Ribs and Fans of Bezier Curves and Surfaces with Applications

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Explains newly found geometric features of Bezier curves and surfaces called "rib and fan.

- Author: Joo-Haeng Lee
- Affiliation: ETRI
- Date: 2007-12-07

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Ribs and Fans of Bezier Curves and Surfaces with Applications

  1. 1. Ribs and Fans ofBézier Curves & Surfaces withApplications<br />Joo-Haeng Lee, PhD<br />Digital Contents Division<br />ETRI, KOREA<br />2007-12-07<br />
  2. 2. Agenda<br />Theory<br />Ribs and Fans of BézierCurves and Surfaces<br />Properties<br />Application<br />Geometric Morphology<br />Development<br />Transformation<br />Q & A<br />
  3. 3. Geometric Morphology<br />Morphological Development<br />Morphological Transformation<br />
  4. 4. Basic Theory<br />RFD<br />Rib-and-Fan Decomposition<br />A Bézier curve/surface can be decomposed into <br />Ribs: curves/surfaces<br />Fans: vector fields<br />Reference<br />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)<br />
  5. 5. RFD of a Bézier Curve<br />Definition<br />Decomposition<br />Rib<br />Fan<br />Control points of ribs<br />Control vectors of fans<br />
  6. 6. Example: A quartic Bézier curve<br />RFD of a Bézier Curve<br />
  7. 7. RFD of a Bézier Curve<br />Rib and its Control Points<br />
  8. 8. RFD of a Bézier Curve<br />Fan Control Vectors<br />
  9. 9. RFD of a Bézier Curve<br />Scaled Fan<br />
  10. 10. RFD of a Bézier Curve<br />Decomposition<br />
  11. 11. RFD of a Bézier Curve<br />Decomposition, further<br />
  12. 12. RFD of a Bézier Curve<br />Fan Lines<br />
  13. 13. RFD of a Bézier Curve<br />Fan Curves<br />
  14. 14. RFD of a Bézier Curve<br />Sampled Fan Curves and Ribs<br />
  15. 15. Example &gt; Curve RFD (1) -&gt; Globe Curve<br />
  16. 16. Example &gt; Curve RFD (1)<br />
  17. 17. Example &gt; Curve RFD (2)<br />
  18. 18. Example &gt; Curve RFD (3)<br />
  19. 19. Example &gt; Curve RFD (4)<br />
  20. 20. Example &gt; Curve RFD (5)<br />
  21. 21. Example &gt; Curve RFD (6)<br />
  22. 22. RFD of a Bézier Surface<br />Bézier Surface (9,9)<br />
  23. 23. RFD of a Bézier Surface<br />Ribs<br />
  24. 24. RFD of a Bézier Surface<br />Ribs<br />
  25. 25. RFD of a Bézier Surface<br />Decomposition<br />
  26. 26. RFD of a Bézier Surface<br />Definition<br />
  27. 27. RFD of a Bézier Surface<br />Definition (Continued)<br />
  28. 28. Example &gt; Surface RFD<br />Bézier Surface (11,11)<br />
  29. 29. Example &gt; Surface RFD<br />
  30. 30. Properties of RFD of Bézier Curves<br />Three properties of RFD of Bézier curves:<br />Composite Fan<br />Rib-Invariant Deformation<br />Fan continuity in subdivision<br />Reference<br />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.<br />
  31. 31. Composite Fan &gt; Introduction<br />Definition<br />
  32. 32. Composite Fan &gt; Decomposition<br />Control vectors of a composite fan<br />Key idea: degree elevation of fans<br />
  33. 33. Composite Fan &gt; Construction<br />Construction of a Bézier curve of degree n<br />Generally, it requires specification of (n+1) control points<br />We propose a new method using<br />A base line segment <br />Defined by 2 end-points<br />Equivalent to the rib of degree 1<br />A composite fan<br />Defined by (n-1) control vectors<br />
  34. 34. Composite Fan &gt; Example (1)<br />
  35. 35. Composite Fan &gt; Example (2)<br />
  36. 36. Composite Fan &gt; Construction<br />Derivation of (n+1) control points from<br />2 end-points and<br />(n-1) control vectors<br />
  37. 37. Composite Fan &gt; Decomposition<br />Control vectors of a composite fan<br />More explicit expression<br />
  38. 38. Composite Fan &gt; Curve Development<br />
  39. 39. Rib-Invariant Deformation<br />Motivation<br />Ribs as guides or constraints in the course of curve deformation<br />An example of use-case<br />“The rib of degree d should be invariant during the deformation of a curve of degree n.”<br />
  40. 40. Rib-Invariant Deformation &gt; Example (1)<br />
