Tracking Point-Curve Critical Distances Xianming Chen, Elaine Cohen, Richard Riesenfeld School of Computing, University of Utah

Critical Distance (CD)

Type Discriminant D

Example: Min Dist

Example: Max Dist

Example: Degenerate CD

Higher Order Degenerate CD

Distance Tracking Problem Given critical distances of P to the curve If P is perturb on the plane by –Create any new CDs if any –Annihilate any old CDs if any –Evolve the rest of CDs Distance tracking without global searching?

CD as a Space Point

Normal Bundle

Lifted Normal Bundle implicit surface = Locus of CDs

Lifting the Perturbation

Sectional Curve

Tangent Vector Field

Evolution

Transition

Transition Type Classification

An Example

2-Stage Detection Algorithm Line hits bounding box of evolute Line intersect diagonal of hit box

Extended Evolute From one of our other work

C 1 situation Extra transition events at C 1 breaks But only evolution algorithm is required Evolve CD wrt left and right segment –Keep valid result, and discard invalid one Resulting annihilation or creation accordingly

C 0 situation Convert to 2 collapsed C 1 breaks, connected by an imaginary arc of (positive or negative) infinite curvature

Conclusion Solve dynamic critical distances of plane point to static plane C 0 curve. –Implicit surface formulation, i.e., lifted normal bundle –Construct vector field T –Evolve CD following T –Track topology of CD by 2 nd order computation Covariant derivative of T wrt T –Classification of transition via pre-computed kk´ sign –Detection via intersecting line segment to evolute. Distance tracking without global searching

Thank you!