Ribs and Fans of Bezier Curves and Surfaces with Applications

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

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

Rib and its Control Points

Fan Control Vectors

Scaled Fan

Decomposition

Decomposition, further

Fan Lines

Fan Curves

Sampled Fan Curves and Ribs

Example > Curve RFD (1) -> Globe Curve

Example > Curve RFD (1)

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

Ribs

Decomposition

Definition

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

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)

Relation between two ribs of different degrees:
Degree n of the given curve
Degree d (<n) of the lower rib

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

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

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.

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

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
.

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
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.

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

Simply, it means morphing or metamorphosis between two Bézier curves

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.

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

Three Methods
TLI
TCE
Cubic Blending and Linear Extrapolation
TDE
Development, Quadratic Blending, and Extrapolation
Trajectory: Curve

Morphological Transformation > TLI

Shape Blending by Direction Map [Lee:2003]

Morphological Transformation > TCE
Cubic Blending

Cubic Blending
Dynamic sequence, but curves are relatively small.
Linear
Cubic

Cubic Blending and Extrapolation

Increase the size through extrapolation.
Linear
Cubic
Cubic & Extrapolate

Actually, re-parameterized linear interpolation!
Linear
Cubic
Cubic & Extrapolate

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

Development (DFL/DFC) & Quadratic Blending
Immaturity in size and features
Linear
DFC & Quad. Blend

Development & Quad Blend & Extrapolation

Development & Quad Blend & Extrapolation
Over-growth
Linear
DFC & Quad. Blend
TDE

Development + Quad Blend + Extrapolation
Control of over-growth
Revision of extrapolation ratio
Selection of base transformation

Linear
TDE (ß=3.0)
TDE (ß=2.5)

Selection of base Transformation

Selection of base Transformation
Linear
TDE (1,1; ß=3.0)
TDE (5,5; ß=3.0)

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

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

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

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

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

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

Evolutionary Tree or Phylogenetic Tree

Evolutionary Tree: Plantae

Evolutionary Tree: Animalia

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