  41. 41. Rib-Invariant Deformation<br />Relation between two ribs of different degrees:<br />Degree n of the given curve<br />Degree d (&lt;n) of the lower rib<br />
  42. 42. Rib-Invariant Deformation<br />Procedure<br />Specify d<br />the degree of a invariant rib<br />Initially, we have an under-constrained linear system<br />Known: (d-1) control points of the rib of degree d<br />Unknown: (n-1) control points of the curve of degree n<br />Specify (n-d) control points of the given curve<br />Now, we have a (d-1)x(d-1) linear system<br />Known: (d-1) control points of the rib of degree d<br />Unknown: (d-1) control points of the curve of degree n<br />Solve the linear system <br />To compute the unknown control points of the curve of degree n<br />
  43. 43. Rib-Invariant Deformation &gt; Example (2)<br />
  44. 44. Rib-Invariant Deformation &gt; Example (3)<br />
  45. 45. Fan Continuity<br />Subdivision of a Bézier curve of degree n<br />Cncontinuity at the joint<br />Motivation<br />What happens to the ribs and the fans of subdivided segments, especially in the sense of continuity?<br />
  46. 46. Fan Continuity<br />Ribs and fans of the subdivided curves<br />Ribs<br />C0 continuity at most<br />Hence, we are not interested in them<br />Fans<br />Fans of the subdivided curves are directionally continuous at the joint<br />Moreover, they directionally coincide with the subdivided fans<br />
  47. 47. Fan Continuity<br />Mathematical description of the property<br />
  48. 48. Fan Continuity &gt; Example (1)<br />Fans of <br />subdivided curves<br />Subdivision of <br />the topmost fan<br />
  49. 49. Fan Continuity &gt; Example (2)<br />Fans of <br />subdivided curves<br />Subdivision of <br />the topmost fan<br />
  50. 50. Geometric Morphology<br />Summary<br />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).<br />Reference<br />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.<br />
  51. 51. Morphology<br />Definition<br /> 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<br />WordNet 1.7.1, Edition. Copyright 2001 by Princeton University. All rights reserved.This electronic edition published by Hanmesoft Corp. All rights reserved.<br />
  52. 52. Geometric Morphology<br />Morphological Development<br />Morphological Transformation<br />
  53. 53. Morphological Development<br />Morphological Development<br />From a simple linear line segment: i.e., parameter domain or a base rib<br />To a high-degree Bézier curve with a complex shape and features<br />MorphologicalRegression<br />Vice versa<br />
  54. 54. Morphological Development<br />Common Formulation<br />Input<br />A given Bézier curve<br />Development Path<br />Interpolating end conditions at 0 and 1<br />Intermediate trajectory determines the developmental pattern of a curve<br />. <br />
  55. 55. Morphological Development<br />Three Methods of Development<br />Linear Interpolation<br />Trajectory: Straight Line<br />Composite Fans (DCF)<br />Piecewise Linear Interpolation<br />Trajectory: Poly Line<br />Fan Lines (DFL)<br />Smooth Curve<br />Trajectory: Bézier Curve<br />Fan Curves (DFC)<br />
  56. 56. Morphological Development &gt; DCF<br />Development by Composite Fan (DCF)<br />
  57. 57. Morphological Development &gt; DFL<br />Development by Fan Lines (DFL)<br />
  58. 58. Morphological Development &gt; DFC<br />Development by Fan Curves (DFC)<br />
  59. 59. Morphological Development &gt; Compare!<br />DCF<br />DFL<br />DFC<br />
  60. 60. Composite<br />Fan<br />DCF<br />Fan Lines<br />DFL<br />DFC<br />Fan Curves<br />
  61. 61. Morphological Development &gt; Compare!<br />DFC<br />DCF<br />DFL<br />
  62. 62. Shape # 32<br />Composite<br />Fan<br />DCF<br />Fan Lines<br />DFL<br />DFC<br />Fan Curves<br />
  63. 63. Shape # 20<br />Composite<br />Fan<br />DCF<br />Fan Lines<br />DFL<br />DFC<br />Fan Curves<br />
  64. 64. Shape # 45<br />Composite<br />Fan<br />DCF<br />Fan Lines<br />DFL<br />DFC<br />Fan Curves<br />
  65. 65. Shape # 46<br />Composite<br />Fan<br />DCF<br />Fan Lines<br />DFL<br />DFC<br />Fan Curves<br />
