computer graphics representing curves and surfaces
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Computer Graphics
Representing Curves and Surfaces
Revieweq(11.5)
zzzz
yyyy
xxxx
dtctbtatz
dtctbtaty
dtctbtatx
t
tztytxtQ
23
23
23
)(
)(
)(
10
)()()()(
Review eq(11.6/11.7)
zyx
zyx
zyx
zyx
ddd
ccc
bbb
aaa
ttt
CT
tztytxtQ
1
)()()()(
23
Review eq(11.8)
zzz
yyy
xxx
ctbtatz
ctbtaty
ctbtatx
tztytx
tzdt
dty
dt
dtx
dt
dtQtQ
dt
d
23)('
23)('
23)('
)(')(')('
)()()()(')(
2
2
2
Review eq(11.8)
zzzyyyxxx ctbtactbtactbta
CttCTdt
d
tzdt
dty
dt
dtx
dt
dtQtQ
dt
d
232323
0123
)()()()(')(
222
2
Review eq(11.9)
zyx
zyx
zyx
zyx
ggg
ggg
ggg
ggg
G
G
G
G
G
G
G
G
mmmm
mmmm
mmmm
mmmm
ttt
GMTtztytxtQ
444
333
222
111
4
3
2
1
4
3
2
1
44434241
34333231
24232221
14131211
23 1
)()()()(
Review eq(11.10)
z
z
z
z
y
y
y
y
x
x
x
x
gmtmtmtm
gmtmtmtm
gmtmtmtm
gmtmtmtmtz
gmtmtmtm
gmtmtmtm
gmtmtmtm
gmtmtmtmty
gmtmtmtm
gmtmtmtm
gmtmtmtm
gmtmtmtmtx
444342
243
14
343332
233
13
242322
223
12
141312
213
11
444342
243
14
343332
233
13
242322
223
12
141312
213
11
444342
243
14
343332
233
13
242322
223
12
141312
213
11
)(
)(
)(
)()(
)(
)(
)(
)()(
)(
)(
)(
)()(
Review
Blending function (also called ‘Basis’ function)
MTB
Hermite Curves
以曲線端點 P1.P4以及端點斜率 R1.R4
求曲線方程式
Hermite Curveseq(11.12)
zyx
zyx
zyx
zyx
H
RRR
RRR
PPP
PPP
R
R
P
P
G
GMTtztytxtQ
444
111
444
111
4
1
4
1
)()()()(
Hermite Curveseq(11.13)
zz
yy
xx
HHHH
zzzz
HHHH
yyyy
HHHH
xxxx
HH
GMtttGMT
dtctbtatz
GMtttGMT
dtctbtaty
GMtttGMT
dtctbtatx
GMTtztytxtQ
1
)(
1
)(
1
)(
)()()()(
23
23
23
23
23
23
Hermite Curveseq(11.14)
HH
HHz
HHy
HHx
GMPQ
GMPz
GMPy
GMPx
z
y
x
1000)0(
1000)0(
1000)0(
1000)0(
1
1
1
1
Hermite Curveseq(11.15)
HH
HHz
HHy
HHx
GMPQ
GMPz
GMPy
GMPx
z
y
x
1111)1(
1111)1(
1111)1(
1111)1(
4
4
4
4
Hermite Curves
zz
yy
xx
HHHH
zzz
HHHH
yyy
HHHH
xxx
HH
GMttGMTdt
d
ctbtatz
GMttGMTdt
d
ctbtaty
GMttGMTdt
d
ctbtatx
GMTdt
dtztytxtQtQ
dt
d
0123
23)('
0123
23)('
0123
23)('
)(')(')(')(')(
2
2
2
2
2
2
Hermite Curveseq(11.16)
HH
HHz
HHy
HHx
GMRQ
GMRz
GMRy
GMRx
z
y
x
0100)0('
0100)0('
0100)0('
0100)0('
1
1
1
1
Hermite Curveseq(11.17)
HH
HHz
HHy
HHx
GMRQ
GMRz
GMRy
GMRx
z
y
x
0123)1('
0123)1('
0123)1('
0123)1('
4
4
4
4
Hermite Curveseq(11.18)
HHH GMG
R
R
P
P
0123
0100
1111
1000
4
1
4
1
Hermite Curveseq(11.19)
0001
0100
1233
1122
HM
Hermite Curveseq(11.20)
zyx
zyx
zyx
zyx
HH
RRR
RRR
PPP
PPP
ttt
GMTtztytxtQ
444
111
444
111
23
0001
0100
1233
1122
1
)()()()(
Hermite Curveseq(11.21)
423
123
423
123
4
1
4
1
23232323
23
)()2()32()132(
232132
0001
0100
1233
1122
1
)(
RttRtttPttPtt
R
R
P
P
ttttttttt
Gttt
GBGMTtQ
H
HHHH
Hermite Curve
23232323 232132 tttttttttBH
Hermite Curveeq(11.22)
1,
0,
,
1
1
7
4
7
4
2
4
1
4
1
1
kcontinuityC
kcontinuityG
R
kR
P
P
G
R
R
P
P
G HH
Hermite Curveeq(11.23)
Reduce 6 multiplies and 3 additions to 3 multiplies and 3 additions.
