Korea University Jung Lee, Computer Graphics Laboratory 3D Game Engine Design David H. Eberly 8.3 Special Surfaces 2001/11/13.

Korea UniversityKorea University

Jung Lee, Computer Graphics Jung Lee, Computer Graphics LaboratoryLaboratory

3D Game Engine Design

David H. Eberly

8.3 Special Surfaces

2001/11/13


2


Bezier Rectangle PatchesBezier Rectangle Patches

Popular with game programmers

• Mathematical simplicity

• Ease of use

Equation :

• (0 i0 n0, 0 i1 n1) : 3D control points

• The first-order partial derivatives of the patch

0

0

1

1

1011000 0

,,, )()(),(n

i

n

iiiinin ptBsBtsx

10iiP

))(()(),(10

0

0

1

1

101100 ,

1

0 0,1,,10 ii

n

i

n

iiiinins pptBsBntsx

))!(!/(!);(

)1();()(,inininC

ttinCtB iniin

))(()(),(10

0

0

1

1

101100 ,0

1

0,1,1,1 ii

n

i

n

iiiinint pptBsBntsx


3


Bezier Rectangle Patches Bezier Rectangle Patches EvaluationEvaluation

Trade-off of speed vs. accuracy Bernstein polynomial computation

• n0+1 evaluations of Bn0,i0(s)

• n1+1 evaluations of Bn1,i1(t)

• n0n1 multiplications for pairs of the evaluated polynomials

de Casteljau algorithm• Repeatedly computes convex combinations• Generally stable• Use more floating-point operations


4


Comparison For Bilinear Comparison For Bilinear InterpolationInterpolation

Case where n0=1 and n1=1

Bernstein form

•

• 2 subtractions, 9 additions, and 16

multiplications

de Casteljau form

•

• 2 subtractions, 9 additions, and 18

multiplications

1,10,11,00,0 )1()1()1)(1(

pstptsptspts

))1(())1)((1( 1,10,11,00,0

ptptsptpts


5


Degree ElevationDegree Elevation

Degree-elevated to (n0+1, n1)

•

Degree-elevated to (n0, n1+1)

•

Degree-elevated to (n0+1, n1+1)

•

• Degree-elevated control points

1

0 0,1

0

0,

0

0,,1

0

0

1

1

10101100 1)1

1()()(),(n

i

n

iiiiiinin p

n

ip

n

itBsBtsx

0

0

1

1

101011000

1

01,

1

1,

1

1,1, 1

)1

1()()(),(n

i

n

iiiiiinin p

n

ip

n

itBsBtsx

1

0

1

0,,,

0

0

1

1

101100)()(),(

n

i

n

iiiinin qtBsBtsx

1

11

111

1

1

1

1

1,1,1

1,,

0

0

0

0,

1010

1010

10

n

in

i

pp

pp

n

i

n

iq

iiii

iiiiii


6


Degree ReductionDegree Reduction

Let the degree-reduced surface• m0 n0, m1 n1

• Constraint The endpoints of the two curves are same Two choices for the extension to rectangle patches

The four corner points match between the two patches Least-squares fit to construct the remaining control points

• Process Degree-reduce the four boundary curves first Compute the remaining interior control points

Least-squares fit

0

0

1

1

1011000 0

,,, )()(),(m

i

m

iiiimim qtBsBtsy


7


Bezier Triangle PatchesBezier Triangle Patches

Slightly more complicated than Bezier rectangle patches

Useful for creating models of arbitrary complexity

Equation :• (i0, i1, i2 0, i0 + i1 + i2 = n)

• Evaluation

• Computing the coefficients To minimize arithmetic operations by saving

intermediate products and sums

nI

IIn pwvuBwvux ),,(),,( ,

210 ,, iiip

210

210

!!!

!

