Representation%20and%20Manipulation%20of%20Curves

1

• Representation and Manipulation of Curves

2
Representation and Manipulation of Curves

• ???p?? ??apa??stas??
• ?) ???eß???? ??apa??stas?
• ???f? yf(x) ? g(x,y)0
• ?) ?a?aµet???? ??apa??stas?
• ???f? P(t)(x(t),y(t),z(t))

3
Representation of Curves

• ??e????t?µa pa?aµet????? ??a?t? a??eß?????
  µ??f??
• ?? s??teta?µ??e? t?? s?µe??? t?? ?aµp???? e??a?
  a?e???t?te? µeta?? t??? ?a? e?a?t??ta? µ??? ap?
  µ?a pa??µet?? gt µp????? ?a efa?µ?st??? ?? affine
  µetas??µat?sµ??, ?p?? µetaf???, st??f?, st??ß??s?
  ?a? a??a?? ???µa?a?.
• ?? s??µa t?? pa?aµet????? ?aµp???? e?a?t?ta? µ???
  ap? t? ?e?µet??a t?? s?µe??? ?a? ??? ap? t??
  s??teta?µ??e? t??? se ??p??? s?st?µa
  s??teta?µ????.

4
Representation of Curves

• Ge???? e??s?s? pa?aµet????? ?aµp????
• P(t)Ski0Pibi(t)
• ?p?? Pi ta s?µe?a e?????? t?? ?aµp???? ?a?
• bi(t) ? p??????µ??? ß?s?
• ??a ?aµp??? 3?? ßa?µ?? ?a? ta s?µe?a e?????? t??
  (e?????ta? ??a ?a s??µat?s??? t? p???????
  e??????) e??a?

.
.
.
.
5
Representation of Curves

• Ge????? ?d??t?te? ?a?aµet????? ?aµp????
• Affine Invariance
• Convex Hull
• ??p???? pa?aµet????? ?aµp??e? e??a? ?? Hermite,
  Bezier, B-spline ?a? NURBS.

6
Representation of Curves

• HERMITE ???????S
• Ge???? ???f?
• P(u)P0(1-3u22u3)P1(3u2-2u3)P0(u-2u2u3)P1(-
  u2u3)
• ???f? 3?? ßa?µ?? P(u)a0a1ua2u2a3u3
• ?a ai e??a? se a??eß???? µ??f? gt µetat??p? se
  ?e?µet????
• ?p?te P(u)1-3u22u3 3u2-2u3 u-2u2u3
  -u2u3P0 P1 P0 P1T

P0
.
.
P1
P0
P1
7
Representation of Curves

• BEZIER ???????S
• Ge???? ???f?
• Pn(u)Sni0?i,n(t)Pi, t ap? 0 ?? 1
• ?p?? ?i,n(t)(ni)ti(1-t)n-1
• ? ?aµp??? Bezier 3?? ßa?µ?? e??a? ?
• P(t)(1-t)3P03t(1-t)2P13t2(1-t)P2t3P3

P1
.
.
P2
.
.
P3
P0
8
Representation of Curves

• BEZIER ???????S
• ? ?aµp??? Bezier 3?? ßa?µ?? µp??e? ?a ??afe? se
  µ??f? p??????.
• -1 3 -3 1
• 3 -6 3 0
• -3 3 0 0
• 1 0 0 0
• ???t? ?a???????
• P(t)nSn-1i0?i,n(t)(Pi1-Pi)

P0 P1 P2 P3
P(t)t3 t2 t 1
TMbGb
9
Representation of Curves

• BEZIER ???????S
• ???????µ?? de Casteljau
• T?t??ta? P0i(t)Pi, i0,,n t?te ta e?d??µesa
  s?µe?a p?? p????pt??? e??a?
• Pri(t)(1-t)Pr-1i(t)tPr-1i1(t)
• Se t???????? d??ta?? ??a ??ß??? ?aµp??? Bezier
  ????µe

1-t
P0P00 -----gtP01 -----gtP02 -----gtP03P3(t) P1P01
-----gtP11 -----gtP21 P2P02 -----gtP12 P3P03
t
10
Representation of Curves

• BEZIER ???????S
• ?d??t?te? ?aµp???? Bezier
• Convex Hull
• ??a?????t? se affine µetas??µat?sµ???
• S?µµet??a
• G?aµµ??? ????ße?a
• ?a?eµß??? ???a??? S?µe???
• ?a?µ?? ?aµp???? ?????? S?µe??? ??????? 1
• ?e?te?e? ?a??????? sta ???a
• ?e?d?-??p???? ??e????

