Computer Graphics

Curves and Surfaces

Aleksandra Pizurica

Ghent University

Telecommunications and Information Processing

Image Processing and Interpretation Group

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'272$:&*+71$

BE:)*-*7$L(:/(1(+729&+$

G&17$32%*)*2/$3&/%$&3$-./0($*+$Z>$

y = f (x)

$

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! $[(/9-2)$)*+(1$

! $W*/-)(1$

$

BE7(+1*&+$7&$=>$$

!  y = f (x), z = g (x) '(\+(1$2$-./0($*+$=>$

•  (S,S$(].29&+1$y = ax + b; z = cx + d '(1-/*@($2$)*+($*+$=>D$@.7$-2++&7$

/(:/(1(+7$2$)*+($*+$2$:)2+($5*7"$-&+172+7$x

! $H"($3&/%$z = f (x,y) '(\+(1$2$1./32-($<@.7$-2++&7$/(:/(1(+7$2$1:"(/(?$

x

y

x

y

z

L09_5

I%:)*-*7$L(:/(1(+729&+$$

H5&$'*%(+1*&+2)$-./0($*+$*%:)*-*7$3&/%R$

f (x,y) = 0

G&17$-./0(1$2+'$1./32-(1$7"27$5($5&/6$5*7"$"20($*%:)*-*7$3&/%1R$

! $T))$)*+(1$$ax + by + c = 0

! $W*/-)(1$$$$x2 + y2 - r2 = 0

I+$7"/(($'*%(+1*&+1$$f (x,y,z) = 0 '(\+(1$2$1./32-($

! $(S,SD$2+#$:)2+(R$ax + by + cz +d = 0D$5*7"$-&+172+71$a, b, c 2+' d

! $2$1:"(/($-(+7(/('$27$&/*,*+D$5*7"$/2'*.1$rR$$$$$x2 + y2 + z2 - r2 = 0$

W./0(1$*+$=>$2/($+&7$1&$(21*)#$/(:/(1(+7('$*+$*%:)*-*7$3&/%S$$

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kji zyxzyxf

2≤++ kji

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*+'(:(+'(+7$02/*2@)(D$-2))('$7"($:2/2%(7(/R$

x = x(u),

y = y(u),

z = z(u)

p(u)=[x(u), y(u), z(u)]T

p(u)

p(umin)

p(umax)

dp(u)

du=dx(u)

du

dy(u)

du

dz(u)

du

T;*0(1$7"($0()&-*7#$5*7"$5"*-"$7"($

-./0($*1$7/2-('$&.7$2+'$:&*+71$*+$

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M2/2%(7/*-$_*+(1$

_*+($-&++(-9+,$75&$:&*+71$p0$$2+'$p1$$

p(u)=(1-u)p0+up1

!($-2+$+&/%2)*Q($u$7&$@($&0(/$7"($*+7(/02)$<YDX?$

p(0) = p0

p(1)= p1

L2#$3/&%$p0$*+$7"($'*/(-9&+$d

p(u)=p0+ud

p(0) = p0

p(1)= p0 +d

d

L09_9

M2/2%(7/*-$U./32-(1$

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x = x(u,v),

y = y(u,v),

z = z(u,v)

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p(u,v) = [x(u,v), y(u,v), z(u,v)]T$

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x

y

z p(u,0)

p(1,v) p(0,v)

p(u,1)

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1./32-($

;

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∂∂

∂∂

∂∂

=∂

∂

uvuz

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uvux

u

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∂∂

∂∂

∂∂

=∂

∂

vvuz

vvuy

vvux

v

vu

/),(

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v

vu

u

vu

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∂

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M2/2%(7/*-$M)2+(1$

p(u,v)=p0+u q+v r

q

r

p0

n

p0

n

p1

p2

q = p1 – p0

r = p2 – p0

M&*+7a0(-7&/$3&/%$

H"/((a:&*+7$3&/%$

rqn ×=

rqn ×=

L09_12

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W&+172+7 φ R$-*/-)(1$&3$-&+172+7$)297.'($

M2/2%(7/*-$U:"(/($

x (θ, φ) = ρ cos θ sin φ

y (θ, φ) = ρ sin θ sin φ

z (θ, φ) = ρ cos φ

360 ≥ θ ≥ 0

180 ≥ φ ≥ 0

H"($:&*+71$&+$2$1:"(/($5*7"$/2'*.1$ρ$

>*`(/(+927($7&$1"&5$$n = p

L09_13

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p(u)

q(u) p(0)

q(1)

A&*+$:&*+7$p(1) = q(0)

L09_14

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,)(0

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kk∑

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2+2)#9-2))#$

p(u)

q(u)

A&*+$:&*+7$p(1) = q(0)

@.7 p’(1) ≠ q’(0)

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•  N = M = L = 3D$*1$-&+1*'(/('$21$2$,&&'$@2)2+-($@(75((+$(21($&3$

(02).29&+$2+'$8(E*@*)*7#$*+$'(1*,+$

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L09_18

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kk∑

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p(u,v) = [x(u,v), y(u,v), z(u,v)]T$D$5"(/($

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ij∑∑= =

=3

0

3

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p *1$2+#$&3$xD y 2+' z.

