1. Gergely Klár, Gábor Valasek
Conference of PhD Students in Computer Science Employing Pythagorean Hodograph Curves for Artistic Patterns
Gergely Klár, Gábor Valasek
Eötvös Loránd University
Faculty of Informatics
June 29 - July 2, 2010
Szeged, Hungary 1

2. Goal Create a tool to aid the design of aesthetical, fair curves
In particular design element creator for vines, swirls, swooshes and floral components

3. Previous work
Floral components are popular elements in both ornamental and contemporary abstract design
Tools can aid among other things:
Generation of ornamental elements
Generation of ornamental patterns

4. Previous work
Plants generated with L-systems, proposed by Prusinkiewicz and Lindenmayer

5. Previous work
Wong et. al. proposed a method for filling a region of interest with ornaments using proxy objects that can be replaced by arbitrary ornamental elements

6. Previous work
Xu and Mould created ornamental patters by simulating a charged particle's movement in a magnetic field (magnetic curves)

7. Our focus We concentrate on using polynomial curves for element design
Let us presume that a pleasing curve has a smooth and monotone curvature
Farin's definition:
A curve is fair if its curvature plot is continuous and consists of only a few monotone pieces
A fair curve should only have curvature extrema where the designer explicitly wishes so

8. Our focus
To satisfy the fairness conditions we use G2 splines, that consist of spiral segments
A spiral is a curved line segment whose curvature varies monotonically with arc-length

9. Designer control An intuitive way to control our curves is required
Use hiearchy of circles:

10. Designer control – an alternative
If we let the user specify the curve segment’s starting- and endpoints on the control circles, we can formulate the problem as geometric Hermite interpolation

11. Designer control – an alternative
Given are position, tangent, and curvature data at each the endpoint
Find a Bézier curve which reconstructs these quantities at it’s endpoints
These are 2x4 scalar constraints on each segment
Position: 2 scalar
Tangent: 1 scalar
Curvature: 1 scalar

12. Designer control – an alternative Cubic Bézier solution for GH
A cubic Bézier curve has 8 scalar degrees of freedom
A depressed quartic equation results from

13. Designer control – an alternative Cubic Bézier solution for GH
With appropriate geometric constraints on position, tangent and curvature the following system has positive real roots
No spiral (cubic Bézier spirals have 5 degrees of freedom)

14. Designer control
Let us use Pythagorean Hodograph spirals for the transition curves
curve
curve’s hodograph

15. Pythagorean Hodographs
Let the parameterization be such that
For some integral polynomial
The arc-length can be expressed in closed-form
Hodograph: the derivate treated as a curve on its own 15

16. Pythagorean Hodographs
Theorem: the Pythagorean condition for polynomials holds if and only if they can be expressed in terms of other polynomials as where u(t) and v(t) are relatively prime.
Hodograph: the derivate treated as a curve on its own
Farouki and Sakkalis arrived at this by using Kubota’s results on Pythagorean triplets in unique factorization domains (~” a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements (or irreducible elements) “) 16

17. Pythagorean Hodographs
PH curves’ hodographs satisfy:
PH curves of degree n have n+3 degrees of freedom
General polynomials of degree n have 2n + 2 degrees of freedom
N + 3 – 3 (choosing the origion + coordinate axes orientation) – 2 (freedoms of parameterization parameterization) 17

18. Pythagorean Hodographs
Arc-length is a polynomial
Offset of a degree n PH curve is a rational polynomial of degree 2n-1
For practical usage
Cubic PH curves cannot have an inflection point
We use quintic PH curves
We need to find u(t), v(t) quadratic polynomials in Bézier form
Cubic: u and v are linear
Quintic: u and v are quadratic 18

19. Method Let the user create a hierarchy of control circles
Create spiral segments between a node and its descendants
Three cases are possible:
Circles can be connected by an S-shaped circle-to-circle curve
Circles can be connected by a C-shaped circle-in-circle curve
The circles cannot be connected

20. Circle-to-circle transition
The circles have to be non-touching and non-overlapping
We used the work of Walton and Meek to define the quintic PH curve’s control points

21. Circle-to-circle transition

22. Circle-to-circle transition
The circle centres have to be within a certain distance (depending on their radii)
We have to solve
Where

23. Circle-in-circle transition
A fully contained circle is joined to its ancestor if such transition is possible
The conditions and the derivation of control points can be found in Habib and Sakai’s work

24. Circle-in-circle transition

25. Circle-in-circle transition
Constraints on the radius of the smaller circle and its distance from the big circle
In our tool the user only specifies that a circle is needed within a given control circle, it’s position and radius will be computed automatically
The resulting smaller circle can be adjusted within the valid range of solutions

26. Export
Since most vector graphics systems support cubic Bézier curve’s we provide export in such format
The quintic Bézier curve is approximated by cubic Bézier segments

27. Design process

28. Future work Integration into vector graphics systems
More streamlined workflow
Use of improved transition curves

29. Pythagorean Hodographs
Theorem (Farouki): It is not possible to parameterize any plane curve, other than a straight line, by rational functions of its arc length.
No unit-speed parameterization
Arc-length in closed form:
Degree 2: possible with logarithmic terms
Degree 3: possible, but only with elliptic functions
Let us use a subset of polynomial curves that has closed form for arc-length
We cannot represent curves with rational polynomials with arc-length parameter
Even though we cannot equate the parametric speed to unity, we can at least try to turn it into something that can be integrated => arc-length computation becomes possible 29

30. Rational polynomials with rational offsets
Pottmann introduced the RPRO set of curves
The rational curve is obtained as the envelope of its tangent line:
Where h(t) is the signed distance of the tangent line from the origin and n(t) is a rational unit normal vector to the tangent line g(t) given by
We use the normal equation of the tangent lines
Generalization by using the projective dual representation of rational Bezier curves 30

31. Rational polynomials with rational offsets
The envelope of g(t) family can be found by solving a linear system for g(t) and g’(t) for x and y as a function of t
This subset of rational polynomials is closed under offsetting
The offset curve’s degree is the same as that of the curve to be offset
Arc-length is not a rational function, in general

32. Rational polynomials with rational offsets
Evolute: where
Theorem: the rational curves c(t), whose arc- length parameter s(t) is a rational function of t, are exactly the rational offsets of the RPRO curves’ evolutes.
The evolutes of polynomial or rational polynomial curves are always rational polynomials

33. Previous work Definition of “fair” is not obvious:
Farin: "A curve is fair if its curvature plot is continuous and consists of only a few monotone pieces."
A spiral's curvature varies monotonically with arc-length

35. Ábrák Circle to circle Circle in circle Valami egész ábrás bigyó
Esetleg egy visszafogott, oldalra kirakandó ábra, pl. sarokba vmi növény?

36. Motiváció, korábbi munkák Saját munka:
Bevezetés
Miről lesz szó, nagyjából a poszter lényege
Motiváció, korábbi munkák
A cikkek amiket küldtél meg még?
Saját munka:
Kör hierarchia
Körök közti különböző átmenetek
Gyakorlati megfontolások (köbös Bézier konverzió meg hasonlók)
További teendők

37. Our goal In general: In particular:
creation of aesthatically pleasing curves
In particular:
smoothly curving design element creation

38. Our goal Transition curves between two circular arcs
For design of highways: G2 with few curvature extrema
Farin: "A curve is fair if its curvature plot is continuous and consists of only a few monotone pieces."

40. Introduction and background
Pythagorean hodograph
curves with no curvature extrema for an S-shaped transition
single curvature extremum for a C-shaped transition
are suitable for the design of fair curves

42. Motivation
Ezek voltak: ...