the differential geometry of parametric primitives

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THE DIFFERENTIAL GEOMETRY OF PARAMETRIC PRIMITIVES
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
Apple Technical Report No. KT-23
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:
The first fundamental form is defined as:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 1

G = JJ
t
=
∂ x
∂ u

•
∂ x
∂u
∂x
∂u
•
∂ x
∂v
∂ x
∂v
•
∂ x
∂u
∂x
∂ v
•
∂ x
∂v











H =

∂
2
x, y,z
(
)
∂ u, v
(
)
∂ u,v
(
)
=
∂
2
x
∂u
2
∂
2
y
∂u
2
∂
2
z
∂u
2







∂
2
x
∂u∂v
∂
2
y
∂u∂v
∂
2
z
∂u∂v






∂
2
x
∂v∂u
∂
2
y
∂ v∂u
∂

2
z
∂ v∂u






∂
2
x
∂ v
2
∂
2
y
∂ v
2
∂
2
z
∂v
2



















=
∂
2
x
∂u
2
∂
2
x
∂u∂v
∂
2
x
∂v∂u
∂
2
x
∂v

2













J =
∂ x,y,z
(
)
∂ u,v
(
)
=
∂
x
∂ u
∂ y
∂ u
∂ z
∂ u
∂ x

∂ v
∂ y
∂ v
∂ z
∂ v










=
∂x
∂u
∂x
∂v











u = u v
[
]

x = x y z
[
]

x = F u
(
)
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:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 2

˙˙
x • n =
˙
uD
˙
u

t

˙
u

κ
n
=
˙
uD
˙
u
t
˙
uG
˙
u
t

D = n• H =
n•
∂
2
x
∂u
2
n •
∂
2
x

∂u∂v
n •
∂
2
x
∂ v∂u
n •
∂
2
x
∂v
2













S = G
(
)
R
∫∫

1
2
dudv

δ S≈ G
(
)
1
2
δ uδv

δu,δv
(
)

s =
ds
dt
t
0
t
1
∫
dt =
˙
x
t
0
t
1

∫
dt =
˙
x
t
0
t
1
∫
dt =
˙
uG
˙
u
t
(
)
t
0
t
1
∫
1
2
dt

u = u t
(
)
,

t
0
< t < t
1

dx
(
)
2
= du
(
)
G du
(
)
t
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:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 3

′D = PDP
T

′G = PGP
T

Q =
∂ u,v
(
)
∂ ′u
2
∂
u,v
(
)
∂ ′u ∂ ′v
∂ u,v
(
)
∂ ′v ∂ ′u
∂ u,v
(
)
∂ ′v
2












′H = PHP
T
+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
(
)

′u = ′u u,v
(
)
and ′
v = ′v 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,
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 4

C =
r 0 0
0 r 0
0 0 r










= r I

′D = ′H • ′n =
HC
(
)
• nC
−1t
(

)
nC
−1t
C
−1
n
t
(
)
1
2
=
HCC
−1
n
t
nC
−1t
C
−1
n
t
(
)
1
2
=
H •n
nC
−1t

C
−1
n
t
(
)
1
2
=
D
nC
−1t
C
−1
n
t
(
)
1
2

′G = ′J ′J
t
= JCC
t
J
t

′n =
nC

−1t
nC
−1t
C
−1
n
t
(
)
1
2

′J = JC, ′H = HC

C =
x
x
y
x
z
x
x
y
y
y
z
y
x
z
y

z
z
z











A =
x
x
y
x
z
x
x
y
y
y
z
y
x
z
y

z
z
z
x
o
y
o
z
o













C =
∂ ′x
∂ x
=
∂ ′x
∂ x
∂ ′y
∂ x

∂ ′z
∂ x
∂ ′x
∂ y
∂ ′y
∂ y
∂ ′z
∂ y
∂ ′x
∂ z
∂ ′y
∂ z
∂ ′z
∂ z

















′x = ′x x
(
)
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 :
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 5

′J = JC, ′H = HC, ′n =
nC
C
(
)
1
3
, ′G = C
(
)
2
3
G, ′D = C
(
)
1
3

D

C = r R

′J = rJR, ′H = rHR, ′n = nR, ′G = r
2
G, ′D = rD

′J = JR, ′H = HR, ′n = nR, ′G = G, ′D =D

′J = rJ, ′H = rH, ′n = n, ′G = r
2
G, ′D = r 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:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 6

