Ken Turkowski
Media Technologies: Graphics Software
Advanced Technology Group
Apple Computer, Inc.
(Draft Friday, May 18, 1990)
Abstract: We derive the expressions for first and second derivatives, normal, metric matrix and curvature matrix for spheres, cones, cylinders, and tori.
26 January 1990
The Differential Geometry of Parametric Primitives
Ken Turkowski 26 January 1990
Differential Properties of Parametric Surfaces
A parametric surface is a function:
where
is a point in affine 3-space, and
is a point in affine 2-space.
The Jacobian matrix is a matrix of partial derivatives that relate changes in u and v to changes in x, y, and z:
The Hessian is a tensor of second partial derivatives:
and establishes a metric of differential length:
so that the arc length of a curve segment, is given by:
The differential surface area enclosed by the differential parallelogram is approximately:
so that the area of a region of the surface corresponding to a region R in the u-v plane is:
The second fundamental matrix measures normal curvature, and is given by:
The normal curvature is defined to be positive a curve u on the surface turns toward the positive direction of the surface normal by:
The deviation (in the normal direction) from the tangent plane of the surface, given a differential displacement of is:
˙˙
x•n=uD˙ u˙t
˙
u
κn=
˙
uDu˙t
˙
uGu˙t D=n•H=
n• ∂ 2
x
∂u2 n•
∂2 x
∂u∂v
n• ∂ 2
x
∂v∂u n• ∂2
x
∂v2
S=
( )
GR
∫∫
12dudvδS≈
( )
G 1 2δuδvδu,δv
( )
s= ds dt
t0 t1
∫
dt= x˙t0 t1
∫
dt= x˙t0 t1
∫
dt=(
uG˙ u˙t)
t0t1
∫
1 2
dt
u=u( )t , t0<t<t1 dx
( )2
Reparametrization
If the parametrization of the surface is transformed by the equations:
then the chain rule yields:
or
where
is the new Jacobian matrix of the surface with respect to the new parameters and , and
is the Jacobian matrix of the reparametrization. The new Hessian is given by
where
.
The new fundamental matrix is given by:
and the new curvature matrix is given by: ′
D =PDPT
′
G =PGPT
Q=
∂(u,v) ∂ ′u2
∂( )u,v ∂ ′u∂ ′v ∂(u,v)
∂ ′v∂ ′u
∂( )u,v ∂ ′v2
′
H =PHPT+QJ P= ∂(u,v)
∂ ′(u,v′)= ∂u ∂ ′u
∂v ∂ ′u ∂u ∂ ′v
∂v ∂ ′v
′ v ′ u ′
J =∂(x,y,z)
∂ ′(u,v′) ′
J =PJ
∂(x,y,z) ∂ ′(u,v′) =
∂(u,v) ∂ ′(u,v′)
∂(x,y,z) ∂( )u,v ′
Change of Coordinates
For simplicity, we have defined several primitives with unit size, located at the origin. Related to the reparametrization is the change of coordinates , with associated Jacobian:
When the change of coordinates is represented by the affine transformation:
the Jacobian is simply the submatrix:
Regardless, the Jacobian and Hessian transform as follows:
The normal is transformed as:
The denominator arises from the desire to have a unit normal. The first and second fundamental matrices are then calculated as:
Not very pretty. But certain types of transformations can be applied easily. For a uniform scale with arbitrary translations,
so that
For rotations (and arbitrary translations), the Jacobian matrix C=R is orthogonal, so the inverse is equal to the transpose, yielding:
Combining the two, we have the results for a transformation that includes translations, rotations and uniform scale:
or in terms of the composite matrix :
′
J =JC, H′=HC, n′= nC C
( )
13, G′=
( )
C 23G, D′=
( )
C 1 3D C=rR′
J =rJR, H′=rHR, n′=nR, G′=r2G, D′=rD
′
J =JR, H′=HR, n′=nR, G′=G, D′=D
′
Sphere
Given the spherical coordinates:
we have the Jacobian matrix:
the Hessian tensor:
the first fundamental form:
the normal:
and the second fundamental form:
Unit Sphere
Angle Parametrization
Given the unit spherical coordinates with , we parametrize the sphere:
This yields the Jacobian matrix:
the Hessian tensor:
the first fundamental form:
the normal:
and the second fundamental form:
Angle Parametrization
With the reparametrization , we have the Jacobian:
Applying the chain rule, we have:
Changing coordinates to yield a sphere of arbitrary radius, we find that the expressions for the Jacobian, the Hessian, and the metric matrix remain the same, because x, y, and z scale linearly with r. The curvature matrix changes to:
Duv= −
4π2 x2+
y2
(
)
r 0
0 −π2r
Duv=
−4π2
(
x2+y2)
0 0 −π2
Guv=
4π2
(
x2+y2)
0 0 π2
Huv=
4π2
[
−x −y 0]
2π − yz x2 +y2
xz
x2+
y2 0
2π − yz x2+y2
xz
x2+ y2 0
π2 −
x −y −z
[ ]
Cone
Angle Parametrization
Given the unit conical parametrization:
we have the Jacobian matrix:
the Hessian tensor:
the first fundamental form:
the normal:
and the second fundamental form:
Unit Parametrization
For the parametrization:
we have:
Duv= −
4π2rz 1+h2
0
0 0
nuv= 1
1+h2
h2
x rz
h2
y rz −1
Guv=
4π2
(
x2+y2)
00 h
2
x2+y2+z2
(
)
z2
Cylinder
Angle Parametrization
Given the cylindrical parametrization:
we have the Jacobian matrix:
the Hessian tensor:
the first fundamental form:
the normal:
and the second fundamental form:
Unit Parametrization
With the parametrization:
we have the Jacobian matrix:
the Hessian tensor:
Huv= −4π 2
x −4π2y 0
[
]
[
0 0 0]
0 0 0
[
]
[
0 0 0]
Juv= −2πy 2πx 0
0 0 h
x y z
[ ] =[rcos2πu rsin2πu hv]
Dθφ = −1 0
0 0
n=[x y 0]
Gθφ = 1 0
0 1
Hθφ = [−x −y 0] [0 0 0]
0 0 0
[ ] [0 0 0]
Jθφ= −y x 0
0 0 1
x y z
and the second fundamental form:
Duv=
−4π2r 0 0 0
n= x
r y r 0
Torus
Angle Parametrization
Given the torus parametrization:
we have the Jacobian matrix:
the Hessian tensor:
the first fundamental form:
the normal:
and the second fundamental form:
using the torus’s implicit equation:
