Generalized Partial Differential Equations for Interactive Design

Abstract

This paper presents a method for interactive design by means of extending the PDE based approach for surface generation. The governing partial differential equation is generalized to arbitrary order allowing complex shapes to be designed as single patch PDE surfaces. Using this technique a designer has the flexibility of creating and manipulating the geometry of shape that satisfying an arbitrary set of boundary conditions. Both the boundary conditions which are defined as curves in 3-space and the spine of the corresponding PDE are utilized as interactive design tools for creating and manipulating geometry intuitively. In order to facilitate interactive design in real time, a compact analytic solution for the chosen arbitrary order PDE is formulated. This solution scheme even in the case of general boundary conditions satisfies exactly the boundary conditions where the resulting surface has an closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical examples.

A PDE patch-based spectral method for progressive mesh compression and mesh denoising

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A PDE‐based head visualization method with CT data

“…If C 2 continuity is demanded, a sixth‐order PDE has to be used, leading to a higher computational cost. Moreover, it has been proved that the BWPDE method can be extended to an arbitrary order allowing complex shape design with a single PDE .…”

Manipulation of Parametric Surfaces Through a Simple Deformation Algorithm

“…How Bézier surfaces can be generated from boundary information through a general 4th-order PDE was tackled by Monterde and Ugail (2006). Generalizing the governing partial differential equation to arbitrary order, complex shapes were designed as single patch by Ugail (2007). Incorporating dynamic effects into a fourth order PDE, You and Zhang studied creation of 3D deformable moving surfaces (2003).…”