Point-Normal Subdivision Curves and Surfaces

TL;DR

This paper introduces point-normal subdivision schemes that incorporate control normals to produce smooth, circle-reproducing curves and surfaces, extending linear schemes to nonlinear, visually smooth models with guaranteed convergence.

Contribution

It generalizes linear subdivision schemes to nonlinear point-normal schemes, enabling circle and sphere reproduction, convergence, and smoothness comparable to linear schemes, even with non-flat extraordinary points.

Findings

Reproduces circles, cylinders, and spheres using control points and normals.

Converges with the same smoothness order as linear schemes.

Generates visually C^2 surfaces with non-flat extraordinary points.

Abstract

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted averages of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that generate $C^2$ subdivision surfaces with flat extraordinary points can generate visually $C^2$ subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.