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: Min Dist

Example: Max Dist

Example: Max Dist

Example: Degenerate CD

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 2nd 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!