11
Representation of Curves

• BEZIER ???????S
• ?a?µ?t?? ?aµp??e? Bezier
• Bezier µe wi ß???? e?????? t?? s?µe??? Pi.
• ???f? P(u)
• ?etaß?????ta? t? wi µetaß?????µe t?? ?aµp???.
  S???e???µ??a
• ?e???? ß???? s??e? t?? ?aµp??? p??? t? s?µe??
  e?????? ?a? a?t?st??fa
• ?????0 gt a???e? ? ?aµp??? t? s?µe?? e??????
• ?????lt0 gt ? ?aµp??? ap?µa????eta? ap? t? s?µe??
  e??????
• ????? ?pe??? gt ? ?aµp??? s?µp?pte? µe t?
  p??????? e??????

Sni0Bni(u)wiPi)
Sni0Bni(u)wi)
12
Representation of Curves

• B-SPLINE ???????S
• ???f?
• P(u)Sni0PiNi,k(u), tk-1ltulttn1
• ?p?? Ni,k(u)((u-ti)Ni,k-1(u))/(tik-1-ti)((tik
  -u)Ni1,k-1(u))/(tik-ti1)
• 1, tiltultti1
• ?a? Ni,1(u)
• 0, a?????
• ? ?aµp??? µeta?? Pi ?a? Pi1 e??a?

Pi-1 Pi Pi1 Pi2

• -1 3 -3 1
• -6 3 0
• -3 0 3 0
• 1 4 1 0

P(t)t3 t2 t 1
TMhGh
13
Representation of Curves

• B-SPLINE ???????S
• S???es?
• P1(u)(1-u)2 Pou(1-u)(2-u)u/2 P1(u2/2)P2,
  0ltult1
• P2(u)((2-u)2/2) P1(u(2-u)/2)((3-u)(u-1)/2)P
  2((u-1)2/2)P3, 1ltult2
• P3(u)((3-u)2/2)P21/2(-2u210u-11)P3((u-2)2/2
  )P4, 2ltult3
• P4(u)((4-u)2/2)P31/2(-3u220u-32)P4(u-3)2P5
  , 3ltult4

P3
.
P5
P0
.
.
P3(u)
.
.
P4(u)
P1(u)
P2(u)
u2
u3
.
.
.
.
P1
u1
P2
P4
14
Representation of Curves

• B-SPLINE ???????S
• ??µß?? (Knots)
• Periodic
• tii-k, 0ltiltnk
• Nonperiodic
• 0, 0ltiltk
• ti i-k1, kltiltn
• n-k2, nltiltnk
• Cox-de Boor a??????µ??
• ?pa?a??pt??? µ???d?? ??a ?p?????sµ? s?µe??? t??
  ?aµp????.
• Ge????? t?p?? P(u)Slil-kr-1PriNi,k-r(u)
  ?p??
• PriBoor point((u-ti)/(tik-r-ti))Pr-1i(1-(u-ti
  )/(tik-r-ti))Pr-1i-1

15
Representation of Curves

• B-SPLINE ???????S
• ?a???????
• d/duP(u)Slil-k2 P1i Ni,k-1(u) ,
  tlltulttl1
• ?p?? P1i(k-1)(Pi-Pi-1)/(tik-1- ti)
• ?d??t?te?
• ??p???? ??e????
• Convex Hull
• G?aµµ??? ????ße?a
• ?a?eµß??? ???a??? S?µe???
• S????e?a t???? 2
• ??a?????t? se affine µetas??µat?sµ???

16
Representation of Curves

• NURBS ???????S
• NURBSNon-Uniform Rational B-Splines
• ???f?
• P(u)(Sni0PiNi,k(u)hi)/(Sni0Ni,k(u)hi),
  tk-1ltulttn1
• µe
• xh Sni0Ni,k(u)(xihi)
• yh Sni0Ni,k(u)(yihi)
• zh Sni0Ni,k(u)(zihi)
• h Sni0Ni,k(u)hi

17
Representation of Curves

• NURBS ???????S
• ?a???????
• d/duP(u)
• d/duSni0PiNi,k(u)hi)Sni0Ni,k(u)hi-Sni0Pi
  Ni,k(u)hi)d/duSni0Ni,k(u)hi /
  (Sni0Ni,k(u)hi)2)
• S???es?
• ?d?a µ???d?? µe t?? B-spline.

