The differential Geometry of parametric primitives
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
Turkowski
The Differential Geometry of Parametric Primitives
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:
x = F(u)
where
x = [x
y z]
is a point in affine 3-space, and
u = [ u v]
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:
∂ x ∂ y
∂ ( x,y,z) ∂ u ∂ u
J=
=
∂x ∂ y
∂ (u,v)
∂ v ∂ v
∂ z ∂x
∂ u ∂u
=
∂z
∂x
∂ v ∂v
The Hessian is a tensor of second partial derivatives:
∂ 2 x ∂ 2 y ∂ 2z
∂ 2x
∂ 2y
∂ 2z
2
2
2
∂ 2( x, y,z)
∂u ∂u ∂u
∂u∂ v ∂u∂ v ∂u∂ v
H=
= 2
2
2
∂ 2 x ∂ 2 y ∂ 2z
∂ ( u, v)∂ (u,v) ∂ x
∂ y
∂ z
2
2
2
∂v∂u ∂ v∂u ∂ v∂u
∂ v ∂ v ∂v
∂ 2x
2
∂u
= 2
∂
x
∂v∂u
∂ 2x
∂u∂v
∂ 2x
∂v2
The first fundamental form is defined as:
∂ x ∂ x
∂ u • ∂u
t
G = JJ =
∂x ∂x
•
∂v ∂u
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∂x ∂ x
•
∂u ∂v
∂x ∂ x
•
∂ v ∂v
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
and establishes a metric of differential length:
( dx) = ( du) G( du)
t
2
so that the arc length of a curve segment, u = u( t) , t0 < t < t1 is given by:
1
t1
t1
t1
ds
t 2
s = ∫ dt = ∫ ẋ dt = ∫ ẋ dt = ∫ (u̇G u̇ ) dt
t 0 dt
t0
t0
t0
t1
The differential surface area enclosed by the differential parallelogram ( δu,δv) is approximately:
1
δ S≈ ( G ) 2δ uδv
so that the area of a region of the surface corresponding to a region R in the u-v plane is:
1
S = ∫∫ ( G ) dudv
2
R
The second fundamental matrix measures normal curvature, and is given by:
∂ 2x
∂ 2x
n•
n
•
∂u2
∂u∂ v
D = n• H =
2
∂ 2x
n • ∂ x
n• 2
∂v
∂ v∂u
The normal curvature is defined to be positive a curve u on the surface turns toward the positive
direction of the surface normal by:
κn =
u̇Du̇t
t
u̇G u̇
The deviation (in the normal direction) from the tangent plane of the surface, given a differential
displacement of u̇ is:
˙˙ • n = u̇Du̇ t
x
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Reparametrization
If the parametrization of the surface is transformed by the equations:
u′ = u′(u,v)
and
v′ = v′( u,v)
then the chain rule yields:
∂ ( x,y,z)
∂ ( u′, v′)
=
∂ ( u,v) ∂ ( x,y,z)
∂ ( u′, v′) ∂ (u,v)
or
J ′ = PJ
where
J′ =
∂ ( x, y,z)
∂ (u ′, v′)
is the new Jacobian matrix of the surface with respect to the new parameters u′ and v′ , and
∂u
∂ ( u,v) ∂ u ′
P=
=
∂ (u ′, v′) ∂u
∂ v′
∂v
∂ u′
∂v
∂ v′
is the Jacobian matrix of the reparametrization.
The new Hessian is given by
H′ = PHP +QJ
T
where
∂ (u, v)
∂ u ′2
Q=
( )
∂ u, v
∂ v′∂ u′
∂ (u,v)
∂ u′∂ v′
∂ (u,v)
∂ v′ 2 .
