Shape Morphing of Complex Geometries Using Partial Differential Equations

Abstract

Abstract-An alternative technique for shape morphing using a surface generating method using partial differential equations is outlined throughout this work. The boundaryvalue nature that is inherent to this surface generation technique together with its mathematical properties are hereby exploited for creating intermediate shapes between an initial shape and a final one. Four alternative shape morphing techniques are proposed here. The first one is based on the use of a linear combination of the boundary conditions associated with the initial and final surfaces, the second one consists of varying the Fourier mode for which the PDE is solved whilst the third results from a combination of the first two. The fourth of these alternatives is based on the manipulation of the spine of the surfaces, which is computed as a by-product of the solution. Results of morphing sequences between two topologically nonequivalent surfaces are presented. Thus, it is shown that the PDE based approach for morphing is capable of obtaining smooth intermediate surfaces automatically in most of the methodologies presented in this work and the spine has been revealed as a powerful tool for morphing surfaces arising from the method proposed here.

Facial geometry parameterisation based on Partial Differential Equations

“…One major use of the Bloor et al's PDE method was to develop computer aided manufacturing tools by which many industrial models, such as the marine propeller [7], the swirl port [31] and aircraft geometry [6], can be easily mimicked. Moreover, it has also been demonstrated that the PDE method can be applied to shape morphing [8], geometric modelling for web visualisation [9], mesh reconstruction [10], etc.…”

“…Moreover, the behaviour of PDE surfaces has been proven compatible with those underlying tensor-product surfaces, such as Bezier surface [2], B-splines [3], etc. Taken together, these advantages have contributed to a widespread adoption of the PDE methods in a range of disciplines, such as free-form surface design [4], solid modelling [5], computer aided manufacturing [6,7], shape morphing [8], web visualisation [9], mesh reconstruction [10], etc. In this paper, we address a prolonged topic, facial geometry parameterisation, by exploring and examining the feasibility of one PDE method in face modelling and facial animation.…”

Surface Modeling Using Partial Differential Equations: A Survey

“…In addition to the work carried out by Bloor and Wilson, using PDE-based surface modelling in generating three-dimensional models was also greatly promoted by Ugail et al Their work applied PDE-based approaches in optimisation of thin-walled structures [42], modelling of oedemous limbs and venous ulcers [43], reconstruction of 3D human facial images [44], shape morphing between different geometric objects [45], parametric design of aircraft geometry [46], surface profiling of micro-scale structures [47], modelling of the compaction behaviour of cylindrical pharmaceutical tablets [48], and facial geometry parameterisation [49].…”

A new surface joining technique for the design of shoe lasts

“…Castro and Ugail [19] showed a classification of morphing techniques based into two main groups that are essentially distinguished by the kind of approach employed in their development, as 'volume-based' and 'boundary-based' approaches. The first kind regarded the entire surface representing the object as a set of specific control points that can be modified.…”