This book contains the Proceedings of the Ninth Mathematics of Surfaces Conference organised by the Institute of Mathematics and its Applications, and held in Cambridge, UK, on 4th - 6th September 2000. The papers describe the mathematical construction, representation, approximation, recognition, and manipulation of surfaces, with an emphasis on computational methods. Highlights include invited papers from M. Floater (SNTEF, Norway), O. Faugeras (INRIA, France), P. Giblin (Liverpool University, UK), M.-S. Kim (Seoul National University, Korea), J. Koenderink (University of Utrecht, Netherlands), N. Patrikalakis (MIT, USA), H. Pottmann (Technical University of Vienna, Austria) and R. Schaback (University of GAttingen, Germany).

This volume collects the papers accepted for presentation at the 10th IMA C- ference on the Mathematics of Surfaces, held at the University of Leeds, UK, September 15-17, 2003. As with all earlier conferences in the series, contributors to this volume come from a wide variety of countries in Asia, Europe and North America. The papers presented here re?ect the continued relevance of the s- ject, and, by comparison with the contents of earlier volumes in the series, the changing nature of application areas and mathematical techniques. To give the casual browser of these proceedings an idea of the diversity of the subject area, the papers in the present volume cover such topics and techniques as digital geometry processing, computer graphics, surface topology, medical applications, subdivision surfaces, surface reconstruction, surface triangulation, waterma- ing, data compression, data smoothing, human-computer interaction, extracting shapeformshading,surfaceheightrecovery,reverseengineering,box-splines,the Plateau Problem, splines (a variety of papers), trans?nite blending, and a?ne arithmetic. There is also a paper in memory of the late Prof. Josef Hoschek of theTechnischeUniversit. atDarmstadt,co-authoredbyaformerresearchstudent, Prof. Bert Juttler, .. on the subject of using line congruences for parameterizing special algebraic surfaces. We would like to thank all those who attended the conference and helped to makeitasuccess. Inparticularwethankthosewhocontributedtotheseproce- ings.

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

The papers in the present volume include seven invited papers, as well as 16 submitted papers. The topics covered include subdivision schemes and their continuity, polar patchworks, compressive algorithms for PDEs, surface invariant functions, swept volume parameterization, Willmore flow, computational conformal geometry, heat kernel embeddings, and self-organizing maps on manifolds, mesh and manifold construction, editing, flattening, morphing and interrogation, dissection of planar shapes, symmetry processing, morphable models, computation of isophotes, point membership classification and vertex blends. Surface types considered encompass polygon meshes as well as parametric and implicit surfaces.