The new fundamental matrix is given by:
G′ = PGP
T
and the new curvature matrix is given by:
D′ = PDP
T
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
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 x ′ = x ′( x) , with associated Jacobian:
∂ x′
∂x
∂ x ′ ∂ x′
C=
=
∂x ∂ y
∂ x′
∂z
∂ y′
∂x
∂ y′
∂y
∂ y′
∂z
∂ z′
∂x
∂ z′
∂ y
∂ z′
∂z
When the change of coordinates is represented by the affine transformation:
xx
x
y
A=
x
z
x
o
yx zx
yy zy
yz zz
yo zo
the Jacobian is simply the submatrix:
xx yx zx
C = xy yy zy
xz yz zz
Regardless, the Jacobian and Hessian transform as follows:
J ′ = JC, H ′ = HC
The normal is transformed as:
n′ =
nC−1 t
1
(nC −1t C −1nt) 2
The denominator arises from the desire to have a unit normal.
The first and second fundamental matrices are then calculated as:
G′ = J′ J′ = JCC J
t
D′ = H′ • n′ =
t
t
( HC) • ( nC −1 t)
1
=
HCC− 1n t
1
=
H •n
1
=
D
1
( nC−1t C− 1n t) 2 (nC −1t C −1nt) 2 (nC−1 t C −1nt) 2 (nC−1t C −1nt) 2
Not very pretty. But certain types of transformations can be applied easily. For a uniform scale
with arbitrary translations,
r
C= 0
0
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0 0
r 0 =rI
0 r
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
so that
J ′ = rJ, H′ = rH, n′ = n, G′ = r G, D ′ = r D
2
For rotations (and arbitrary translations), the Jacobian matrix C=R is orthogonal, so the inverse
is equal to the transpose, yielding:
J ′ = JR, H ′ = HR, n′ = nR, G′ = G, D′ =D
Combining the two, we have the results for a transformation that includes translations, rotations
and uniform scale:
J ′ = rJR, H′ = rHR, n ′ = nR, G ′ = r G, D ′ = rD
2
or in terms of the composite matrix C = r R :
J ′ = JC, H ′ = HC, n′ =
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nC
2
1
G′ = ( C ) 3G, D′ = ( C ) 3 D
1,
( C)3
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Sphere
Given the spherical coordinates:
[ x y z] = [ r sinφ cosθ
r sin φ sinθ
r cosφ ]
we have the Jacobian matrix:
−y
= xz
∂ (θ ,φ ) x 2 + y2
x
yz
∂ ( x,y,z)
x 2 + y2
− x +y
0
2
2
the Hessian tensor:
[− x −y 0]
2
∂ ( x, y,z)
=
∂ ( θ,φ )∂ (θ ,φ )
yz
xz
−
2
2
2
2
x +y
x +y
yz
− 2
x + y2
0
xz
x2 + y2
[− x −y − z]
0
the first fundamental form:
x2 + y2 0
G=
r 2
0
the normal:
x
n=
r
y
r
z
r
and the second fundamental form:
x2 + y2
−
D=
r
0
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0
−r
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Unit Sphere
Angle Parametrization
Given the unit spherical coordinates with 0 ≤ θ < 2π, 0 ≤ ϕ < π , we parametrize the sphere:
[ x y z] = [sin φ cosθ
sin φ sinθ
cosφ ]
This yields the Jacobian matrix:
−y
Jθφ = xz
2
2
x +y
x
yz
− x +y
0
2
x2 + y2
2
the Hessian tensor:
[ − x −y 0]
Hθφ =
yz
xz
−
2
2
2
2
x +y
x +y
yz
− 2
x + y2
0
xz
x2 + y2
[ − x −y −z]
0
the first fundamental form:
x2 + y2 0
Gθφ =
1
0
the normal:
n = [x
y z]
and the second fundamental form:
−( x2 + y2) 0
Dθφ =
