A shape grammar interpreter for curved shapes Iestyn Jowers University of Leeds Design Computing and Cognition 2010 Workshop on Shape Grammar Implementation 11 th 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, C 1 (t) and C 2 (u) their shapes can be compared according to κ 1 (t) = λ -1 κ 2 [u(t)] τ 1 (t) = λ -1 τ 2 [u(t)] for some constant λ (≠ 0) and some continuous function u(t) C2(u)C2(u) C1(t)C1(t)

An intrinsic solution If two curve segments C 1 (t) and C 2 (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