Explains newly found geometric features of Bezier curves and surfaces called "rib and fan. - Author: Joo-Haeng Lee - Affiliation: ETRI - Date: 2007-12-07

1. Ribs and Fans ofBézier Curves & Surfaces withApplications Joo-Haeng Lee, PhD Digital Contents Division ETRI, KOREA 2007-12-07

2. Agenda Theory Ribs and Fans of BézierCurves and Surfaces Properties Application Geometric Morphology Development Transformation Q & A

3. Geometric Morphology Morphological Development Morphological Transformation

4. Basic Theory RFD Rib-and-Fan Decomposition A Bézier curve/surface can be decomposed into Ribs: curves/surfaces Fans: vector fields Reference Joo-Haeng Lee and Hyungjun Park, “Ribs and Fans of Bézier Curves and Surfaces,” Computer-Aided Design and Applications,2 (2005), pp.125-134. (Proc. of CAD’05, Bangkok, Thailand)

5. RFD of a Bézier Curve Definition Decomposition Rib Fan Control points of ribs Control vectors of fans

6. Example: A quartic Bézier curve RFD of a Bézier Curve

7. RFD of a Bézier Curve Rib and its Control Points

8. RFD of a Bézier Curve Fan Control Vectors

9. RFD of a Bézier Curve Scaled Fan

10. RFD of a Bézier Curve Decomposition

11. RFD of a Bézier Curve Decomposition, further

12. RFD of a Bézier Curve Fan Lines

13. RFD of a Bézier Curve Fan Curves

14. RFD of a Bézier Curve Sampled Fan Curves and Ribs

15. Example > Curve RFD (1) -> Globe Curve

16. Example > Curve RFD (1)

17. Example > Curve RFD (2)

18. Example > Curve RFD (3)

19. Example > Curve RFD (4)

20. Example > Curve RFD (5)

21. Example > Curve RFD (6)

22. RFD of a Bézier Surface Bézier Surface (9,9)

23. RFD of a Bézier Surface Ribs

24. RFD of a Bézier Surface Ribs

25. RFD of a Bézier Surface Decomposition

26. RFD of a Bézier Surface Definition

27. RFD of a Bézier Surface Definition (Continued)

28. Example > Surface RFD Bézier Surface (11,11)

29. Example > Surface RFD

30. Properties of RFD of Bézier Curves Three properties of RFD of Bézier curves: Composite Fan Rib-Invariant Deformation Fan continuity in subdivision Reference Joo-Haeng Lee and Hyungjun Park, “Geometric Properties of Ribs and Fans of a Bézier Curve,” J. of Comp. Sci. & Tech, 21(2), pp.279—283, 2006.

31. Composite Fan > Introduction Definition

32. Composite Fan > Decomposition Control vectors of a composite fan Key idea: degree elevation of fans

33. Composite Fan > Construction Construction of a Bézier curve of degree n Generally, it requires specification of (n+1) control points We propose a new method using A base line segment Defined by 2 end-points Equivalent to the rib of degree 1 A composite fan Defined by (n-1) control vectors

34. Composite Fan > Example (1)

35. Composite Fan > Example (2)

36. Composite Fan > Construction Derivation of (n+1) control points from 2 end-points and (n-1) control vectors

37. Composite Fan > Decomposition Control vectors of a composite fan More explicit expression

38. Composite Fan > Curve Development

39. Rib-Invariant Deformation Motivation Ribs as guides or constraints in the course of curve deformation An example of use-case “The rib of degree d should be invariant during the deformation of a curve of degree n.”

40. Rib-Invariant Deformation > Example (1)

41. Rib-Invariant Deformation Relation between two ribs of different degrees: Degree n of the given curve Degree d (<n) of the lower rib

42. Rib-Invariant Deformation Procedure Specify d the degree of a invariant rib Initially, we have an under-constrained linear system Known: (d-1) control points of the rib of degree d Unknown: (n-1) control points of the curve of degree n Specify (n-d) control points of the given curve Now, we have a (d-1)x(d-1) linear system Known: (d-1) control points of the rib of degree d Unknown: (d-1) control points of the curve of degree n Solve the linear system To compute the unknown control points of the curve of degree n

43. Rib-Invariant Deformation > Example (2)

44. Rib-Invariant Deformation > Example (3)

45. Fan Continuity Subdivision of a Bézier curve of degree n Cncontinuity at the joint Motivation What happens to the ribs and the fans of subdivided segments, especially in the sense of continuity?

