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Ribs and Fans of Bezier Curves and Surfaces with Applications
 

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.

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 Ribs and Fans of Bezier Curves and Surfaces with Applications Presentation Transcript

    • Ribs and Fans ofBézier Curves & Surfaces withApplications
      Joo-Haeng Lee, PhD
      Digital Contents Division
      ETRI, KOREA
      2007-12-07
    • Agenda
      Theory
      Ribs and Fans of BézierCurves and Surfaces
      Properties
      Application
      Geometric Morphology
      Development
      Transformation
      Q & A
    • Geometric Morphology
      Morphological Development
      Morphological Transformation
    • 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)
    • RFD of a Bézier Curve
      Definition
      Decomposition
      Rib
      Fan
      Control points of ribs
      Control vectors of fans
    • Example: A quartic Bézier curve
      RFD of a Bézier Curve
    • RFD of a Bézier Curve
      Rib and its Control Points
    • RFD of a Bézier Curve
      Fan Control Vectors
    • RFD of a Bézier Curve
      Scaled Fan
    • RFD of a Bézier Curve
      Decomposition
    • RFD of a Bézier Curve
      Decomposition, further
    • RFD of a Bézier Curve
      Fan Lines
    • RFD of a Bézier Curve
      Fan Curves
    • RFD of a Bézier Curve
      Sampled Fan Curves and Ribs
    • Example > Curve RFD (1) -> Globe Curve
    • Example > Curve RFD (1)
    • Example > Curve RFD (2)
    • Example > Curve RFD (3)
    • Example > Curve RFD (4)
    • Example > Curve RFD (5)
    • Example > Curve RFD (6)
    • RFD of a Bézier Surface
      Bézier Surface (9,9)
    • RFD of a Bézier Surface
      Ribs
    • RFD of a Bézier Surface
      Ribs
    • RFD of a Bézier Surface
      Decomposition
    • RFD of a Bézier Surface
      Definition
    • RFD of a Bézier Surface
      Definition (Continued)
    • Example > Surface RFD
      Bézier Surface (11,11)
    • Example > Surface RFD
    • 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.
    • Composite Fan > Introduction
      Definition
    • Composite Fan > Decomposition
      Control vectors of a composite fan
      Key idea: degree elevation of fans
    • 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
    • Composite Fan > Example (1)
    • Composite Fan > Example (2)
    • Composite Fan > Construction
      Derivation of (n+1) control points from
      2 end-points and
      (n-1) control vectors
    • Composite Fan > Decomposition
      Control vectors of a composite fan
      More explicit expression
    • Composite Fan > Curve Development
    • 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.”
    • Rib-Invariant Deformation > Example (1)
    • Rib-Invariant Deformation
      Relation between two ribs of different degrees:
      Degree n of the given curve
      Degree d (<n) of the lower rib
    • 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
    • Rib-Invariant Deformation > Example (2)
    • Rib-Invariant Deformation > Example (3)
    • 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?
    • 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
    • Fan Continuity
      Mathematical description of the property
    • Fan Continuity > Example (1)
      Fans of
      subdivided curves
      Subdivision of
      the topmost fan
    • Fan Continuity > Example (2)
      Fans of
      subdivided curves
      Subdivision of
      the topmost fan
    • 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.
    • 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.
    • Geometric Morphology
      Morphological Development
      Morphological Transformation
    • 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
    • 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
      .
    • 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)
    • Morphological Development > DCF
      Development by Composite Fan (DCF)
    • Morphological Development > DFL
      Development by Fan Lines (DFL)
    • Morphological Development > DFC
      Development by Fan Curves (DFC)
    • Morphological Development > Compare!
      DCF
      DFL
      DFC
    • Composite
      Fan
      DCF
      Fan Lines
      DFL
      DFC
      Fan Curves
    • Morphological Development > Compare!
      DFC
      DCF
      DFL
    • Shape # 32
      Composite
      Fan
      DCF
      Fan Lines
      DFL
      DFC
      Fan Curves
    • Shape # 20
      Composite
      Fan
      DCF
      Fan Lines
      DFL
      DFC
      Fan Curves
    • Shape # 45
      Composite
      Fan
      DCF
      Fan Lines
      DFL
      DFC
      Fan Curves
    • Shape # 46
      Composite
      Fan
      DCF
      Fan Lines
      DFL
      DFC
      Fan Curves
    • Shape # 56
      Composite
      Fan
      DCF
      Fan Lines
      DFL
      DFC
      Fan Curves
    • Shape # 60
