Moving Least Squares Coordinates. Josiah Manson and Scott Schaefer Texas A&M University

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1 Movg Leas Squares Coordaes Josah Maso ad Sco Schaefer Teas A&M Uversy

2 Barycerc Coordaes olygo Doma p 4 p 3 p 2 p 0 p

3 Barycerc Coordaes olygo Doma

4 Barycerc Coordaes olygo Doma f 4 f 3 f 2 f 0 f

5 Barycerc Coordaes olygo Doma

6 Barycerc Coordaes olygo Doma Boudary Ierpolao ˆ

7 Barycerc Coordaes olygo Doma Boudary Ierpolao ˆ

8 Barycerc Coordaes olygo Doma Boudary Ierpolao Bass ucos f 4 f 3 f 2 ˆ b f f 0 f

9 Barycerc Coordaes olygo Doma Boudary Ierpolao Bass ucos ˆ b f

10 Barycerc Coordaes olygo Doma Boudary Ierpolao Bass ucos ˆ b f

11 Barycerc Coordaes olygo Doma Boudary Ierpolao Bass ucos ˆ b f

12 Barycerc Coordaes olygo Doma Boudary Ierpolao Bass ucos ˆ b f

13 Barycerc Coordaes olygo Doma Boudary Ierpolao Bass ucos ˆ b f

14 Barycerc Coordaes olygo Doma Boudary Ierpolao Bass ucos Lear recso L b L p

15 Barycerc Coordaes olygo Doma Boudary Ierpolao Bass ucos Lear recso L b L p

16 Barycerc Coordaes olygo Doma Boudary Ierpolao Bass ucos Lear recso b

17 Oher roperes Desrable eaures Smoohess Closed-form soluo osvy Eeded Coordaes olyomal Boudary alues olyomal recso Ierpolao of Dervaves Curved Boudares

18 Applcaos e Eleme Mehods [Wachspress 975]

19 Applcaos Boudary alue roblems [Ju e al. 2005]

20 Applcaos ree-orm Deformaos [Sederberg e al. 986], [MacCracke e al. 996], [Ju e al. 2005], [Josh e al. 2007]

21 Applcaos Surface arameerzao [Horma e al. 2000], [Desbru e al. 2002]

22 Comparso of Mehods Wachspress Mea al. os. Mea al. Ma Eropy Harmoc Herme MC Movg Leas Sqr.

23 Movg Leas Squares Coordaes A ew famly of barycerc coordaes Solves a leas squares problem Soluo depeds o po of evaluao

24 a olyomal o os argm C 2 2 p C p ˆ C

25 a olyomal o os

26 a olyomal o os

27 Ierpolag os

28 Ierpolag os 2, p p W 2, argm C p C p p W C ˆ

29 Ierpolag os

30 Ierpolag Le Segmes,,2

31 Ierpolag Le Segmes,2,,2,

32 Ierpolag Le Segmes,2, 2 ', W,2,

33 Ierpolag Le Segmes,2, d C W C 2 0, argm 2 ', W,2,

34 Le Bass ucos d W A T 0, d W A C T 0,

35 Le Bass ucos ˆ C

36 Le Bass ucos d W A T 0, C ˆ

37 Le Bass ucos d W A T 0, C ˆ d W A T,2, 0,

38 Le Bass ucos d W A T 0, C ˆ d W A T,2, 0,,2, 0, T d W A

39 Le Bass ucos d W A T 0, C ˆ d W A T,2, 0,,2, 0, T d W A,2,,2, B B

40 olygo Bass ucos ˆ, B, B,2,2

41 olygo Bass ucos ˆ, B, B,2,2 f,,2 b B, B, 2

42 olygo Bass ucos

43 olygo Bass ucos

44 olygo Bass ucos ˆ, B, B,2,2 f,,2 b B, B, 2 ˆ b f

45 olyomal Boudary alues,2,,3,2, 2 2 2,4,3,2,

46 olyomal Boudary alues

47 olyomal Boudary alues

48 olyomal recso

49 olyomal recso Lear Quadrac

50 Ierpolao of Dervaves argm W, C d C 0 2

51 Ierpolao of Dervaves d C G W 2 0,, d C W C 2 0, argm

52 Ierpolao of Dervaves d C G W 2 0,, d C W C 2 0, argm, 2 G

53 Ierpolao of Dervaves

54 Ierpolao of Dervaves

55 Ierpolao of Dervaves

56 Soluos are Closed-orm or polygos s lear ' ad are cosa olyomal umeraor Deomaor quadrac o power 2α Iegrals have closed-form soluos

57 Curved Boudares,2,,3,2, 2 2 2,4,3,2,

58 Comparso o Oher Mehods

59 Comparso o Oher Mehods

60 b 3D Deformao p

61 3D Deformao b p ˆ b ˆ p

62 Cocluso New famly of barycerc coordaes Corolled by parameer α olyomal boudares olyomal precso Dervave erpolao Ope polygos Closed-form