We build a surface between two complex closed spatial spline curves. Our algorithm allows the input curves to have differing degree, parameterization, and shape. The construction of this surface is useful in geometric modeling including applications for filleting, hole filling methods, and generation of parting surfaces for injection mold constructions.

Joel DanielsElaine Cohen

University of Utah

Algorithm Overview

Stage 1: Solving the navigation problem

ACKNOWLEDGEMENTSThis work was supported in part by NIH 673996. All opinions, findings, conclusions or recommendations expressed on this poster are those of the author and do not necessarily reflect the views of the sponsoring agencies.

Stage 2: Building a planar mesh (step 1)

Stage 3: Solving boundary continuity and parameterization challenges (steps 2-8) we resolve the brown (overlapping) regions and white (void) regions)

Stage 4: Computing height components (steps 9-12)

3. Assign these fields to projected curves’ skeleton (medial axis approximation)

4. Trace vertex trajectories from all control points through the new field without local minima!

5. Add path repulsion to avoid intersections with neighbor trajectories

Results1. Create surfaces between two spatial curves.

A. Novel solution to the vertex trajectory problem within curve deformations.1. Guarantees avoidance of local and global

self-intersections.B. Novel solution to surface generation.

1. Approach considers surface generation as a constrained deformation to avoid intersections between two intermediate curve shapes.

2. Broader applications:A. Robot path planning, hole filling procedures, filleting for

modeling by example applications, parting surface generation, and other geometric modeling challenges.

1. Standard repulsion fields trap vertex paths at local minima!

2. Define new local vector field functions:a. Attraction

(blue)b. Flow

(green)

Stage 2 Stage 3Stage 1

Stage 4

Research Challenges• Navigational problems due to local minima in gradient descent algorithms.• Quad mesh extraction without self-intersections.• Continuous parameterization and degree changes across the boundary of two surfaces.• Curve deformations without self-intersections.• Curve deformations between curves with dissimilar degree, parameterization, and shape.• Generation of a smooth height field surface from a planar parameterized spline surface.• Deformations between spatial spline curves.

Problem Statement