      Composite
      Fan
      DCF
      Fan Lines
      DFL
      DFC
      Fan Curves
    • Shape # 62
      Composite
      Fan
      DCF
      Fan Lines
      DFL
      DFC
      Fan Curves
    • 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.
    • Inspiration from Biology
    • Inspiration from Biology
    • 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
    • Morphological Transformation
      Simply, it means morphing or metamorphosis between two Bézier curves
    • 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.
    • Morphological Transformation
      Common Formulation
      Input
      Two curves
      Output
      One-parameter family of curves representing the intermediate shapes
      .
    • 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
    • Morphological Transformation > TLI
      Linear Interpolation
    • Morphological Transformation
      Shape Blending by Direction Map [Lee:2003]
    • Morphological Transformation > TCE
      Cubic Blending
    • Morphological Transformation > TCE
      Cubic Blending
      Dynamic sequence, but curves are relatively small.
      Linear
      Cubic
    • Morphological Transformation > TCE
      Cubic Blending and Extrapolation
    • Morphological Transformation > TCE
      Cubic Blending and Extrapolation
    • Morphological Transformation > TCE
      Cubic Blending and Extrapolation
      Increase the size through extrapolation.
      Linear
      Cubic
      Cubic & Extrapolate
    • Morphological Transformation > TCE
      Cubic Blending and Extrapolation
      Actually, re-parameterized linear interpolation!
      Linear
      Cubic
      Cubic & Extrapolate
    • Morphological Transformation > TDE
      Development (DFL/DFC) & Quadratic Blending
    • Morphological Transformation > TDE
      Development (DFL/DFC) & Quadratic Blending
      Immaturity in size and features
      Linear
      DFC & Quad. Blend
    • Morphological Transformation > TDE
      Development & Quad Blend & Extrapolation
    • Morphological Transformation > TDE
      Development & Quad Blend & Extrapolation
      Over-growth
      Linear
      DFC & Quad. Blend
      TDE
    • Morphological Transformation > TDE
      Development + Quad Blend + Extrapolation
      Control of over-growth
      Revision of extrapolation ratio
      Selection of base transformation
    • Morphological Transformation > TDE
      Development + Quad Blend + Extrapolation
      Revision of extrapolation ratio
    • Morphological Transformation > TDE
      Development + Quad Blend + Extrapolation
      Revision of extrapolation ratio
      Linear
      TDE (ß=3.0)
      TDE (ß=2.5)
    • Morphological Transformation > TDE
      Development + Quad Blend + Extrapolation
      Selection of base Transformation
    • Morphological Transformation > TDE
      Development + Quad Blend + Extrapolation
      Selection of base Transformation
      Linear
      TDE (1,1; ß=3.0)
      TDE (5,5; ß=3.0)
    • Morphological Transformation > TDE
      Test Set: 71
      Shapes: 48 & 62
      TDE (k,k; ß=3.0)
      k=(1…8)
    • 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
    • Morphological Transformation > TDE
      Examples
    • 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
    • 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
    • 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
    • Inspiration from Biology
      Different Developmental Process
      * Life: The Science of Biology (William K. Purves, et al., 2004)
    • Inspiration from Biology
      Different Developmental Process
      * Life: The Science of Biology (William K. Purves, et al., 2004)
    • Inspiration from Biology
      Different Developmental Process
      Recapitulation theory (Earnst Haeckel, 1866)
      * http://en.wikipedia.org/wiki/Ontogeny_recapitulates_phylogeny
    • Inspiration from Biology
      Evolutionary Tree or Phylogenetic Tree
      * Life: The Science of Biology (William K. Purves, et al., 2004)
    • Inspiration from Biology
      Evolutionary Tree: Plantae
      * Life: The Science of Biology (William K. Purves, et al., 2004)
    • Inspiration from Biology
      Evolutionary Tree: Animalia
      * Life: The Science of Biology (William K. Purves, et al., 2004)
    • 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
    • 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
    • Concluding Remarks
      Future works
      Extension
      Bézier surfaces
      B-spline
      Interpretation/simulation of natural phenomena
      Evolution
      Morphological diversity
      Evolution of Geometry
      Geometric Gene?
    • Concluding Remarks
      Future works
      Extension to Bézier surfaces
      Extension to piece-wise curves/surfaces
      Interpretation/simulation of natural phenomena
      Comparison with other methods
    • Q & A
      Thank you!
      Questions/Comments
      E-mail: joohaeng@gmail.com