Employing Pythagorean Employing Pythagorean Hodograph Curves for Artistic Hodograph Curves for Artistic

PatternsPatterns

Conference of PhD Students in Computer Science

June 29 - July 2, 2010Szeged, Hungary

Gergely Klár, Gábor Valasek

Eötvös Loránd UniversityFaculty of Informatics

GoalGoal Create a tool to aid the design of

aesthetical, fair curves In particular design element creator

for vines, swirls, swooshes and floral components

Previous workPrevious 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

Previous workPrevious work Plants generated with L-systems,

proposed by Prusinkiewicz and Lindenmayer

Previous workPrevious 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

Previous workPrevious work Xu and Mould created ornamental

patters by simulating a charged particle's movement in a magnetic field (magnetic curves)

OurOur focusfocus 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 states

Our focusOur focusTo 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

Designer controlDesigner control An intuitive way to control our

curves is required Use hiearchy of circles:

Designer control – an Designer control – an alternativealternativeIf 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

Designer control – an Designer control – an alternativealternativeGiven are position, tangent, and

curvature data at each the endpointFind a Bézier curve which

reconstructs these quantities at it’s endpoints

These are 2x4 scalar constraints on each segment

Position: 2 scalarTangent: 1 scalarCurvature: 1 scalar

Designer control – an alternative Designer control – an alternative

Cubic Bézier solution for GHCubic Bézier solution for GHA cubic Bézier curve has 8 scalar

degrees of freedomA quadratic equation system

results from

Designer control – an alternative Designer control – an alternative

Cubic Bézier solution for GHCubic Bézier solution for GHWith appropriate geometric constraints on position, tangent and curvature the following system has positive real roots

Resulting curve is not a spiral

Designer controlDesigner controlLet us use Pythagorean Hodograph

spirals for the transition curves

curve curve’s hodograph

Let the parameterization be such that

For some integral polynomial The arc-length

can be expressed in closed-form

Pythagorean HodographsPythagorean 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 relative primes.

Pythagorean HodographsPythagorean Hodographs

Pythagorean HodographsPythagorean HodographsPH 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

Pythagorean HodographsPythagorean HodographsArc-length is a polynomialOffset of a degree n PH curve is a

rational polynomial of degree 2n-1For practical usage

◦Cubic PH curves cannot have an inflection point

◦We use quintic PH curves

MethodMethodLet the user create a hierarchy of

control circlesCreate 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

Circle-to-circle transitionCircle-to-circle transitionThe circles have to be non-

touching and non-overlappingWe used the work of Walton and

Meek to define the quintic PH curve’s control points

Circle-to-circle transitionCircle-to-circle transition

Circle-to-circle transitionCircle-to-circle transitionThe circle centres have to be

within a certain distance (depending on their radii)

We have to solve

Where

Circle-in-circle transitionCircle-in-circle transitionA 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

Circle-in-circle transitionCircle-in-circle transition

Circle-in-circle transitionCircle-in-circle transitionConstraints 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

ExportExportSince 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

Design processDesign process

Future workFuture workIntegration into vector graphics

systemsMore streamlined workflow Use of improved transition curves