vector function of twoparameters R(u, v). Suppose that

R(u, v) = x(u, v)+ y(u, v)+ z(u, v)k

is a vector function defined on a parameter domain D (in theuv-plane). Any particular choice of (u, v) D gives a point(x, y, z) such that

x = x(u, v) y = y(u, v) z = z(u, v).

The set of all such points as (u, v) varies over D is called aparametric surface.

The parametric surface istraced out by the tip of themoving vector R(u, v) as (u, v)varies over D.


Parametric Surfaces

Example

Identify the surface given by R(u, v) = u, cos v, sin v.

Solution. The corresponding set of parametric equations are

x = u, y = cos v, z = sin v.

Note that for any point (x, y, z) on the surface,

y2 + z2 = cos2 v + sin2 v = 1

This means that for constant x, the cross sections parallel tothe x-axis are circles of radius 1.

Hence, the surface is a rightcircular cylinder


Grid Curves

Consider a surface S given by a vector function R(u, v).

If we hold u = u0, constant, thenR(u0, v) becomes a vectorfunction of a single parameter vwhich traces a curve C1 on S.

If we hold v = v0, constant, thenR(u, v0) becomes a vectorfunction of a single parameter uwhich traces a curve C2 on S.

We call these curves the grid curves of S.


Grid Curves

The grid curves of R(u, v) = u, cos v sin v:


Parametric Surfaces

Example

Describe the parametric surface given by the parametricequations x = u cos v, y = u sin v, z = 4 u2. Identify gridcurves with constant u and grid curve with v = 0.

Solution. To eliminate the parameters u and v note that

x2 + y2 = u2 cos2 v + u2 sin2 v = u2,

but z = 4 u2 u2 = 4 z. Hence,x2 + y2 = u2 = 4 z

z = 4 x2 y2, a paraboloid.If u = u0, a constant, then z = 4 u20 is constant andx2 + y2 = u20. Thus, we have circles parallel to thexy-plane.

If v = 0, the parametric equation becomesx = u, y = 0, z = 4 u2

which gives the parabola z = 4 x2 on the xz-plane.Math 55 Parametric Surfaces 7/ 15

Surfaces of Revolution

Let f(x) 0 for a x b and S be the surface obtained whenthe curve y = f(x) is revolved about the x-axis. Let be theangle of rotation as shown below:

If (x, y, z) is a point on S, then

x = x, y = f(x) cos , z = f(x) sin

which is a parametric surface in the parameters x and .



If S is a surface obtained by revolving y = f(x) or z = f(x)about the x-axis, then S has parametric equations

x = x, y = f(x) cos , z = f(x) sin .

If S is a surface obtained by revolving x = f(y) or z = f(y)about the y-axis, then S has parametric equations

x = f(y) cos , y = y, z = f(y) sin .

If S is a surface obtained by revolving x = f(z) or y = f(z)about the z-axis, then S has parametric equations

x = f(z) cos , y = f(z) sin , z = z.



Example

Give a set of parametric equations for the surface obtained byrevolving y = ex about the

1 x-axis. 2 y-axis.

Solution.1 The generating curve isy = f(x) = ex.Hence, the surface ofrevolution is given by

x = x

y = ex cos

z = ex sin

2 Note: y = ex x = ln y, sothe generating curve isx = f(y) = ln y. Hence, thesurface of revolution is givenby

x = ln y cos

y = y

z = ln y sin


Tangent Plane to Parametric Surfaces

Consider a surface S given by R(u, v) = x(u, v), y(u, v), z(u, v)and a point P0 in S with position vector R(u0, v0).

If C1 is the grid curve obtained by setting u = u0, then thetangent vector to C1 at P0 is

Rv(u0, v0) = xv(u0, v0), yv(u0, v0), zv(u0, v0) .Similarly, if C2 is the grid curve obtained by setting v = v0,then the tangent vector to C2 at P0 is

Ru(u0, v0) = xu(u0, v0), yu(u0, v0), zu(u0, v0) .



If Ru Rv 6= 0, the surface S is called smooth.For a smooth surface S, the tangent plane is the planecontaining the vectors Ru and Rv. Clearly, Ru Rv is normalto the tangent plane.



Example

Find the equation of the tangent plane to the surface given by

x = sin v, y = v 1, z = euat the point P0(0,1, 1).

Solution. First, note that P0 is generated when u = v = 0. Next,we solve the partial derivatives:

Ru(u, v) = 0, 0, eu Ru(0, 0) = 0, 0, 1Rv(u, v) = cos v, 1, 0 Rv(0, 0) = 1, 1, 0

The normal vector to the tangent plane at (0,1, 1) is

0, 0, 1 1, 1, 0 = det k0 0 1

1 1 0

= 1, 1, 0Hence, the equation of the tangent plae at (0,1, 1) is

x+ y + 1 = 0Math 55 Parametric Surfaces 13/ 15

Exercises

1 Identify the surface given by the vector equationR(u, v) = u+ v, 3 v, 1 + 4u+ 5v.

2 Find a set of parametric equations for the surface obtained byrevolving the circle x2 + y2 = a2 about the y-axis.

3 Find the equation of the tangent plane to the given parametricsurface at the specified point.

a. x = u+ v, y = 3u2, z = u v; (2, 3, 0)b. R(u, v) = u2 + 2u sin v+ u cos vk; u = 1, v = 0

4 Show that the parametric equations

x = a sinu cos v, y = b sinu sin v, z = c cosu,

where a, b and c are constants, represent an ellipsoid.

5 Find a set of parametric equations for thetorus, obtained by rotating the circle on thexz-plane centered at (b, 0, 0) about the z-axis.(Hint: Take the parameters and ,as shown.)


References

1 Stewart, J., Calculus, Early Transcendentals, 6 ed., ThomsonBrooks/Cole, 2008

2 Leithold, L., The Calculus 7, Harper Collins College Div., 1995

3 Dawkins, P., Calculus 3, online notes available athttp://tutorial.math.lamar.edu/


05 parametric surfaces - handout

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