Section 17.5Parameterized Surfaces

Recall Parametric curves in 3-space• In rectangular coordinates (x,y,z) we have

• We used these to make a helixbtathztgytfx )(),(),(

Parameterized curves in spherical coordinates• In spherical coordinates we have (ρ, , θ)• As parametric equations we let C be the curve

• We used these to make a spherical helix

btathtgtf ),(),(),(

• Now we are going to move on to parameterized surfaces

• What would we get from the following set of parametric equations?

• Let’s take a look with Maple

1) cos , sin , 0x t y t z

3) cos , sin ,x t y t z z

2) cos , sin , 3x t y t z

• In the first two cases we have a parameterized curve in 3 space

• In the third case we have a parameterized surface as our parametric equations were in terms of two variables

• In rectangular coordinates, parameterized surfaces are given by

where a ≤ u ≤ b and c ≤ v ≤ d• Both u and v are object parameters• Can do in cylindrical and spherical also

( , ), ( , ), ( , )x f u v y g u v z h u v

• Let’s see how we can parameterize a sphere– In rectangular coordinates– In spherical coordinates

• How about a Torus– One way of thinking

about a Torus is

a circle rotated

around the z-axis– Let’s take a look

in maple

Generalized Torus• Say we want to create a Torus that is not

circular• Essentially we want to rotate some region in a

plane around the third axis

(5,1)

(12,6)

(8,14)

x

z

Generalized Torus• We need to make parametric equations for the

triangular region• This will be easier to do in cylindrical

coordinates

(5,1)

(12,6)

(8,14)

x

z

Parameterizing Planes• The plane through the point with position

vector and containing the two nonparallel vectors and is given by

• So if

• The parametric equations are

0r

1v

2v

0 1 2( , )r s t r sv tv

0 0 0 0 1 1 2 3

2 1 2 3

, , , , ,

and , ,

r x y z v a a a

v b b b

0 1 1 0 2 2

0 3 3

, ,x x sa tb y y sa tb

z z sa tb

Parameterizing Surfaces of Revolution• We can create surfaces that have an axis rotational

symmetry and circular cross sections to that axis• For example, how about a cone that has a base that is a

circle of radius 3 in the xy-plane and a height of 10.• The following structure can be used to revolve a curve

around the z axis

• This can be modified to revolve around other axes as well

( ) cos( ), ( )sin( ),f z f z z

Parameter Curves• Parameter curves are obtained by setting one

of the parameters to a constant and letting the other vary

• Take the following parametric equations

– What do they give us?– What would we get if z is held constant?– What would we get if t is held constant?

• These parameter curves are cross sections of our parameterized surface

cos ,sin ,t t z