Parametric Surface Patches
As with parametric curves, define a vector-valued function
Derivatives are tangent to surface
•not necessarily orthogonal
And the unit normal can be computed as
[]
(,) (,) (,) (,)uv xuv yuv zuv
=
p
(,)
(,)
u
v
xyz
uv uuu
xyz
uv vvv
∂
∂∂
⎡⎤
=⎢⎥
∂∂∂
⎣⎦
∂∂∂
⎡⎤
=⎢⎥
∂∂∂
⎣⎦
p
p
(,) uv
uv
uv ×
=×
pp
npp
Example: Parametric Paraboloid
Described by equation
Horizontal cross sections are ellipses
(,)uv u v u v
κκ
22
12
⎡⎤
=+
⎣⎦
p
x
y
z
κ
1
κ
2
Exercise: Parametric Cylinders
Cylinders have circular profiles and some fixed height
So how to parameterize one?
(,) ?
(,) ?
(,) ?
xuv
yuv
zuv
=
=
=
x
y
z
1
r2
Exercise: Parametric Cylinders
Cylinders have circular profiles and some fixed height
The natural choice for two parameters is
Can easily write down coordinate functions
(,) cos
(,) sin
(,)
xhr
yhr
zhh h
θ
θ
θ
θθπ
θ
=
=0≤≤2
=0≤≤1 x
y
z
the angle around the base
location along the cylinder
u
vh
θ
=
=
1
r2
