A shape grammar interpreter for curved shapes

Iestyn JowersUniversity of Leeds

Design Computing and Cognition 2010Workshop on Shape Grammar Implementation

11th July 2010

The problem with curves…

A nice thing about lines is they all the same…

… everywhere

This means they can all be embedded in each other

Iestyn Jowers, University of Leeds

The problem with curves…

This is not true for curves…… so need a way to compare embedding properties

Iestyn Jowers, University of Leeds

An intrinsic solution

The shape of a curve can be defined by its curvature (κ) and torsion (τ):

- κ describes how much a curve turns in a plane- τ describes how much a curve twists out of a plane

Can use these properties to compare the shape of infinite curve segments…

Iestyn Jowers, University of Leeds

An intrinsic solution

Given two parametric curves of infinite extent, C1(t) and C2(u)

their shapes can be compared according to

κ1(t) = λ-1 κ2 [u(t)]τ1(t) = λ-1 τ2 [u(t)]

for some constant λ (≠ 0) and some continuous function u(t)

C2(u)

C1(t)

An intrinsic solution

If two curve segments C1(t) and C2(u) are embedded in infinite curves of the same shape then their end points can be compared according to u(t) to determine if one can be embedded in the other

Iestyn Jowers, University of Leeds

A curved interpreter

This intrinsic comparison has been implemented for shapes composed of quadratic Bezier curves:

- parametric curves defined by three control points- planar curves so only need to compare κ- segments of parabolic curves which are symmetric

Iestyn Jowers, University of Leeds

A curved interpreter

Demo…

Iestyn Jowers, University of Leeds

An example

QI has been used to develop a shape grammar to generate Celtic knotwork patterns:

- consists of 29 rules- designs are “grown” from an initial 8 knot- each rule application results in a valid knot- braiding is maintained and closure is maintained

Iestyn Jowers, University of Leeds

An example

Demo…

Iestyn Jowers, University of Leeds

An example

Some generated designs:

Iestyn Jowers, University of Leeds