# Minimizing Curvature Variation for Aesthetic Surface Design

### Pushkar Prakash Joshi

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2008-129

October 7, 2008

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-129.pdf

We investigate the usability of functional surface optimization for the design of free-form shapes. The optimal shape is subject to only a few constraints and is influenced largely by the choice of the energy functional. Among the many possible functionals that could be minimized, we focus on third-order functionals that measure curvature variation over the surface.

We provide a simple explanation of the third-order surface behavior and decompose the curvature-variation function into its Fourier components. We extract four geometrically intuitive, parameterization-independent parameters that completely define the third order shape at a surface point. We formulate third-order energy functionals as functions of these third-order shape parameters.

By computing the energy minimizers for a number of canonical input shapes, we provide a catalog of diverse functionals that span a reasonable domain of aesthetic styles. The functionals can be linearly combined to obtain new functionals with intermediate aesthetic styles. Our side-by-side tabular comparison of functionals helps to develop an intuition for the preferred aesthetic styles of the functionals and to predict the aesthetic styles preferred by a new combination of the functionals.

To compare the shapes preferred by the functionals, we built a robust surface optimization system. We represent shapes using Catmull--Clark subdivision surfaces, with the control mesh vertices acting as degrees of freedom for the optimization. The energy is minimized by an off-the-shelf implementation of a quasi-Newton method. We discuss some future work for further improving the optimization system and end with some conclusions on the use of optimization for aesthetic design.

**Advisor:** Carlo H. Séquin

BibTeX citation:

@phdthesis{Joshi:EECS-2008-129, Author = {Joshi, Pushkar Prakash}, Title = {Minimizing Curvature Variation for Aesthetic Surface Design}, School = {EECS Department, University of California, Berkeley}, Year = {2008}, Month = {Oct}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-129.html}, Number = {UCB/EECS-2008-129}, Abstract = {We investigate the usability of functional surface optimization for the design of free-form shapes. The optimal shape is subject to only a few constraints and is influenced largely by the choice of the energy functional. Among the many possible functionals that could be minimized, we focus on third-order functionals that measure curvature variation over the surface. We provide a simple explanation of the third-order surface behavior and decompose the curvature-variation function into its Fourier components. We extract four geometrically intuitive, parameterization-independent parameters that completely define the third order shape at a surface point. We formulate third-order energy functionals as functions of these third-order shape parameters. By computing the energy minimizers for a number of canonical input shapes, we provide a catalog of diverse functionals that span a reasonable domain of aesthetic styles. The functionals can be linearly combined to obtain new functionals with intermediate aesthetic styles. Our side-by-side tabular comparison of functionals helps to develop an intuition for the preferred aesthetic styles of the functionals and to predict the aesthetic styles preferred by a new combination of the functionals. To compare the shapes preferred by the functionals, we built a robust surface optimization system. We represent shapes using Catmull--Clark subdivision surfaces, with the control mesh vertices acting as degrees of freedom for the optimization. The energy is minimized by an off-the-shelf implementation of a quasi-Newton method. We discuss some future work for further improving the optimization system and end with some conclusions on the use of optimization for aesthetic design.} }

EndNote citation:

%0 Thesis %A Joshi, Pushkar Prakash %T Minimizing Curvature Variation for Aesthetic Surface Design %I EECS Department, University of California, Berkeley %D 2008 %8 October 7 %@ UCB/EECS-2008-129 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-129.html %F Joshi:EECS-2008-129