dtctbat
dctbtattf
))((
)( 23
Bezier Curves
以曲線端點 P1.P4以及控制點 P2.P3
求曲線方程式曲線端點斜率為
)(3)1('
)(3)0('
344
121
PPQR
PPQR
Bezier Curveseq(11.24)
3
)0,3(
)0,3(),0,2(),0,1(),0,0(
)(
)(
4114
4321
344
121
RRPP
PPPP
PPR
PPR
Bezier Curveseq(11.25)
4
3
2
1
P
P
P
P
GB
Bezier Curveseq(11.26)
BHBH GM
P
P
P
P
R
R
P
P
G
PPR
PPR
4
3
2
1
4
1
4
1
344
121
3300
0033
1000
0001
)(3
)(3
Bezier Curveseq(11.27/11.28)
0001
0033
0363
1331
3300
0033
1000
0001
0001
0100
1233
1122
)(
)(
)(
HBHB
BB
BHBH
BHBH
HH
MMM
GMT
GMMT
GMMT
GMTtQ
Bezier Curveseq(11.29)
43
32
22
13
444
333
222
111
23
)1(3)1(3)1(
0001
0033
0363
1331
1
)(
PtPttPttPt
PPP
PPP
PPP
PPP
ttt
GBGMTtQ
zyx
zyx
zyx
zyx
BBBB
Bezier Curves 3223
3232323
)1(3)1(3)1(
33363133
tttttt
tttttttttBB
Bezier Curves
Define n as the order of Bezier curves.Define i as control point.
inin
i
n
iini
inini
tuinC
PtBtQ
knk
nknC
ttinCtB
)1(),(
)()(
)!(!
!),(
)1(),()(
0
0,
,
Bezier Curves
33333,3
22323,2
21313,1
30303,0
)1(!0!3
!3)(
)1(3)1(!1!2
!3)(
)1(3)1(!2!1
!3)(
)1()1(!3!0
!3)(
ttttB
tttttB
tttttB
ttttB
Bezier Curves
BB GMT
P
P
P
P
ttt
P
P
P
P
tttttt
PtPttPttPttQ
0001
0033
0363
1331
1
)1(3)1(3)1(
)1(3)1(3)1()(
3
2
1
0
23
3
2
1
0
3223
33
22
12
03
Bezier Curves
De Casteljau iterations
],0[
],1[
)1( 11
1
jni
nj
tPPtP ji
ji
ji
Bezier Curves
Linear Bezier splinesControl points: P0, P1
10
)1()()( 10010
t
tPPttPPPtP
Bezier Curves
Quadratic Bezier splinesControl points: P0, P1, P2
10
)1()(
)1()(
211211
1
100101
0
t
tPPttPPPP
tPPttPPPP
Bezier Curves
Quadratic Bezier splinesControl points: P0, P1, P2
2
210
2
2110
11
10
10
11
10
)1(2)1(
)1()1()1(
)1(
)()(
PtPttPt
tPPtttPPtt
tPPt
tPPPtP
Bezier Curves
Cubic Bezier splinesControl points: P0, P1, P2, P3
322321
2
211211
1
100101
0
)1()(
)1()(
)1()(
tPPttPPPP
tPPttPPPP
tPPttPPPP
Bezier Curves
Cubic Bezier splinesControl points: P0, P1, P2, P3
3
221
2
3221
12
11
11
12
11
21
22
102
2110
11
10
10
11
10
20
)1(2)1(
)1()1()1(
)1()(
)1(2)1(
)1()1()1(
)1()(
PtPttPt
tPPtttPPtt
tPPttPPPP
PtPttPt
tPPtttPPtt
tPPttPPPP
Bezier Curves
Cubic Bezier splinesControl points: P0, P1, P2, P3
33
22
12
03
32
212
22
102
21
20
20
21
20
)1(3)1(3)1(
)1(2)1(
)1(2)1()1(
)1(
)()(
PtPttPttPt
PtPttPtt
PtPttPtt
tPPt
tPPPtP
Bezier Curves
BB
zyx
zyx
zyx
zyx
GMT
PPP
PPP
PPP
PPP
ttt
P
P
P
P
tttttt
PtPttPttPttP
0001
0033
0363
1331
1
)1(3)1(3)1(
)1(3)1(3)1()(
333
222
111
000
23
3
2
1
0
3223
33
22
12
03
Bezier Curves
http://www.ibiblio.org/e-notes/Splines/Bezier.htm
Spline
Natural cubic splineC0, C1, C2 continuous.Interpolates(passes through) the control points.Moving any one control point affects the entire curve.