0
−1
Angle Parametrization
With the reparametrization θ = 2π u, ϕ = π v , we have the Jacobian:
2π
P=
0
0
π
Applying the chain rule, we have:
−2π y
Juv = πxz
2
2
x +y
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2π x
πyz
−π x2 + y2
0
x2 + y2
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Turkowski
The Differential Geometry of Parametric Primitives
4π 2[ −x −y 0]
Huv =
xz
2π − yz
2
2
2
2
x +y
x +y
yz
2π −
x2 + y2
0
26 January 1990
xz
x2 + y2
π [ − x −y −z]
2
0
4π 2( x2 + y2) 0
Guv =
2
0
π
−4π 2( x 2 + y2) 0
Duv =
2
0
−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:
4π 2( x2 + y2)
Duv = −
r
0
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0
−π 2r
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Cone
Angle Parametrization
Given the unit conical parametrization:
[ x y z] = [ zcosθ
zsinθ
z]
we have the Jacobian matrix:
− y x 0
Jθ z = x y
1
z z
the Hessian tensor:
y x
[ −x −y 0] − z z 0
Hθ z =
y x
−
0
0
0
0
[
]
z z
the first fundamental form:
x2 + y2
Gθ z =
0
2
0
z 0
x2 + y2 + z2 =
0 2
2
z
the normal:
x
nθ z =
z 2
y
z 2
−
1
2
and the second fundamental form:
z
−
Dθ z = 2
0
0
0
Unit Parametrization
For the parametrization:
[ x y z] = [ rvcos2 π u rv sin2 πu vh]
we have:
−2π y 2π x 0
hy
Juv = hx
h
rz
rz
2
4π [ −x −y 0]
Huv =
2π h
[ −y x 0]
rz
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2π h
[ −y x 0]
rz
[ 0 0 0]
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Turkowski
The Differential Geometry of Parametric Primitives
4π 2( x2 + y2)
Guv =
0
nuv =
h2x
1+ h2 rz
1
4π 2 rz
−
Duv = 1 + h2
0
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26 January 1990
2
2
2
2
h(x + y + z)
z2
0
h2y
rz
−1
0
0
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Cylinder
Angle Parametrization
Given the cylindrical parametrization:
[ x y z] = [ cosθ
z]
sinθ
we have the Jacobian matrix:
− y x 0
Jθφ =
0 0 1
the Hessian tensor:
[ − x −y 0] [ 0 0 0]
Hθφ =
[ 0 0 0]
[0 0 0]
the first fundamental form:
1
Gθφ =
0
0
1
the normal:
n = [x
y 0]
and the second fundamental form:
−1 0
Dθφ =
0 0
Unit Parametrization
With the parametrization:
[ x y z] = [ r cos2 π u r sin2 πu hv]
we have the Jacobian matrix:
−2π y 2π x 0
Juv =
0
h
0
the Hessian tensor:
[ −4π 2 x −4π 2y 0] [0 0 0]
Huv =
[0 0 0]
[0 0 0]
the first fundamental form:
4π 2 r 2 0
Guv =
h2
0
the normal:
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x
n=
r
The Differential Geometry of Parametric Primitives
y
r
26 January 1990
0
and the second fundamental form:
−4π 2r
Duv =
0
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0
0
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Torus
Angle Parametrization
Given the torus parametrization:
[ x y z] = [ (R + r cosφ ) cosθ (R + r cosφ ) sin θ r sin φ ]
we have the Jacobian matrix:
−y
Jθφ = − xz
2
2
x +y
x2 + y2 − R
x
yz
−
0
x2 + y2
the Hessian tensor:
[ − x −y 0]
Hθφ =
xz
yz
− 2 2
2
2
x +y
x + y
yz
xz