46. Fan Continuity Ribs and fans of the subdivided curves Ribs C0 continuity at most Hence, we are not interested in them Fans Fans of the subdivided curves are directionally continuous at the joint Moreover, they directionally coincide with the subdivided fans

47. Fan Continuity Mathematical description of the property

48. Fan Continuity > Example (1) Fans of subdivided curves Subdivision of the topmost fan

49. Fan Continuity > Example (2) Fans of subdivided curves Subdivision of the topmost fan

50. Geometric Morphology Summary We present techniques to generate a sequence of curves that represent the morphological development and transformation of Bézier curves based on the rib-and-fan decomposition (RFD). Reference Joo-Haeng Lee and Hyungjun Park, “A Note on Morphological Development and Transformation of Bézier Curves based on Ribs and Fans,” ACM Symposium on Solid and Physical Modeling (2007), pp. 379-385, Beijing, China, 2007.

51. Morphology Definition 1 (morphology) the branch of biology that deals with the structure of animals and plants2 (morphology) studies of the rules for forming admissible words3 (morphology, sound structure, syllable structure, word structure) the admissible arrangement of sounds in words4 (morphology, geomorphology) the branch of geology that studies the characteristics and configuration and evolution of rocks and land forms WordNet 1.7.1, Edition. Copyright 2001 by Princeton University. All rights reserved.This electronic edition published by Hanmesoft Corp. All rights reserved.

52. Geometric Morphology Morphological Development Morphological Transformation

53. Morphological Development Morphological Development From a simple linear line segment: i.e., parameter domain or a base rib To a high-degree Bézier curve with a complex shape and features MorphologicalRegression Vice versa

54. Morphological Development Common Formulation Input A given Bézier curve Development Path Interpolating end conditions at 0 and 1 Intermediate trajectory determines the developmental pattern of a curve .

55. Morphological Development Three Methods of Development Linear Interpolation Trajectory: Straight Line Composite Fans (DCF) Piecewise Linear Interpolation Trajectory: Poly Line Fan Lines (DFL) Smooth Curve Trajectory: Bézier Curve Fan Curves (DFC)

56. Morphological Development > DCF Development by Composite Fan (DCF)

57. Morphological Development > DFL Development by Fan Lines (DFL)

58. Morphological Development > DFC Development by Fan Curves (DFC)

59. Morphological Development > Compare! DCF DFL DFC

60. Composite Fan DCF Fan Lines DFL DFC Fan Curves

61. Morphological Development > Compare! DFC DCF DFL

62. Shape # 32 Composite Fan DCF Fan Lines DFL DFC Fan Curves

63. Shape # 20 Composite Fan DCF Fan Lines DFL DFC Fan Curves

64. Shape # 45 Composite Fan DCF Fan Lines DFL DFC Fan Curves

65. Shape # 46 Composite Fan DCF Fan Lines DFL DFC Fan Curves

66. Shape # 56 Composite Fan DCF Fan Lines DFL DFC Fan Curves

67. Shape # 60 Composite Fan DCF Fan Lines DFL DFC Fan Curves

68. Shape # 62 Composite Fan DCF Fan Lines DFL DFC Fan Curves

69. Inspiration from Biology Bluefin Tuna 참다랑어 北方蓝鳍金枪鱼 クロマグロ Morphological Development Miyashita, S., Sawada, Y., Okada, T., Murata, O., and Kumai, H., Morphological development and growth of laboratory-reared larval and juvenile Thunnus Thynnus (Pisces: Scombridae), Fishery Bulletin, Vol. 99, No. 4, pp. 601-616, 2001.

70. Inspiration from Biology

71. Inspiration from Biology

72. Morphological Development Characteristics of Intermediate Shapes Proposed Method (TFL/TFC) Features appears gradually Intermediate curves are relatively smooth Analogous to morphological development in biology Linear Interpolation (TLI) Early appearance of shape features in immature curves More likely to have wiggles and cusps

73. Morphological Transformation Simply, it means morphing or metamorphosis between two Bézier curves

74. Morphological Transformation Metamorphosis 1 (metamorphosis, metabolism) the marked and rapidtransformation of a larva into an adult that occurs in some animals 2 (transfiguration, metamorphosis) a striking change in appearance or character or circumstances 3 (metamorphosis) a completechange of physical form or substance especially as by magic or witchcraft WordNet 1.7.1, Edition. Copyright 2001 by Princeton University. All rights reserved.This electronic edition published by Hanmesoft Corp. All rights reserved.

75. Morphological Transformation Common Formulation Input Two curves Output One-parameter family of curves representing the intermediate shapes .