Spline
B-splineLocal control.Moving a control point affects only a small part of a curve.Do not interpolate their control points.Sharing control points between segments.
B-spline
m+1 control points P0, …, Pm, m≥3m-2 curve segments Q3, Q4, …, Qm
For each i≥4, there is a join point or knot between Qi-1 and Qi at the parameter value ti.
B-splineeq(11.43/11.44)
)()(
)()()(
)()()(
,0)(
,1)(
4,4,11
4,224,33
1,11
1,1
,
1,
11,
tBPtBP
tBPtBPtQ
tBtt
tttB
tt
tttB
otherwisetB
ttttB
iiii
iiiii
diidi
didi
idi
idi
i
iii
B-splineeq(11.44)
)()()(
)()()(
)()()(
3,114
43,
34,
2,113
32,
23,
1,112
21,
12,
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
Uniform Nonrational B-spline
‘Uniform’ means that the knots are spaced at equal intervals of the parameter t.‘Nonrational’ is used to distinguish these splines from rational cubic polynomial curves, see Section 11.2.5We assume that t3=0 and the interval ti+1-ti=1
Uniform Nonrational B-spline
eq(11.32/11.33/11.34)
0141
0303
0363
1331
6
1
3,
,)(
1
2
3
1
Bs
i
i
i
i
B
iiBBSii
M
mi
P
P
P
P
G
tttGMTtQ
Si
Si
Uniform Nonrational B-spline
eq(11.35)
.10,1333463)1(6
1
13334631336
1
323233
3232323
123 0
tttttttt
ttttttttt
BBBBMTB BsBsBsBsBsBs
Uniform Nonrational B-spline
eq(11.36)
iiii
iBiBiBiBBsBs
BsBsii
Pt
Pttt
Ptt
Pt
PBPBPBPBGB
GMTtQ
SSSSi
i
66
1333
6
463
6
)1(
)(
3
1
23
2
23
3
3
123 0123
1 ii ttt
Uniform Nonrational B-spline
eq(11.37)
)()(
)()(
)()(
112
2
12
2
111
111
iiii
iiii
iiii
txdt
dtx
dt
d
txdt
dtx
dt
d
txtx
Uniform Nonrational B-spline
eq(11.38)
012
2
12
2
011
011
1
1
1
||
||
||
ii
ii
ii
ttitti
ttitti
ttitti
xdt
dx
dt
d
xdt
dx
dt
d
xx
Uniform Nonrational B-spline
iiii
iiiiii
iiiiii
iiiiii
tPPtPtPt
Pt
Pt
Pt
Pt
ttQ
Pt
Ptt
Ptt
Ptt
ttQ
Pt
Pttt
Ptt
Pttt
ttQ
123
123
2
1
2
2
2
3
2
3
1
23
2
23
3
32
)13()23()1( 6
6
6
618
6
1218
6
66)(''
6
3
6
369
6
129
6
363)('
66
1333
6
463
6
331)(
Uniform Nonrational B-spline
112
11211
1
22
1
2
2
2
11
1
323
1
23
2
32
11
)13()23()1( 6
6
6
618
6
1218
6
66)(''
6
3
6
369
6
129
6
363)('
66
1333
6
463
6
331)(
iiii
iiiiii
iiiiii
iiiiii
tPPtPtPt
Pt
Pt
Pt
Pt
ttQ
Pt
Ptt
Ptt
Ptt
ttQ
Pt
Pttt
Ptt
Pttt
ttQ
Uniform Nonrational B-spline
eq(11.39/11.40/11.41)
xxxii
xx
ii
xxx
ii
iiiiittitti
iiiittitti
iii
iittitti
PPPQQxdt
dx
dt
d
PPQQx
dt
dx
dt
d
PPPQQxx
121012
2
12
2
21011
12
1011
2)0()1(||
2)0()1(||
6
4)0()1(||
1
1
1
Uniform Nonrational B-spline
eq(11.42)The curve can be forced to be interpolate specific points by replicating control points.