−
0
2
2
x2 + y2
x +y
R
R
0 − x 1− 2
−
y
1
−
−z
2
2
2
x +y
x +y
the first fundamental form:
x2 + y2 0
Gθφ =
r 2
0
the normal:
R
1− 2
2
x +y
n= x
r
1−
y
R
x +y
2
2
r
z
r
and the second fundamental form:
Dθφ
x2 + y2
R
−
1− 2
0
2
=
r
x +y
0
−r
R2 − x 2 − y2 + z2 − r 2
=
2r
0
0
−r
using the torus’s implicit equation:
(
)
2
x + y − R +z = r
2
2
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2
2
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Page 13
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
Turkowski
The Differential Geometry of Parametric Primitives
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:
x = F(u)
where
x = [x
y z]
is a point in affine 3-space, and
u = [ u v]
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:
∂ x ∂ y
∂ ( x,y,z) ∂ u ∂ u
J=
=
∂x ∂ y
∂ (u,v)
∂ v ∂ v
∂ z ∂x
∂ u ∂u
=
∂z
∂x
∂ v ∂v
The Hessian is a tensor of second partial derivatives:
∂ 2 x ∂ 2 y ∂ 2z
∂ 2x
∂ 2y
∂ 2z
2
2
2
∂ 2( x, y,z)
∂u ∂u ∂u
∂u∂ v ∂u∂ v ∂u∂ v
H=
= 2
2
2
∂ 2 x ∂ 2 y ∂ 2z
∂ ( u, v)∂ (u,v) ∂ x
∂ y
∂ z
2
2
2
∂v∂u ∂ v∂u ∂ v∂u
∂ v ∂ v ∂v
∂ 2x
2
∂u
= 2
∂
x
∂v∂u
∂ 2x
∂u∂v
∂ 2x
∂v2
The first fundamental form is defined as:
∂ x ∂ x
∂ u • ∂u
t
G = JJ =
∂x ∂x
•
∂v ∂u
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∂x ∂ x
•
∂u ∂v
∂x ∂ x
•
∂ v ∂v
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
and establishes a metric of differential length:
( dx) = ( du) G( du)
t
2
so that the arc length of a curve segment, u = u( t) , t0 < t < t1 is given by:
1
t1
t1
t1
ds
t 2
s = ∫ dt = ∫ ẋ dt = ∫ ẋ dt = ∫ (u̇G u̇ ) dt
t 0 dt
t0
t0
t0
t1
The differential surface area enclosed by the differential parallelogram ( δu,δv) is approximately:
1
δ S≈ ( G ) 2δ uδv
so that the area of a region of the surface corresponding to a region R in the u-v plane is:
1
S = ∫∫ ( G ) dudv
2
R
The second fundamental matrix measures normal curvature, and is given by:
∂ 2x
∂ 2x
n•
n
•
∂u2
∂u∂ v
D = n• H =
2
∂ 2x
n • ∂ x
n• 2
∂v
∂ v∂u
The normal curvature is defined to be positive a curve u on the surface turns toward the positive
direction of the surface normal by:
κn =
u̇Du̇t
t
u̇G u̇
The deviation (in the normal direction) from the tangent plane of the surface, given a differential
displacement of u̇ is:
˙˙ • n = u̇Du̇ t
x
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Reparametrization
If the parametrization of the surface is transformed by the equations:
u′ = u′(u,v)
and
v′ = v′( u,v)
then the chain rule yields:
∂ ( x,y,z)
∂ ( u′, v′)
=
∂ ( u,v) ∂ ( x,y,z)
∂ ( u′, v′) ∂ (u,v)
or
J ′ = PJ
where
J′ =
∂ ( x, y,z)
∂ (u ′, v′)
is the new Jacobian matrix of the surface with respect to the new parameters u′ and v′ , and
∂u
∂ ( u,v) ∂ u ′
P=
=
∂ (u ′, v′) ∂u
∂ v′
∂v
∂ u′
∂v
∂ v′
is the Jacobian matrix of the reparametrization.
The new Hessian is given by
H′ = PHP +QJ
T
where
∂ (u, v)
∂ u ′2
Q=
( )
∂ u, v
∂ v′∂ u′
∂ (u,v)
∂ u′∂ v′
∂ (u,v)
∂ v′ 2 .
