Homotopic Morphing of Planar Curves Nadav Dym Anna

Homotopic Morphing of Planar Curves Nadav Dym, Anna Shtengel and Yaron Lipman Weizmann Institute of Science

Morphing of planar curves

This is how it looks

Guaranteed morphing: Global intersection prevention [Iben et al. 2009], [Gotsman and Surazhsky 2001] [Iben et al. 2009]

Our Goal: Local intersection prevention (Regular homotopy) angle-length ours

Same guarantees-different problem Robust Fairing [Crane et al. 2013]: Regular homotopic fairing

Regular polygons

Regular homotopy: Definition non-regular

Is regular homotopy always possible? non-regular homotopy Is regular homotopy possible for the example above?

Turning number: The angle accumulated by the tangent field when traversing the curve

Theorem (Whitney-Graustein) Regular homotopy

Our Goal: Regular Homotopic Morphing No local intersections, degenerate edges Aesthetic animation

Main result Guaranteed regular homotopic morphing. Obtained by (more details later): • Convex representation of the space of regular curves • Choosing optimal regular curve with respect to fitting energy

Method

Intrinsic coordinates •

Reconstruction from intrinsic coordinates •

Curve reconstruction • … Terms we met earlier, in these coordinates:

Closing condition •

Regular polygonal curves

Angle-length method [Sederberg et al. 1993] •

Angle-length method [Sederberg et al. 1993] • • Convex problem

Our method

Feasibility theorem

Choosing an energy • Relative error Length element

Dealing with the open constraint

Optimization • Second order cone programming. • Available solvers (e. g. , Mosek)

Well defined, continuous • Smooth vertex path

Mission accomplished •

Summary: Main result • We showed how an optimal regular homotopy can be found by convex optimization over the space of regular curves. • Let’s see some examples:

Comparison with angle-length ours

angle-length ours

ours angle-length

Additional results: Briefly

Convex morphing angle-length ours

Morphing piecewise smooth curves • Special cases: Polygons, smooth curves • For smooth curves: Modification of curvature interpolation methods [Surazhsky and Elber 2002], [Saba et al. 2014]

Partial extension to polygon mesh morphing

Homotopic morphing of b-spline curves • Can we homotopically morph b-spline curves by morphing their control polygons? • Not always. • We give an easily checkable sufficient condition.

Curves with different turning number •

Problem: Flips always occur at first vertex! (for angle-length also)

Results for unmodified algorithm

Solution: Automatic selection of “correct” flipping location.

Results with automatic flip location

THE END