A survey of partial differential equations in geometric design

Castro, Gabriela; Ugail, Hassan; Willis, Philip; Palmer, Ian J.

doi:10.1007/s00371-007-0190-z

Cited by 28 publications

(5 citation statements)

References 46 publications

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“…You et al present a dynamic skin deformation method based on a timedependent partial differential equation [26]. Castro et al make a comprehensive survey on partial differential equation-based geometric design [27]. Sheng et al propose a patchwise partial differential equation function representation and use it to replace large polygon meshes [28].…”

Section: Pde-based Geometric Modellingmentioning

confidence: 99%

Improving Realism of Facial Interpolation and Blendshapes with Analytical Partial Differential Equation-Represented Physics

Day,

Xiao,

Chaudhry

et al. 2024

Axioms

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How to create realistic shapes by interpolating two known shapes for facial blendshapes has not been investigated in the existing literature. In this paper, we propose a physics-based mathematical model and its analytical solutions to obtain more realistic facial shape changes. To this end, we first introduce the internal force of elastic beam bending into the equation of motion and integrate it with the constraints of two known shapes to develop the physics-based mathematical model represented with dynamic partial differential equations (PDEs). Second, we propose a unified mathematical expression of the external force represented with linear and various nonlinear time-dependent Fourier series, introduce it into the mathematical model to create linear and various nonlinear dynamic deformations of the curves defining a human face model, and derive analytical solutions of the mathematical model. Third, we evaluate the realism of the obtained analytical solutions in interpolating two known shapes to create new shape changes by comparing the shape changes calculated with the obtained analytical solutions and geometric linear interpolation to the ground-truth shape changes and conclude that among linear and various nonlinear PDE-based analytical solutions named as linear, quadratic, and cubic PDE-based interpolation, quadratic PDE-based interpolation creates the most realistic shape changes, which are more realistic than those obtained with the geometric linear interpolation. Finally, we use the quadratic PDE-based interpolation to develop a facial blendshape method and demonstrate that the proposed approach is more efficient than numerical physics-based facial blendshapes.

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Section: Pde-based Geometric Modellingmentioning

confidence: 99%

Improving Realism of Facial Interpolation and Blendshapes with Analytical Partial Differential Equation-Represented Physics

Day,

Xiao,

Chaudhry

et al. 2024

Axioms

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show abstract

“…The second subgroup of surface generation techniques consists of automatic methods that produce free form surfaces from user drawn curve clouds in one step. These studies are based on different concepts such as energy minimization [2,15,16], partial differential equations (PDE) [17][18][19], potential field information [4,20,21]. While creating high quality surfaces, these approaches may require users to understand some of the underlying mathematical operations.…”

Section: Related Workmentioning

confidence: 99%

Free Form Surface Skinning of 3D Curve Clouds for Conceptual Shape Design

Arisoy¹,

Orbay²,

Kara

2012

Journal of Computing and Information Science in Engineering

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In product design, designers often create a multitude of concept sketches as part of the ideation and exploration process. Transforming such sketches to 3D digital models requires special expertise due to a lack of intuitive computer aided design (CAD) tools suitable for rapid modeling. Recent advances in sketch-based user interfaces and immersive environments have introduced novel curve design methods that facilitate the transformation of such sketches into 3D digital models. However, rapid surfacing of such data remains an open challenge. Based on the observation that a sparse network of curves is reasonably sufficient to convey the intended geometric shape, we propose a new method for creating approximate surfaces on curve clouds automatically. A notable property of our method is that it relieves many topological and geometric restrictions of 3D conventional networks such as the curves do not need to be connected to one another or gently drawn. Our method calculates a 3D guidance vector field in the space that the curve cloud appears. This guidance vector field helps drive a deformable closed surface onto the curves. During this deformation, surface smoothness is controlled through a set of surface smoothing and subdivision operations. The resulting surface can be further beautified by the user manually using selective surface modification and fairing operations. We demonstrate the effectiveness of our approach on several case examples. Our studies have shown that the proposed technique can be particularly useful for rapid visualization.

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“…Second group of surface modeling techniques consists of automatic methods that produce free form surfaces. These studies are based on different concepts such as optimization [2,10,11], partial differential equations (PDE) [12][13][14], parametric patches [15], subdivision surfaces [16], potential field information [4,17,18]. While creating high quality surfaces, these approaches require the user to understand and apply the boundary conditions for the desired outcomes.…”

Section: Related Workmentioning

confidence: 99%

Approximate Surfacing of Curve Clouds for Conceptual Shape Creation and Evaluation

Arisoy

Orbay

Kara

2011

Volume 5: 37th Design Automation Conference, Parts a and B

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In product design, designers often create a multitude of concept sketches as part of the ideation process. Transforming such sketches to 3D digital models usually require special expertise and effort due to a lack of suitable Computer Aided Design (CAD) tools. Although recent advances in sketch-based user interfaces and immersive environments (such as augmented/virtual reality) have introduced novel curve design tools, rapid surfacing of such data remains an open challenge. To this end, we propose a new method that enables a quick construction of approximate surfaces from a cloud of 3D curves that need not be connected to one another. Our method first calculates a vector field by discretizing the space in which the curve cloud appears into a voxel image. This vector field drives a deformable surface onto the 3D curve cloud thus producing a closed surface. The surface smoothness is achieved through a set of surface smoothing and subdivision operations. Our studies show that the proposed technique can be particularly useful for early visualization and assessment of design ideas.

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A survey of partial differential equations in geometric design

Cited by 28 publications

References 46 publications

Improving Realism of Facial Interpolation and Blendshapes with Analytical Partial Differential Equation-Represented Physics

Improving Realism of Facial Interpolation and Blendshapes with Analytical Partial Differential Equation-Represented Physics

Free Form Surface Skinning of 3D Curve Clouds for Conceptual Shape Design

Approximate Surfacing of Curve Clouds for Conceptual Shape Creation and Evaluation

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