18
Manipulation of Curves

• Intersection
• Interpolation
• ?a?eµß??? µe Hermite ?aµp???
• ?a?eµß??? µe B-spline ?aµp???

19
Manipulation of Curves

• INTERSECTION
• ?st? P(u), Q(v) d?? ?p??esd?p?te pa?aµet?????
  ?aµp??e?.
• ?? pa?aµet????? t?µ?? p?? a?t?st?????? sta s?µe?a
  d?ast????s?? ??????ta? ap?
• P(u) - Q(v)0
• ????µe t?e?? e??s?se?? (??a x,y,z) ?a? d??
  a???st??? (u,v). ?p??????µe ta x,y
• Px(u)-Qx(v)0
• Py(u)-Qy(v)0
• ?????µe ?? p??? u, v (???s? µ?a? a???µ?t????
  µe??d?? ?p?? Newton Raphson) ?a? ???s?µ?p????µe
  t?? e??s?s? ?? p??? z ??a epa???e?s? t?? u, v.

20
Manipulation of Curves

• INTERPOLATION
• Interpolation Using Hermite Curve
• ?st? ta P0,,Pn s?µe?a ?a? ?????µe ?a ß???µe t??
  n Hermite ?aµp??e?
• P1(u),,Pn(u).
• Pi(u)Pi-1Pi-1u3(Pi-Pi-1)-2Pi-1-Piu22(Pi-
  1-Pi)Pi-1Piu3, 0ltult1
• G?a ?a ????µe s????e?a 2?? ßa?µ?? p??pe?
• d2Pi(u)/du2u1 d2Pi1(u)/du2u0 gt
  Pi-14PiPi13Pi1-3Pi-1, i1,,n1
• ??t??a??st??ta? t?? t?µ?? t?? I ap? 1 ?? n-1
  pa?????µe t?? e??? p??a?a

21
Manipulation of Curves

• INTERPOLATION
• Interpolation Using Hermite Curve
• ???es? P0,Pn
• Clamped-end condition
• Free-end condition

3P2-3P0-P0 3P3-3P1 3Pn-3P2 .
. 3Pn-3Pn-2-Pn
P1 . . Pn-1
4 1 1 4 1 1 4 1 1 4
0
.
.
.

0
22
Manipulation of Curves

• INTERPOLATION
• Interpolation Using Hermite Curve
• Free-end condition
• ?st? P0Pn0.
• ??te
• d2P1(u)/du2u0 -3P03P1-2P0-P1
• d2Pn(u)/du2u1 23(Pn-Pn-1)-2Pn-1-Pn62(Pn
  -1-Pn)Pn-1Pn0
• ?
• 2P0P13P1-3P0 ?a? 2PnPn13Pn-3Pn-1
  
• ? e??s?s? ???eta?

3P1-3P0 3P2-3P0 3P3-3P1 .
. .
P0 . . Pn
2 1 1 4 1 1 4 1 1 2
0
.
.
.

0
23
Manipulation of Curves

• INTERPOLATION
• Interpolation Using B-spline Curve
• ?st? ta Q0,,Qn s?µe?a.
• ?a???????µe t? p????? t?? s?µe??? e??????
  (s?????? 4).
• ?a???????µe t?? t?µ?? t?? knots (nk-1).
• ti0 , i0,,k-1
• ti1(Si-2ji-kdj)/(Sn1mkSm-2jm-kdj) ,
  ik,,n
• ti1 ,in1,,nk
• djsqrt(abs(Qj1-Qj))

24
Manipulation of Curves

• INTERPOLATION
• Interpolation Using B-spline Curve
• S?????? QjSni0PiNi,k(uj), j0,,n (?)
• µe
• uj(tj1tj2tjk-1)/(k-1), j0,,n (?)
• ?p?te

P0 . . Pn
. Ni,k(u0) . . . Ni,k(u0) . . . Ni,k(u0) . . .
Ni,k(u0) . .
Q0 . . Qn