The new fundamental matrix is given by:
G′ = PGP
T
and the new curvature matrix is given by:
D′ = PDP
T
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
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 x ′ = x ′( x) , with associated Jacobian:
∂ x′
∂x
∂ x ′ ∂ x′
C=
=
∂x ∂ y
∂ x′
∂z
∂ y′
∂x
∂ y′
∂y
∂ y′
∂z
∂ z′
∂x
∂ z′
∂ y
∂ z′
∂z
When the change of coordinates is represented by the affine transformation:
xx
x
y
A=
x
z
x
o
yx zx
yy zy
yz zz
yo zo
the Jacobian is simply the submatrix:
xx yx zx
C = xy yy zy
xz yz zz
Regardless, the Jacobian and Hessian transform as follows:
J ′ = JC, H ′ = HC
The normal is transformed as:
n′ =
nC−1 t
1
(nC −1t C −1nt) 2
The denominator arises from the desire to have a unit normal.
The first and second fundamental matrices are then calculated as:
G′ = J′ J′ = JCC J
t
D′ = H′ • n′ =
t
t
( HC) • ( nC −1 t)
1
=
HCC− 1n t
1
=
H •n
1
=
D
1
( nC−1t C− 1n t) 2 (nC −1t C −1nt) 2 (nC−1 t C −1nt) 2 (nC−1t C −1nt) 2
Not very pretty. But certain types of transformations can be applied easily. For a uniform scale
with arbitrary translations,
r
C= 0
0
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0 0
r 0 =rI
0 r
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
so that
J ′ = rJ, H′ = rH, n′ = n, G′ = r G, D ′ = r D
2
For rotations (and arbitrary translations), the Jacobian matrix C=R is orthogonal, so the inverse
is equal to the transpose, yielding:
J ′ = JR, H ′ = HR, n′ = nR, G′ = G, D′ =D
Combining the two, we have the results for a transformation that includes translations, rotations
and uniform scale:
J ′ = rJR, H′ = rHR, n ′ = nR, G ′ = r G, D ′ = rD
2
or in terms of the composite matrix C = r R :
J ′ = JC, H ′ = HC, n′ =
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nC
2
1
G′ = ( C ) 3G, D′ = ( C ) 3 D
1,
( C)3
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Sphere
Given the spherical coordinates:
[ x y z] = [ r sinφ cosθ
r sin φ sinθ
r cosφ ]
we have the Jacobian matrix:
−y
= xz
∂ (θ ,φ ) x 2 + y2
x
yz
∂ ( x,y,z)
x 2 + y2
− x +y
0
2
2
the Hessian tensor:
[− x −y 0]
2
∂ ( x, y,z)
=
∂ ( θ,φ )∂ (θ ,φ )
yz
xz
−
2
2
2
2
x +y
x +y
yz
− 2
x + y2
0
xz
x2 + y2
[− x −y − z]
0
the first fundamental form:
x2 + y2 0
G=
r 2
0
the normal:
x
n=
r
y
r
z
r
and the second fundamental form:
x2 + y2
−
D=
r
0
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0
−r
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Unit Sphere
Angle Parametrization
Given the unit spherical coordinates with 0 ≤ θ < 2π, 0 ≤ ϕ < π , we parametrize the sphere:
[ x y z] = [sin φ cosθ
sin φ sinθ
cosφ ]
This yields the Jacobian matrix:
−y
Jθφ = xz
2
2
x +y
x
yz
− x +y
0
2
x2 + y2
2
the Hessian tensor:
[ − x −y 0]
Hθφ =
yz
xz
−
2
2
2
2
x +y
x +y
yz
− 2
x + y2
0
xz
x2 + y2
[ − x −y −z]
0
the first fundamental form:
x2 + y2 0
Gθφ =
1
0
the normal:
n = [x
y z]
and the second fundamental form:
−( x2 + y2) 0
Dθφ =
0
−1
Angle Parametrization
With the reparametrization θ = 2π u, ϕ = π v , we have the Jacobian:
2π
P=
0
0
π
Applying the chain rule, we have:
−2π y
Juv = πxz
2
2
x +y
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2π x
πyz
−π x2 + y2
0
x2 + y2