76. Morphological Transformation Three Methods TLI Linear Interpolation Trajectory: Straight Line TCE Cubic Blending and Linear Extrapolation Trajectory: Straight Line TDE Development, Quadratic Blending, and Extrapolation Trajectory: Curve

77. Morphological Transformation > TLI Linear Interpolation

78. Morphological Transformation Shape Blending by Direction Map [Lee:2003]

79. Morphological Transformation > TCE Cubic Blending

80. Morphological Transformation > TCE Cubic Blending Dynamic sequence, but curves are relatively small. Linear Cubic

81. Morphological Transformation > TCE Cubic Blending and Extrapolation

82. Morphological Transformation > TCE Cubic Blending and Extrapolation

83. Morphological Transformation > TCE Cubic Blending and Extrapolation Increase the size through extrapolation. Linear Cubic Cubic & Extrapolate

84. Morphological Transformation > TCE Cubic Blending and Extrapolation Actually, re-parameterized linear interpolation! Linear Cubic Cubic & Extrapolate

85. Morphological Transformation > TDE Development (DFL/DFC) & Quadratic Blending

86. Morphological Transformation > TDE Development (DFL/DFC) & Quadratic Blending Immaturity in size and features Linear DFC & Quad. Blend

87. Morphological Transformation > TDE Development & Quad Blend & Extrapolation

88. Morphological Transformation > TDE Development & Quad Blend & Extrapolation Over-growth Linear DFC & Quad. Blend TDE

89. Morphological Transformation > TDE Development + Quad Blend + Extrapolation Control of over-growth Revision of extrapolation ratio Selection of base transformation

90. Morphological Transformation > TDE Development + Quad Blend + Extrapolation Revision of extrapolation ratio

91. Morphological Transformation > TDE Development + Quad Blend + Extrapolation Revision of extrapolation ratio Linear TDE (ß=3.0) TDE (ß=2.5)

92. Morphological Transformation > TDE Development + Quad Blend + Extrapolation Selection of base Transformation

93. Morphological Transformation > TDE Development + Quad Blend + Extrapolation Selection of base Transformation Linear TDE (1,1; ß=3.0) TDE (5,5; ß=3.0)

94. Morphological Transformation > TDE Test Set: 71 Shapes: 48 & 62 TDE (k,k; ß=3.0) k=(1…8)

95. Morphological Transformation > TDE Linear TDE (1,1; ß=3.0) TDE (5,5; ß=3.0) TDE (1,1; ß=2.5) TDE (5,5; ß=2.5) Dev + Quad Blend

96. Morphological Transformation > TDE Examples

97. Morphological Transformation > TDE > Ex 1 Linear TDE (1,1; ß=3.0) TDE (4,4; ß=3.0) TDE (1,1; ß=2.4) TDE (4,4; ß=2.4) Dev + Quad Blend

99. Morphological Transformation > TDE > Ex 2 Linear TDE (1,1; ß=3.0) TDE (3,3; ß=3.0) TDE (1,1; ß=2.5) TDE (3,3; ß=2.5) Dev + Quad Blend

101. Inspiration from Biology Evolutionary Developmental Biology* (evolution of development or informally, 'evo-devo') is a field of biology that compares the developmental processes of different animals in an attempt to determine the ancestral relationship between organisms and how developmental processes evolved. The discovery ofgenes regulating development in model organisms allowed for comparisons to be made with genes and genetic networks of related organisms. * WikiPedia

102. Inspiration from Biology Different Developmental Process * Life: The Science of Biology (William K. Purves, et al., 2004)

103. Inspiration from Biology Different Developmental Process * Life: The Science of Biology (William K. Purves, et al., 2004)

104. Inspiration from Biology Different Developmental Process Recapitulation theory (Earnst Haeckel, 1866) * http://en.wikipedia.org/wiki/Ontogeny_recapitulates_phylogeny

105. Inspiration from Biology Evolutionary Tree or Phylogenetic Tree * Life: The Science of Biology (William K. Purves, et al., 2004)

106. Inspiration from Biology Evolutionary Tree: Plantae * Life: The Science of Biology (William K. Purves, et al., 2004)

107. Inspiration from Biology Evolutionary Tree: Animalia * Life: The Science of Biology (William K. Purves, et al., 2004)

108. Morphological Transformation Characteristics of Intermediate Shapes Proposed Method (TDE) Intermediate curves are neutral to given curves Analogous to evolutionary developmental biology Further control by choosing the degrees of the initial shapes Linear Interpolation (TLI, TCE) Simultaneous mixture of features of two curves Static shape change No further control except re-parameterization

109. Concluding Remarks Novel approach to deal with geometric morphology of Bézier curves based on ribs and fans Analogous to biological phenomena Morphological development in biology Evolutionary developmental biology Development Developmental patterns are generated along trajectories based on intrinsic, internal structure of Bézier curves such as fan lines and fan curves Transformation Based on the assumption that inter-curve transformation happens in the early developmental stage rather than the mature curves alone Extrapolation of immature shapes to control size

110. Concluding Remarks Future works Extension Bézier surfaces B-spline Interpretation/simulation of natural phenomena Evolution Morphological diversity Evolution of Geometry Geometric Gene?

111. Concluding Remarks Future works Extension to Bézier surfaces Extension to piece-wise curves/surfaces Interpretation/simulation of natural phenomena Comparison with other methods

112. Q & A Thank you! Questions/Comments E-mail: joohaeng@gmail.com