This book constitutes the refereed proceedings of the 13th IMA International Conference on the Mathematics of Surfaces held in York, UK in September 2009.

Computing Isophotes on Free-Form Surfaces Based on Support Function Approximation.- Swept Volume Parameterization for Isogeometric Analysis.- Numerical Checking of C 1 for Arbitrary Degree Quadrilateral Subdivision Schemes.- The Invariant Functions of the Rational Bi-cubic Bézier Surfaces.- Crazy Cuts: Dissecting Planar Shapes into Two Identical Parts.- Piecewise Rational Manifold Surfaces with Sharp Features.- Deriving Box-Spline Subdivision Schemes.- Geometric Characterizations of Graphs Using Heat Kernel Embeddings.- Compressive Algorithms-Adaptive Solutions of PDEs and Variational Problems.- Symmetry-Aware Mesh Processing.- Recent Advances in Computational Conformal Geometry.- Finite Curvature Continuous Polar Patchworks.- A New Approach to Point Membership Classification in B-rep Solids.- Probabilistic Modeling and Visualization of the Flexibility in Morphable Models.- Parameterizing Singularities of Positive Integral Index.- Two Step Time Discretization of Willmore Flow.- Surface Triangulation and the Downstream Effects on Flattening.- On Mesh Editing, Manifold Learning, and Diffusion Wavelets.- Gradient Approximation on Uniform Meshes by Finite Differences and Cubic Spline Interpolation.- Metric Methods in Surface Triangulation.- Setback Vertex Blends in Digital Shape Reconstruction.- Learning a Self-organizing Map Model on a Riemannian Manifold.- Surface Quasi-Conformal Mapping by Solving Beltrami Equations.