  66. 66. Shape # 56<br />Composite<br />Fan<br />DCF<br />Fan Lines<br />DFL<br />DFC<br />Fan Curves<br />
  67. 67. Shape # 60<br />Composite<br />Fan<br />DCF<br />Fan Lines<br />DFL<br />DFC<br />Fan Curves<br />
  68. 68. Shape # 62<br />Composite<br />Fan<br />DCF<br />Fan Lines<br />DFL<br />DFC<br />Fan Curves<br />
  69. 69. Inspiration from Biology<br />Bluefin Tuna<br />참다랑어<br />北方蓝鳍金枪鱼<br />クロマグロ<br />Morphological Development<br />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.<br />
  70. 70. Inspiration from Biology<br />
  71. 71. Inspiration from Biology<br />
  72. 72. Morphological Development<br />Characteristics of Intermediate Shapes<br />Proposed Method (TFL/TFC)<br />Features appears gradually<br />Intermediate curves are relatively smooth<br />Analogous to morphological development in biology<br />Linear Interpolation (TLI)<br />Early appearance of shape features in immature curves <br />More likely to have wiggles and cusps<br />
  73. 73. Morphological Transformation<br />Simply, it means morphing or metamorphosis between two Bézier curves<br />
  74. 74. Morphological Transformation<br />Metamorphosis<br />1 (metamorphosis, metabolism) the marked and rapidtransformation of a larva into an adult that occurs in some animals<br />2 (transfiguration, metamorphosis) a striking change in appearance or character or circumstances<br />3 (metamorphosis) a completechange of physical form or substance especially as by magic or witchcraft<br />WordNet 1.7.1, Edition. Copyright 2001 by Princeton University. All rights reserved.This electronic edition published by Hanmesoft Corp. All rights reserved.<br />
  75. 75. Morphological Transformation<br />Common Formulation<br />Input<br />Two curves<br />Output<br />One-parameter family of curves representing the intermediate shapes<br />. <br />
  76. 76. Morphological Transformation<br />Three Methods<br />TLI<br />Linear Interpolation<br />Trajectory: Straight Line<br />TCE<br />Cubic Blending and Linear Extrapolation<br />Trajectory: Straight Line<br />TDE<br />Development, Quadratic Blending, and Extrapolation<br />Trajectory: Curve<br />
  77. 77. Morphological Transformation &gt; TLI<br />Linear Interpolation<br />
  78. 78. Morphological Transformation<br />Shape Blending by Direction Map [Lee:2003]<br />
  79. 79. Morphological Transformation &gt; TCE<br />Cubic Blending<br />
  80. 80. Morphological Transformation &gt; TCE<br />Cubic Blending<br />Dynamic sequence, but curves are relatively small.<br />Linear<br />Cubic<br />
  81. 81. Morphological Transformation &gt; TCE<br />Cubic Blending and Extrapolation<br />
  82. 82. Morphological Transformation &gt; TCE<br />Cubic Blending and Extrapolation<br />
  83. 83. Morphological Transformation &gt; TCE<br />Cubic Blending and Extrapolation<br />Increase the size through extrapolation.<br />Linear<br />Cubic<br />Cubic & Extrapolate<br />
  84. 84. Morphological Transformation &gt; TCE<br />Cubic Blending and Extrapolation<br />Actually, re-parameterized linear interpolation!<br />Linear<br />Cubic<br />Cubic & Extrapolate<br />
  85. 85. Morphological Transformation &gt; TDE<br />Development (DFL/DFC) & Quadratic Blending<br />
  86. 86. Morphological Transformation &gt; TDE<br />Development (DFL/DFC) & Quadratic Blending<br />Immaturity in size and features<br />Linear<br />DFC & Quad. Blend<br />
  87. 87. Morphological Transformation &gt; TDE<br />Development & Quad Blend & Extrapolation<br />
  88. 88. Morphological Transformation &gt; TDE<br />Development & Quad Blend & Extrapolation<br />Over-growth<br />Linear<br />DFC & Quad. Blend<br />TDE<br />
  89. 89. Morphological Transformation &gt; TDE<br />Development + Quad Blend + Extrapolation<br />Control of over-growth<br />Revision of extrapolation ratio<br />Selection of base transformation<br />
  90. 90. Morphological Transformation &gt; TDE<br />Development + Quad Blend + Extrapolation<br />Revision of extrapolation ratio<br />