iBsBsBsiBsi
iii
PBBBPBtQ
PPP
)()(
,
01233
12
Nonuniform Nonrational B-spline
Parameter interval between successive knot values need not be uniform.Blending functions are no longer the same for each interval.Continuity at selected join points can be reduced from C2 to C1 to C0 to none.
When the curve is C0, the curve interpolates a control point.
Nonuniform NonrationalB-spline
If the continuity is reduced to C0, then the curve interpolates a control point, but without the undesirable effect of uniform B-splines, where the curve segments on either side of the interpolated control point are straight lines.
Nonuniform NonrationalB-spline
m+1 control points P0, P1, …, Pm
nondecreasing sequence of knot valuest0, t1, …, tm+4
Nonuniform NonrationalB-spline
Nonuniform Rational Cubic Polynomial Curve
)()()()()(
)(
)()(,
)(
)()(,
)(
)()(
tWtZtYtXtQ
tW
tZtz
tW
tYty
tW
tXtx
Nonuniform Rational Cubic Polynomial CurveAdvantages
They are invariant under rotation, scaling, translation and perspective transformations of the control points.They can define precisely and of the conic sections.
Catmull-Rom spline
132
021
212
1
PPR
PPR
Catmull-Rom spline
0020
0101
1452
1331
21
1010
0101
0200
0020
0001
0100
1233
1122
21
2102
10
02102
10100
0010
4
3
2
1
4
1
4
1
HCHC
CHCH
MMM
GM
P
P
P
P
R
R
P
P
G
Catmull-Rom splineeq(11.47)
i
i
i
i
CCi
P
P
P
P
T
GMTQ
1
2
3
0020
0101
1452
1331
21
Subdividing Curves
Bezier Curve
2/)(
2/)(
2/)(
2/)(
2/)(
2/
2314
32
433
23
32
212
RLRL
RHR
PPR
HLL
PPH
PPL
Subdividing Curves
4
3
2
1
4
3
2
1
4
3
2
1
4
3
2
1
8000
4400
2420
1331
81
1331
0242
0044
0008
81
P
P
P
P
GD
R
R
R
R
G
P
P
P
P
GD
L
L
L
L
G
BRB
RB
BLB
LB
Subdividing Curves
Uniform B-spline CurveDoubling:Do k subdivision steps (for k-degree B-spline)
33221100 ,,,,,,, PPPPPPPP
2,
8
6,
2,
8
6,
2
4
3,
4
3,
4
3,
4
3,
4
3,
4
3
,2
,,2
,,2
,
323212121010
323221211010
332
221
110
0
PPPPPPPPPPPP
PPPPPPPPPPPP
PPP
PPP
PPP
P
Subdividing Curves
i
i
i
i
BSiSRBSi
RB
i
i
i
i
SiBSLBSi
LB
P
P
P
P
GDG
P
P
P
P
GDG
1
2
3
1
2
3
4400
1610
0440
0161
8
1
1610
0440
0161
0044
8
1
Conversioneq(11.56)
111
22221
2
1122
1122
GMMGGMM
GMGM
GMTGMT
Conversioneq(11.57/11.58)
BBBBsBs
BsBsBsBB
BsBsBB
GGMMG
GGMMG
GMTGMT
6720
0210
0120
0276
1410
0420
0240
0141
6
1
1
1
Drawing Curveseq(11.59/11.60/11.61)
