Cubic Surfaces CGP&P Chapter 11

Modeling surfaces

Extension of parametric cubic curves called “parametric bicubic surfaces”

Idea: infinite # of curves stacked together equations now have 2 parameters Q(s,t)

P1(t)P4(t)

t

s

t=0.25

t=0.75

Matrix representation

A single curve was expressedQ(t) = TMG

Now, the geometric information varies

Q(s,t) = SMG

)(

)(

)(

)(

4

3

2

1

tG

tG

tG

tG

G each Gi(t) is itselfa cubic curve

Matrix representation (cont.)

Since each Gi is a cubic curve, it can be written:

Gi(t) = TMGi

4

3

2

1

i

i

i

i

i

g

g

g

g

G

Matrix representation (cont.)

Substituting, we obtain:

Q(s,t) = SM

= SMTM

4

3

2

1

TMG

TMG

TMG

TMG

4

3

2

1

G

G

G

G

This formulation does not work in terms of matrix dimensions...

Matrix representation (cont.)

So, use the transpose rule:Gi(t) = Gi

T MT TT

Q(s,t) = S M MT TT

= S M MT TT

T

T

T

T

G

G

G

G

4

3

2

1

44434241

34333231

24232221

14131211

gggg

gggg

gggg

gggg

Matrix representation (cont.) Finally, remember that the large

geometry matrix has 3 components (x, y, z) for each gij, so that we get three parametric equations:

x(s,t) = S M Gx MT TT

y(s,t) = S M Gy MT TT

z(s,t) = S M Gz MT TT

Hermite surfaces extension of Hermite curves to

parametric bicubic surfaces four elements of the geometry

matrix are now P1(t), P4(t), R1(t), R4(t)

can be thought of as interpolating the curves Q(s,0) and Q(s,1) orQ(0,t) and Q(1,t)

Hermite surface matrices

x(s,t) = S M GHx MT TT

y(s,t) = S M GHy MT TT

z(s,t) = S M GHz MT TT

44434241

34333231

24232221

14131211

gggg

gggg

gggg

gggg

GxH

P1x (t) TMG1x

G1x

g11

g12

g13

g14

Hermite Surface Matrix

Upper left = x-coordinates of surface Upper right = x derivatives in t at corners Lower left = x derivatives in s at corners Lower right = twist at corners

Rendering surfaces Can use iterative methods in s and t

Solve surface at points Q(s, t) and connect points with quadrilaterals

Expensive because iterating for small s and t results in many cubic surface evaluations

Forward Differencing Our old friend… Because we can differentiate cubic curves

three times, we can increment all derivatives x += x x += 2x 2x += 3x

Surface Rendering Subdivision

As with cubic curves, Bezier cubic surface easily supports subdivision

Subdivision ceases when plane described by one quarter of the surface is nearly coplanar with the other three-fourths

Watch for abutting quadrilaterals that don’t match up

This happens when different levels of subdivision are applied to adjoining patches