Parametric Surfaces CS 418 Intro to Computer Graphics John C. Hart - - PowerPoint PPT Presentation

▶

Sep 25, 2023 158 likes •341 views

Parametric Surfaces CS 418 Intro to Computer Graphics John C. Hart y Space Curves Separate into three coordinate functions z x p ( t ) = ( x ( t ), y ( t ), z ( t )) x ( t ) = (1- t ) 3 x 0 + 3 t (1- t ) 2 x 1 + 3 t 2 (1- t ) x 2 + t 3 x 3 y

SLIDE 1

Parametric Surfaces

CS 418 Intro to Computer Graphics John C. Hart

SLIDE 2

Space Curves

Separate into three coordinate functions p(t) = (x(t), y(t), z(t)) x(t) = (1-t)3 x0 + 3t(1-t)2 x1 + 3t2(1-t) x2 + t3 x3 y(t) = (1-t)3 y0 + 3t(1-t)2 y1 + 3t2(1-t) y2 + t3 y3 z(t) = (1-t)3 z0 + 3t(1-t)2 z1 + 3t2(1-t) z2 + t3 z3 t x t y t z z x y

SLIDE 3

Space Curves

Make your own roller-coaster ride

Camera position along space curve
Look at point is next position along

space curve (tangent)

Binormal is cross product of vector

to next position with vector to previous position

Up direction (normal) is cross

product of binormal with tangent z x y

T N B

SLIDE 4

Extrusion

Two 3-D copies of each 2-D curve
Create a mesh of quads (or tri-strip)

p0(t), p1(t), p0(t+∆t), p1(t+∆t) p0(t) = (x(t),y(t),0) p1(t) = (x(t),y(t),1) p(t) = (x(t),y(t))

SLIDE 5

Generalized Cylinder

Construct a 2-D profile curve

q(s) = (a(s), b(s))

Construct a space curve

p(t) = (x(t), y(t), z(t))

Construct a Frenet frame at each

point along space curve T(t) = p(t+∆t) – p(t) B(t) = T(t) × -T(t – ∆t) N(t) = B(t) × T(t) (all normalized)

Plot 2-D curve in (N,B) space

gc(s,t) = p(t) + a(s) N(t) + b(s) B(t)

p(t) p(t+∆t) p(t–∆t) T(t)

T(t–∆t)

SLIDE 6

Generalized Cylinder

Construct a 2-D profile curve

q(t) = (a(t), b(t)) t a b q(t)

SLIDE 7

Generalized Cylinder

Construct a 2-D profile curve

q(t) = (a(t), b(t))

Construct a space curve

p(s) = (x(s), y(s), z(s))

SLIDE 8

Generalized Cylinder

Construct a Frenet frame at each

point along space curve t(s) = p(s + ∆s) – p(s) b(s) = t(s) × -t(s – ∆s) n(s) = b(s) × t(s) (all normalized)

SLIDE 9

Generalized Cylinder

Construct a 2-D profile curve

q(t) = (a(t), b(t))

Construct a space curve

p(s) = (x(s), y(s), z(s))

Plot 2-D curve in (N,B) space

gc(s,t) = p(s) + a(t) n(s) + b(t) b(s)

SLIDE 10

Generalized Cylinder

Construct a 2-D profile curve

q(t) = (a(t), b(t))

Construct a space curve

p(s) = (x(s), y(s), z(s))

Plot 2-D curve in (N,B) space

gc(s,t) = p(s) + a(t) n(s) + b(t) b(s)

SLIDE 11

Revolution

Construct a 2-D profile curve

q(t) = (a(t), b(t))

Rotate about y axis

p(s,t) = (a(t) cos 2π s, b(t), a(t) sin 2π s) z x y

SLIDE 12

Bezier Patches

Bezier patch

– Tensor product of two Bezier curves

∑∑

= =

=

n j n i ij n i n j

t B s B t s p

1 1

) ( ) ( ) , ( p p00 p01 p02 p03 p10 p11 p12 p13 p20 p21 p22 p23 p30 p31 p32 p33

SLIDE 13

Bezier Patches

Bezier patch

– Tensor product of two Bezier curves – Product of Bernstein polynomials

∑∑

= =

=

n j n i ij n i n j

t B s B t s p

1 1

) ( ) ( ) , ( p

( )

∑∑

= =

=

n j n i ij n i n j

t B s B t s p

1 1

) ( ) ( ) , ( p p00 p01 p02 p03 p10 p11 p12 p13 p20 p21 p22 p23 p30 p31 p32 p33

SLIDE 14

Bezier Patches

Bezier patch

– Tensor product of two Bezier curves – Product of Bernstein polynomials – Bernstein interpolation of Bernstein polynomials

∑∑

= =

=

n j n i ij n i n j

t B s B t s p

1 1

) ( ) ( ) , ( p

( )

∑∑

= =

=

n j n i ij n i n j

t B s B t s p

1 1

) ( ) ( ) , ( p

( )

∑ ∑

= =

      =

n j j i n i n i n j

t B s B t s p

1 1

) ( ) ( ) , ( p p00 p01 p02 p03 p10 p11 p12 p13 p20 p21 p22 p23 p30 p31 p32 p33

SLIDE 15

Bezier Patches

Bezier patch

– Tensor product of two Bezier curves – Product of Bernstein polynomials – Bernstein interpolation of Bernstein polynomials

Works same way for B-splines

∑∑

= =

=

n j n i ij n i n j

t B s B t s p

1 1

) ( ) ( ) , ( p

( )

∑∑

= =

=

n j n i ij n i n j

t B s B t s p

1 1

) ( ) ( ) , ( p

( )

∑ ∑

= =

      =

n j j i n i n i n j

t B s B t s p

1 1

) ( ) ( ) , ( p p00 p01 p02 p03 p10 p11 p12 p13 p20 p21 p22 p23 p30 p31 p32 p33

SLIDE 16

Blossoming Patches

Curves: p(t)  p(t,t,t)
Patches: p(s,t)  p(s,s,s;t,t,t)
Variables not allowed to cross

the semicolon

In patches, bilinear

interpolation replaces linear interpolation in curves

p(000;000) p(001;000) p(011;000) p(111;000) p(000;001) p(000;011) p(000;111) p(001;001)p(011;001) p(111;001) p(001;011) p(011;011) p(111;011) p(001;011) p(011;111) p(111;111)

SLIDE 17

Blossoming Patches

Curves: p(t)  p(t,t,t)
Patches: p(s,t)  p(s,s,s;t,t,t)
Variables not allowed to cross

the semicolon

In patches, bilinear

interpolation replaces linear interpolation in curves

p(00s;00t) p(01s;00t) p(11s;00t) p(00s;01t) p(01s;01t) p(11s;01t) p(00s;11t) p(01s;11t) p(11s;11t)