)(,),(
)()()(
0),()()(
1 nnnnn tffntnfff
tftftf
tftftf
Drawing Curveseq(11.62/11.63)
cbbtaatat
dctbtatdtc
ttbttta
dctbtatdtctbta
tftftf
dctbtattf
2322
23
223223
2323
23
233
)(])(
)2()33([
)(])()()([
)()()(
)(
Drawing Curveseq(11.64/11.65/11.66/11.6
7)
12
1
12
232
2222
2322
2322
2
266
)23(3)2(3
])23(3[
])23)(()(3[
)()())(()(
nnn
nnn
fff
fff
baat
baattta
cbabatat
cbabatta
tftftftf
Drawing Curveseq(11.68/11.69/11.70)
32
21
2
32321
2
3
22
22
23
6
6
6
6)(6
)()(
))(()(
aff
affff
a
tata
tftf
tftf
nn
nnnn
Drawing Curveseq(11.71/11.72)
d
c
b
a
f
f
f
f
D
af
baf
cbaf
df
t
0006
0026
0
1000
6
26
0
3
23
23
03
02
0
0
30
3
230
2
230
0
Comparison
Parametric Bicubic Surfaceseq(11.73)
)(
)(
)(
)(
)(),(
)(
)(
4
3
2
1
tG
tG
tG
tG
MStGMStsQ
GMSsQ
GMTtQ
Parametric Bicubic Surfaces
TT
iiiiT
i
Tiiii
i
i
i
i
ii
TMggggtG
ggggMT
g
g
g
g
MTGMTtG
4321
4321
4
3
2
1
)(
)(
Parametric Bicubic Surfaces
eq(11.74/11.75)
.1,0,
),(
44434241
34333231
24232221
14131211
tsTMGMS
TM
gggg
gggg
gggg
gggg
MStsQ
TT
TT
Parametric Bicubic Surfaceseq(11.76)
TTz
TTy
TTx
TMGMStsz
TMGMStsy
TMGMStsx
),(
),(
),(
Hermite Surfaceseq(11.77)
x
H
HH
tR
tR
tP
tP
MS
tGMStsxx
)(
)(
)(
)(
)(),(
4
1
4
1
Hermite Surfaces
Hermite Surfaceseq(11.78)
x
Hx
x
Hx
x
Hx
x
Hx
g
g
g
g
MTtR
g
g
g
g
MTtR
g
g
g
g
MTtP
g
g
g
g
MTtP
44
43
42
41
4
34
33
32
31
1
24
23
22
21
4
14
13
12
11
1
)(,)(
)(,)(
Hermite Surfaceseq(11.79/11.80/11.81)
TTHH
TTH
x
x
H
THHx
TMGTM
gggg
gggg
gggg
gggg
tR
tR
tP
tP
gggg
gggg
gggg
gggg
G
GMTtRtRtPtP
x
x
x
44434241
34333231
24232221
14131211
4
1
4
1
44434241
34333231
24232221
14131211
4141
)(
)(
)(
)(
)()()()(
Hermite Surfaceseq(11.82/11.83)
TTHHH
TTHHH
TTHHH
TTHHH
TMGMStsQ
TMGMStsz
TMGMStsy
TMGMStsx
z
y
x
),(
),(
),(
),(
Hermite Surfaceseq(11.84)
)1,1()0,1()1,1()0,1(
)1,0()0,0()1,0()0,0(
)1,1()0,1()1,1()0,1(
)1,0()0,0()1,0()0,0(
22
22
xts
xts
xs
xs
xts
xts
xs
xs
xt
xt
xx
xt
xt
xx
GxH
Hermite Surfaces
0,2
1
44434241
24232221
44434241
24232221
kkgkgkgkg
gggg
Patch
gggg
ggggPatch
Normals to Surfaceseq(11.88/11.89)
TT
TT
TT
TT
TT
TT
ttMGMS
Tt
MGMS
TMGMSt
tsQt
TMGMss
TMGMSs
TMGMSs
tsQs
0123
)(
)(),(
0123
)(
)(),(
2
2
Normals to Surfaceseq(11.90)
sttssttsstts
ttt
sss
yxyxxzxzzyzytsQt
tsQs
zyxtsQt
zyxtsQs
),(),(
),,(),(
),,(),(
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