D =
−
x
2
+ y
2
r

0
0 −r









n =
x
r
y
r
z
r







G =
x
2
+ y
2

0
0 r
2







∂
2
x, y,z
(
)
∂ θ,φ
(
)
∂ θ ,φ
(
)
=
− x −y 0
[
]
−
yz
x
2
+ y

2
xz
x
2
+ y
2
0






−
yz
x
2
+ y
2
xz
x
2
+ y
2
0






 − x −y −z
[
]















∂ x,y,z
(
)
∂ θ ,φ
(
)
=
−
y x 0
xz
x
2

+ y
2
yz
x
2
+ y
2
− x
2
+ y
2









x y z
[
]
=
r sinφ cosθ r sin φ sinθ r cosφ
[
]
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:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 7

J
uv
=
−2π y 2π x 0
πxz
x
2
+ y
2
πyz
x
2
+ y
2
−π x
2
+ y
2










P =
2π 0
0 π







θ = 2π u, ϕ = πv

D
θφ
=
− x
2
+ y
2
(
)
0

0 −1







n = x y z
[
]

G
θφ
=
x
2
+ y
2
0
0 1







H
θφ

=
− x −y 0
[
]
−
yz
x
2
+ y
2
xz
x
2
+ y
2
0






−
yz
x
2
+ y
2
xz
x

2
+ y
2
0





 − x −y −z
[
]















J
θφ
=

− y x 0
xz
x
2
+ y
2
yz
x
2
+ y
2
− x
2
+ y
2









x y z
[
]
= sin φ cosθ sin φ sinθ cosφ
[
]

0 ≤ θ < 2π, 0 ≤ ϕ < π
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:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 8

D
uv
=
−
4π
2
x
2
+ y
2
(
)
r
0
0 −π
2
r










D
uv
=
−4π
2
x
2
+ y
2
(
)
0
0 −π
2







G
uv
=
4π
2

x
2
+ y
2
(
)
0
0 π
2







H
uv
=
4π
2
−x −y 0
[
]
2π −
yz
x
2
+ y
2

xz
x
2
+ y
2
0






2π −
yz
x
2
+ y
2
xz
x
2
+ y
2
0





 π

2
− 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:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990

Apple Computer, Inc. Media Technology: Computer Graphics Page 9

H
uv
=
4π
2
−x −y 0
[
]
2π
h
rz
−y x 0
[
]
2π h
rz
−y x 0
[
]
0 0 0
[
]












J
uv
=
−2π y 2π x 0
hx
rz
hy
rz
h









x y z
[
]
=
rvcos2 π u rv sin2 πu vh
[
]

D
θ z
=
−
z
2
0
0 0









n
θz
=
x
z 2
y
z 2
−
1
2








G
θ z
=
x
2
+ y
2
0
0
x
2
+ y
2
+ z
2
z
2









=
z
2
0
0 2







H
θ z
=
−x −y 0
[
]
−
y
z
x
z
0







−
y
z
x
z
0






0 0 0
[
]











J
θ z
=
− y x 0

x
z
y
z
1









x y z
[
]
=
zcosθ zsinθ z
[
]
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 10

D
uv
=
−
4π
2

rz
1 + h
2
0
0 0









n
uv
=
1
1+ h
2
h
2
x
rz
h
2
y
rz
−1








G
uv
=
4π
2
x
2
+ y
2
(
)
0
0
h
2
x
2
+ y
2
+ z
2
(
)
z

2










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:
the first fundamental form:
the normal:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 11

G
uv

=
4π
2
r
2
0
0 h
2







H
uv
=
−4π
2
x −4π
2
y 0
[
]
0 0 0
[
]
0 0 0
[

]
0 0 0
[
]







J
uv
=
−2π y 2π x 0
0 0 h







x y z
[
]
=
r cos2 π u r sin2πu hv
[
]

D
θφ
=
−1 0
0 0







n = x y 0
[
]

G
θφ
=
1 0
0 1







H

θφ
=
− x −y 0
[
]
0 0 0
[
]
0 0 0
[
]
0 0 0
[
]







J
θφ
=
− y x 0
0 0 1








x y z
[
]
= cosθ sinθ
z
[
]
and the second fundamental form:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 12

D
uv
=
−4π
2
r 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:
Turkowski The Differential Geometry of Parametric Primitives 26 January 1990
Apple Computer, Inc. Media Technology: Computer Graphics Page 13

x
2
+ y
2
− R
(
)