No

The aim of this paper is to show how the spine of a PDE surface can be generated and how it can be used to efficiently parameterise a PDE surface. For the purpose of the work presented here an approximate analytic solution form for the chosen PDE is utilised. It is shown that the spine of the PDE surface is then computed as a by-product of this analytic solution. Furthermore, it is shown that a parameterisation can be introduced on the spine enabling intuitive manipulation of PDE surfaces.

No / Manipulation of PDE surfaces using a set of interactively defined parameters is considered. The PDE method treats surface design as a boundary-value problem and ensures that surfaces can be defined using an appropriately chosen set of boundary conditions and design parameters. Here we show how the data input to the system, from a user interface such as the mouse of a computer terminal, can be efficiently used to define a set of parameters with which to manipulate the surface interactively in real time.

Yes / In CAD, when considering the question of new designs of complex mechanical parts, such as engine pistons, a parametric representation of the design is usually defined. However, in general there is a lack of efficient tools to create and manipulate such parametrically defined shapes. In this paper, we show how the geometry of complex mechanical parts can be parameterised efficiently enabling a designer to create and manipulate such geometries within an interactive environment. For surface generation we use the PDE method which allows surfaces to be defined in terms of a relatively small number of design parameters. The PDE method effectively creates surfaces by using the information contained at the boundaries (edges) of the surface patch. An interactively defined parameterisation can then be introduced on the boundaries (which are defined by means of space curves) of the surface. Thus, we show how complex geometries of mechanical parts, such as engine pistons, can be efficiently parameterised for geometry manipulation allowing a designer to create alternative designs.

No / This paper presents a surface generation tool designed for the construction of aircraft geometry. The software generates complex geometries which can be crafted or modified by the user in real time. The surface generation is based on partial differential equations (PDEs). The PDE method can produce different configurations of aircraft shapes interactively. Each surface is generated by a number of curves representing the character lines of a given part of the aircraft shape that can be manipulated in real time. Different surfaces then blend to create the full shape of the airplane. An important function of the proposed tool is its ability to change the aircraft shape through the adjustments of parameters associated with the initial curves. The user can apply linear transformations to the curves generating the airplane through simple input from the computer keyboard and the mouse. The updated curves can then be used to generate the surface leading to different configurations of a given airplane shape. The work presents detailed descriptions on the PDE method, parametric design and manipulation of aircrafts along with graphical demonstrations of its abilities and a series of examples to illustrate the capacity of the methodology implemented.

No

Yes / This paper presents a method to extend PDE surfaces to high dimensional spaces. We review a common existing analytic solution, and show how it can be used straightforwardly to increase the dimension of the space the surface is embedded within. We then further develop a numerical scheme suitable for increasing the number of variables that parametrise the surface, and investigate some of the properties of this solution with a view to future work.

Yes / This paper extends the PDE method of surface generation. The governing partial differential equation is generalised to sixth order to increase its flexibility. The PDE is solved analytically, even in the case of general boundary conditions, making the method fast. The boundary conditions, which control the surface shape, are specified interactively, allowing intuitive manipulation of generic shapes. A compact user interface is presented which makes use of direct manipulation and other techniques for 3D interaction.

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.

Yes