# Minimum Curvature Variation Curves, Networks, and Surfaces for Fair Free-Form Shape Design

### Henry Packard Moreton

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-93-732

March 1993

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/CSD-93-732.pdf

Traditionally methods for the design of free-form curves and surfaces focus on achieving a specific level of inter-element continuity. These methods use a combination of heuristics and constructions to achieve an ultimate shape. Though shapes constructed using these methods are technically continuous, they have been shown to lack fairness, possessing undesirable blemishes such as bulges and wrinkles. Fairness is closely related to the smooth and minimal variation of curvature.

In this work we present a new technique for curve and surface design that combines a geometrically based specification with constrained optimization (minimization) of a fairness functional. The difficult problem of achieving inter-element continuity is solved simply by incorporating it into the minimization via appropriate penalty functions. Where traditional fairness measures are based on strain energy, we have developed a better measure of fairness; the variation of curvature. In addition to producing objects of clearly superior quality, minimizing the variation of curvature makes it trivial to model regular shapes such as, circles and cyclides, a class of surface including: spheres, cylinders, cones, and tori.

In this thesis we introduce: curvature variation as a fairness metric, the minimum variation curve (MVC), the minimum variation network (MVN), and the minimum variation surface (MVS). MVC minimize the arc length integral of the square of the arc length derivative of curvature while interpolating a set of geometric constraints consisting of position, and optionally tangent direction and curvature. MVN minimize the same functional while interpolating a network of geometric constraints consisting of surface position, tangent plane, and surface curvatures. Finally, MVS are obtained by spanning the openings of the MVN while minimizing a surface functional that measures the variation of surface curvature.

We present the details of the techniques outlined above and describe the trade-offs between some alternative approaches. Solutions to difficult interpolation problems and comparisons with traditional methods are provided. Both demonstrate the superiority of curvature variation as a fairness metric and efficacy of optimization as a tool in shape design, albeit at significant computational cost.

**Advisor:** Carlo H. Séquin

BibTeX citation:

@phdthesis{Moreton:CSD-93-732, Author = {Moreton, Henry Packard}, Title = {Minimum Curvature Variation Curves, Networks, and Surfaces for Fair Free-Form Shape Design}, School = {EECS Department, University of California, Berkeley}, Year = {1993}, Month = {Mar}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/5219.html}, Number = {UCB/CSD-93-732}, Abstract = {Traditionally methods for the design of free-form curves and surfaces focus on achieving a specific level of inter-element continuity. These methods use a combination of heuristics and constructions to achieve an ultimate shape. Though shapes constructed using these methods are technically continuous, they have been shown to lack fairness, possessing undesirable blemishes such as bulges and wrinkles. Fairness is closely related to the smooth and minimal variation of curvature. <p>In this work we present a new technique for curve and surface design that combines a geometrically based specification with constrained optimization (minimization) of a fairness functional. The difficult problem of achieving inter-element continuity is solved simply by incorporating it into the minimization via appropriate penalty functions. Where traditional fairness measures are based on strain energy, we have developed a better measure of fairness; the variation of curvature. In addition to producing objects of clearly superior quality, minimizing the variation of curvature makes it trivial to model regular shapes such as, circles and cyclides, a class of surface including: spheres, cylinders, cones, and tori. <p>In this thesis we introduce: curvature variation as a fairness metric, the minimum variation curve (MVC), the minimum variation network (MVN), and the minimum variation surface (MVS). MVC minimize the arc length integral of the square of the arc length derivative of curvature while interpolating a set of geometric constraints consisting of position, and optionally tangent direction and curvature. MVN minimize the same functional while interpolating a network of geometric constraints consisting of surface position, tangent plane, and surface curvatures. Finally, MVS are obtained by spanning the openings of the MVN while minimizing a surface functional that measures the variation of surface curvature. <p>We present the details of the techniques outlined above and describe the trade-offs between some alternative approaches. Solutions to difficult interpolation problems and comparisons with traditional methods are provided. Both demonstrate the superiority of curvature variation as a fairness metric and efficacy of optimization as a tool in shape design, albeit at significant computational cost.} }

EndNote citation:

%0 Thesis %A Moreton, Henry Packard %T Minimum Curvature Variation Curves, Networks, and Surfaces for Fair Free-Form Shape Design %I EECS Department, University of California, Berkeley %D 1993 %@ UCB/CSD-93-732 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/5219.html %F Moreton:CSD-93-732