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p0

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p3

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"$$ ($%.17$\+'$c0 ,c1 ,c2 , c3

L09_22

=

k

k

k

k

z

y

x

p

I+7(/:&)29&+$B].29&+1$

p0=p(0)=c0

p1=p(1/3)=c0+(1/3)c1+(1/3)2c2+(1/3)3c3

p2=p(2/3)=c0+(2/3)c1+(2/3)2c2+(2/3)3c3

p3=p(1)=c0+c1+c2+c3

&/$*+$%27/*E$3&/%$5*7"$p = [p0 p1 p2 p3]T$5($"20($p=Ac!5*7"$

$

=

1111

3

2

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0001

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(].2))#$1:2-('S$U*+-($5($"20($-"&1(+$u$7&$02/#$&0(/$lYDXmD$5($2::)#$

*+7(/:&)29+,$-&+'*9&+1$27$u = 0, 1/3, 2/3, 1

L09_23

I+7(/:&)29&+$G27/*E$

U&)0*+,$3&/$c$5($\+'$7"($*+7(/:&)29&+$%27/*E$

−−

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5.45.135.135.4

5.4185.229

15.495.5

0001

1

AMI

c=MI p

f&7($7"27$MI $'&(1$+&7$'(:(+'$&+$*+:.7$'272$2+'$-2+$@($.1('$3&/$(2-"$

1(,%(+7$*+$x, y 2+' z.$

2+'$5($&@72*+$7"($-&(C-*(+71$c$21$$

L09_24

I+7(/:&)29+,$G.)9:)($U(,%(+71$

.1( p = [p0 p1 p2 p3]T

.1( p = [p3 p4 p5 p6]T

($2-"*(0($-&+9+.*7#$27$A&*+$:&*+71$@.7$+&7$

-&+9+.*7#$&3$'(/*0290(1$$

L09_25

O)(+'*+,$g.+-9&+1$

($-2+$/(5/*7($7"($(].29&+$3&/$p$21$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$p(u)=uTc=uTMIp = b(u)Tp

5"(/($ *1$2$-&).%+$%27/*E$&3$

3&./$@)(+'*+,$:&)#+&%*2)1D$1.-"$7"27$

$

H"(1($@)(+'*+,$3.+-9&+1$3&/$7"($-.@*-$*+7(/:&)29+,$:&)#+&%*2)$2/($$

b0(u) = -4.5(u-1/3)(u-2/3)(u-1)

b1(u) = 13.5u (u-2/3)(u-1)

b2(u) = -13.5u (u-1/3)(u-1)

b3(u) = 4.5u (u-1/3)(u-2/3)

$

uMbT

I

Tububububu == )]()()()([)( 3210

∑=

=+++=3

033221100 )()()()()()(

i

iiububububu b u pppppp

L09_26

O)(+'*+,$g.+-9&+1D$-&+7'S$

H"(1($3.+-9&+1$2/($+&7$1%&&7"$

! N(+-($7"($*+7(/:&)29&+$:&)#+&%*2)$*1$+&7$1%&&7"$

H"($)2-6$&3$1%&&7"+(11$*1$7"($-&+1(].(+-($&3$7"($*+7(/:&)29+,$

/(].*/(%(+7$7"27$7"($-./0($%.17$:211$7"/&.,"$7"($-&+7/&)$:&*+71$

</27"(/$7"2+$A.17$-&%($-)&1($7&$7"(%?$

L09_27

I+7(/:&)29+,$M27-"$

I+7(/:&)29+,$:27-"$*1$2$+27./2)$(E7(+1*&+$&3$7"($*+7(/:&)29+,$-./0(S$$

T$@*-.@*-$1./32-($:27-"$-2+$@($5/*d(+$21$

$

"(/($cij$*1$2$7"/((a()(%(+7$-&).%+$%27/*E$&3$7"($x, y$2+'$z

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($+(('$16$-&+'*9&+1$7&$'(7(/%*+($7"($16$-&(C-*(+71$cijS$W"&&1($27$$$

u,v = 0, 1/3, 2/3, 1

$$

$

$

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∑∑==

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0

3

0

),(j

ji

ji

i

vuvu cp

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16 7"/((a()(%(+7$0(-7&/1$

L09_28

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>(\+(R$$v = [1 v v2 v3]T , C = [cij] P = [pij]

($-2+$/(5/*7($7"($(E:/(11*&+$3&/$p(u,v) 21$

$

$

I3$5($&@1(/0($7"27$3&/$-&+172+7$u$<v?D$5($&@72*+$*+7(/:&)29+,$-./0($*+$$$$$

v$<u?D$5($-2+$1"&5$7"27$

p(u,v) = uTCv

p(u,v) = uTMIPMITv

C = MIPMIT

=>$-&+7/&)$:&*+71$

L09_29

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$

($-2+$@.*)'$2+'$2+2)#Q($1./32-(1$$3/&%$&./$6+&5)(',($&3$-./0(1$

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@($5/*d(+$21$

$

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5&/6$5*7"$3.+-9&+1$&3$u$2+' v *+'(:(+'(+7)#S$

O)(+'*+,$M27-"(1$

)()(),( vuvu gfp =

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