),,;(),,(

210

210,

iii

iiiIn

wvuiii

n

wvuiiinCwvuB

1,0,0,0,),,,( 210210 wvuwvuiiiIiiiI


8


Degree ElevationDegree Elevation

Multiply the original patch by

• Degree-elevated patch

Degree-elevated control points

wvu 1

nI nI

IInIIn qwvuBpwvuBwvuwvux1

,1, ),,(),,()(),,(

)(1

11,,2,1,1,,10 210210210

iiiiiiiiiI pipipin

q


9


Degree ReductionDegree Reduction

Let the degree-reduced surface• m n

• Constraint The endpoints of the two curves are same Two choices for the extension to triangle patches

The three corner points match between the two patches

Least-squares fit to construct the remaining control points

• Process The boundary curves are reduced first Compute the remaining interior control points

Least-squares fit

mI

IIm qwvuBwvuy ),,(),,( ,


10


Bezier Cylinder Surfaces (1 / Bezier Cylinder Surfaces (1 / 2)2)

Bezier rectangle/triangle patches

• More curvature variation than is needed Cylinder/developable surfaces

• Curved in 1D and flat• Two choice of generation

Taking a curve and linearly translating Sweeping operation

Taking a line segment and moving it One endpoint is constrained to the originally specified curve

• Evaluation, degree elevation/reduction can be applied to

Initial curve for cylinder surfaces Boundary curves for generalized cylinder surfaces


11


Bezier Cylinder Surfaces (2 / Bezier Cylinder Surfaces (2 / 2)2)

Cylinder Surface :• Bezier curve :• Linear translation :• The first-order partial derivatives :

Generalized cylinder surface• Specifying two Bezier curves

With the same number of control points• Blending between the two curves• Equation :

Control points : for 0 i n

• The first-order partial derivatives

n

i iin puBuy0 , )()(

D

Dvuyvvux )()1(),( ]1,0[),( vu

yDxyvx vuu ,)1(

n

i

n

i

n

iiiiniiniin qvpvuBquBvpuBvvux

0 0 0,,, )1()()()()1(),(

ii qp

,

1

011,1 )())(1()(

n

iiiiiinu qqvppvuBnx

n

iiiinv pquBx

0, ))((


12


Nonparametric B-Spline Nonparametric B-Spline Rectangle PatchesRectangle Patches

How to interpolate a rectangle lattice of scalars for 0 i0 n0, 0 i1 n1

• Componentwise interpolation B-spline polynomial of degree d

• for 0 k0 d• for 0 k1 d• : intermediate tensor

For applications• The intermediate tensor can be cached

),,(1010 iifii

d

j

d

j

d

k

d

kkkkjkjjdijdiii tXsXMMftsB

0 0 0 02/)1(,2/)1(

0 1 0 1

101100110010)()(),(

0

0)()( 0k

k issX 1

1)()( 1k

k ittX

11001100 2/)1(,2/)1( kjkjjdijdi MMf


13


Quadric SurfacesQuadric Surfaces

Discussion of the general quadratic equation

• A : 33 nonzero symmetric matrix

• : 31 vector

• c : scalar

• : 31 vector that represents the variable quantities A = RTDR

• D : diagonal matrix Diagonal entities are the eigenvalues of A

• R : rotational matrix Rows are corresponding eigenvectors

0

cxbxAxTT

b

x


14


EigendecompositionEigendecomposition

Rewritten quadratic equation• , ,

• Characterize the surface type based on the number of nonzero eigenvalues

Three : elliptic cone, ellipsoid, hyperboloid Two : line, union of two planes, elliptic cylinder,

hyperbolic cylinder, hyperbolic paraboloid One : parabolic cylinder

• Determine that the solution is degenerate from Point, line, plane

0

cyeyDyTT

eD,

xRy

bRe


15


Tube SurfacesTube Surfaces

Swept surface• Sweeping a region of space by a planar object

along a specified central curve Most common planar object : circle

Radius varies with time : r(t) Resulting surface is called a tube surface Equation :

: columns of an orientation matrix R(t) pp.286

t : curve parameter [0, 2)

Surface of revolution The central curve is a straight line

)(tx

)sin)(cos()(),(

BNtrtxtS

BNT ,,

Korea University Jung Lee, Computer Graphics Laboratory 3D Game Engine Design David H. Eberly 8.3 Special Surfaces 2001/11/13.

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