  91. 91. Morphological Transformation &gt; TDE<br />Development + Quad Blend + Extrapolation<br />Revision of extrapolation ratio<br />Linear<br />TDE (ß=3.0)<br />TDE (ß=2.5) <br />
  92. 92. Morphological Transformation &gt; TDE<br />Development + Quad Blend + Extrapolation<br />Selection of base Transformation<br />
  93. 93. Morphological Transformation &gt; TDE<br />Development + Quad Blend + Extrapolation<br />Selection of base Transformation<br />Linear<br />TDE (1,1; ß=3.0)<br />TDE (5,5; ß=3.0)<br />
  94. 94. Morphological Transformation &gt; TDE<br />Test Set: 71<br />Shapes: 48 & 62<br />TDE (k,k; ß=3.0)<br />k=(1…8)<br />
  95. 95. Morphological Transformation &gt; TDE<br />Linear<br />TDE (1,1; ß=3.0)<br />TDE (5,5; ß=3.0)<br />TDE (1,1; ß=2.5) <br />TDE (5,5; ß=2.5) <br />Dev + Quad Blend<br />
  96. 96. Morphological Transformation &gt; TDE <br />Examples<br />
  97. 97. Morphological Transformation &gt; TDE &gt; Ex 1<br />Linear<br />TDE (1,1; ß=3.0)<br />TDE (4,4; ß=3.0)<br />TDE (1,1; ß=2.4) <br />TDE (4,4; ß=2.4) <br />Dev + Quad Blend<br />
  98. 98.
  99. 99. Morphological Transformation &gt; TDE &gt; Ex 2<br />Linear<br />TDE (1,1; ß=3.0)<br />TDE (3,3; ß=3.0)<br />TDE (1,1; ß=2.5) <br />TDE (3,3; ß=2.5) <br />Dev + Quad Blend<br />
  100. 100.
  101. 101. Inspiration from Biology<br />Evolutionary Developmental Biology*<br />(evolution of development or informally, &apos;evo-devo&apos;) 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. <br />* WikiPedia <br />
  102. 102. Inspiration from Biology<br />Different Developmental Process<br />* Life: The Science of Biology (William K. Purves, et al., 2004) <br />
  103. 103. Inspiration from Biology<br />Different Developmental Process<br />* Life: The Science of Biology (William K. Purves, et al., 2004) <br />
  104. 104. Inspiration from Biology<br />Different Developmental Process<br />Recapitulation theory (Earnst Haeckel, 1866)<br />* http://en.wikipedia.org/wiki/Ontogeny_recapitulates_phylogeny<br />
  105. 105. Inspiration from Biology<br />Evolutionary Tree or Phylogenetic Tree<br />* Life: The Science of Biology (William K. Purves, et al., 2004) <br />
  106. 106. Inspiration from Biology<br />Evolutionary Tree: Plantae<br />* Life: The Science of Biology (William K. Purves, et al., 2004) <br />
  107. 107. Inspiration from Biology<br />Evolutionary Tree: Animalia<br />* Life: The Science of Biology (William K. Purves, et al., 2004) <br />
  108. 108. Morphological Transformation <br />Characteristics of Intermediate Shapes<br />Proposed Method (TDE)<br />Intermediate curves are neutral to given curves<br />Analogous to evolutionary developmental biology<br />Further control by choosing the degrees of the initial shapes<br />Linear Interpolation (TLI, TCE)<br />Simultaneous mixture of features of two curves <br />Static shape change<br />No further control except re-parameterization<br />
  109. 109. Concluding Remarks<br />Novel approach to deal with geometric morphology of Bézier curves based on ribs and fans<br />Analogous to biological phenomena<br />Morphological development in biology<br />Evolutionary developmental biology<br />Development<br />Developmental patterns are generated along trajectories based on intrinsic, internal structure of Bézier curves such as fan lines and fan curves<br />Transformation<br />Based on the assumption that inter-curve transformation happens in the early developmental stage rather than the mature curves alone<br />Extrapolation of immature shapes to control size<br />
  110. 110. Concluding Remarks<br />Future works<br />Extension<br />Bézier surfaces<br />B-spline<br />Interpretation/simulation of natural phenomena<br />Evolution<br />Morphological diversity<br />Evolution of Geometry<br />Geometric Gene?<br />
  111. 111. Concluding Remarks<br />Future works<br />Extension to Bézier surfaces<br />Extension to piece-wise curves/surfaces<br />Interpretation/simulation of natural phenomena<br />Comparison with other methods<br />
  112. 112. Q & A<br />Thank you!<br />Questions/Comments<br />E-mail: joohaeng@gmail.com<br />

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