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Turkowski
The Differential Geometry of Parametric Primitives
4π 2[ −x −y 0]
Huv =
xz
2π − yz
2
2
2
2
x +y
x +y
yz
2π −
x2 + y2
0
26 January 1990
xz
x2 + y2
π [ − x −y −z]
2
0
4π 2( x2 + y2) 0
Guv =
2
0
π
−4π 2( x 2 + y2) 0
Duv =
2
0
−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:
4π 2( x2 + y2)
Duv = −
r
0
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0
−π 2r
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Cone
Angle Parametrization
Given the unit conical parametrization:
[ x y z] = [ zcosθ
zsinθ
z]
we have the Jacobian matrix:
− y x 0
Jθ z = x y
1
z z
the Hessian tensor:
y x
[ −x −y 0] − z z 0
Hθ z =
y x
−
0
0
0
0
[
]
z z
the first fundamental form:
x2 + y2
Gθ z =
0
2
0
z 0
x2 + y2 + z2 =
0 2
2
z
the normal:
x
nθ z =
z 2
y
z 2
−
1
2
and the second fundamental form:
z
−
Dθ z = 2
0
0
0
Unit Parametrization
For the parametrization:
[ x y z] = [ rvcos2 π u rv sin2 πu vh]
we have:
−2π y 2π x 0
hy
Juv = hx
h
rz
rz
2
4π [ −x −y 0]
Huv =
2π h
[ −y x 0]
rz
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2π h
[ −y x 0]
rz
[ 0 0 0]
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Turkowski
The Differential Geometry of Parametric Primitives
4π 2( x2 + y2)
Guv =
0
nuv =
h2x
1+ h2 rz
1
4π 2 rz
−
Duv = 1 + h2
0
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26 January 1990
2
2
2
2
h(x + y + z)
z2
0
h2y
rz
−1
0
0
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Cylinder
Angle Parametrization
Given the cylindrical parametrization:
[ x y z] = [ cosθ
z]
sinθ
we have the Jacobian matrix:
− y x 0
Jθφ =
0 0 1
the Hessian tensor:
[ − x −y 0] [ 0 0 0]
Hθφ =
[ 0 0 0]
[0 0 0]
the first fundamental form:
1
Gθφ =
0
0
1
the normal:
n = [x
y 0]
and the second fundamental form:
−1 0
Dθφ =
0 0
Unit Parametrization
With the parametrization:
[ x y z] = [ r cos2 π u r sin2 πu hv]
we have the Jacobian matrix:
−2π y 2π x 0
Juv =
0
h
0
the Hessian tensor:
[ −4π 2 x −4π 2y 0] [0 0 0]
Huv =
[0 0 0]
[0 0 0]
the first fundamental form:
4π 2 r 2 0
Guv =
h2
0
the normal:
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Turkowski
x
n=
r
The Differential Geometry of Parametric Primitives
y
r
26 January 1990
0
and the second fundamental form:
−4π 2r
Duv =
0
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0
0
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Turkowski
The Differential Geometry of Parametric Primitives
26 January 1990
Torus
Angle Parametrization
Given the torus parametrization:
[ x y z] = [ (R + r cosφ ) cosθ (R + r cosφ ) sin θ r sin φ ]
we have the Jacobian matrix:
−y
Jθφ = − xz
2
2
x +y
x2 + y2 − R
x
yz
−
0
x2 + y2
the Hessian tensor:
[ − x −y 0]
Hθφ =
xz
yz
− 2 2
2
2
x +y
x + y
yz
xz
−
0
2
2
x2 + y2
x +y
R
R
0 − x 1− 2
−
y
1
−
−z
2
2
2
x +y
x +y
the first fundamental form:
x2 + y2 0
Gθφ =
r 2
0
the normal:
R
1− 2
2
x +y
n= x
r
1−
y
R
x +y
2
2
r
z
r
and the second fundamental form:
Dθφ
x2 + y2
R
−
1− 2
0
2
=
r
x +y
0
−r
R2 − x 2 − y2 + z2 − r 2
=
2r
0
0
−r
using the torus’s implicit equation:
(
)
2
x + y − R +z = r
2
2
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2
2
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Page 13