Implicit Representation of Open B-Spline Curves Using Thin Offset Polygons

Use Edit Mode to move, add, or delete control points. The gray line is the open control polygon and the red line is the open B-spline. The thin polygon is generated from consistently oriented normals determined by the order of the sampled points.

Edit Mode

Move drags control points. Add places a new point at the end of the open control polygon. Delete removes a clicked control point when the remaining points can support the selected degree.

B-spline degree

Maximum degree is also limited by the number of control points.

Number of samples

The open samples are traversed forward and backward. Timing below shows how much of the range remains interactive.

Ribbon Thickness:

Offsets sampled curve points equally in the positive and negative normal directions.

Normal Smoothing Span:

Span 1 uses one sample before and one after. Larger spans use more widely separated ordered samples to smooth rapid tangent/normal changes.

Boolean Operations

Rvachev α

0.00

4.0

p ranges from 1 to 1000 on an exponential slider, giving much more range at the high end.

Display

Ripple frequency

0.22

Show offset ribbon polygon

Computed monotone formula

Open curve as a thin closed polygon. For each ordered curve sample P_i, a tangent is estimated from samples before and after it. Rotating this consistently oriented tangent by 90° gives a consistent normal N_i. Two new points P_i⁺ = P_i + rN_i and P_i⁻ = P_i − rN_i form the two sides of a ribbon of thickness 2r.

Closed nonzero-area construction. The positive-offset points are traversed forward and the negative-offset points backward. The two connecting edges at the beginning and end automatically form the end caps. For r > 0 this is an ordinary closed polygon with nonzero area, so the standard monotone-formula algorithm can be applied without the zero-area degeneracy.

Normal smoothing. Span 1 estimates each interior tangent from one sample before and one sample after, analogous to the centered tangent used in Catmull–Rom constructions. Increasing the span uses more widely separated ordered samples. This smooths rapid changes in the estimated normal without changing the B-spline itself.

Open-curve limit. Reducing r makes the ribbon thinner. As r approaches zero, the two offset boundaries approach the original open B-spline. Thus the open curve can be studied as the limiting case of an ordinary implicit closed region.

Approximation and performance. Sampling approximates the B-spline geometrically; Akleman–Chen Approximate changes the functional Boolean combinations. These are independent approximations. The displayed computation time lets us determine experimentally how many samples remain practical for dynamic interaction.

Monotone Polygon Representation. D. Dobkin, L. Guibas, J. Hershberger, and J. Snoeyink (1988), “An Efficient Algorithm for Finding the CSG Representation of a Simple Polygon,” Computer Graphics, 22(4), 31–40.

Rvachev Operation. A. Pasko, V. Adzhiev, A. Sourin, and V. Savchenko (1995), “Function Representation in Geometric Modeling: Concepts, Implementations and Applications,” The Visual Computer, 11, 429–446, presenting the Rvachev formulation used here.

Akleman–Chen Operations. E. Akleman and J. Chen (1999), “Constant Time Updateable Operations for Implicit Shape Modeling” (PDF).