# Shape Deformation Models

**DOI:**https://doi.org/10.1007/978-3-319-08234-9_358-1

## Synonyms

## Definition

Shape deformation models have occupied a prominent place in computer graphics and image processing. They possess superior geometric properties of flexible nature which make them an ideal candidate for several applications in graphics and imaging industry. In this article, some basic properties of shape deformation models are presented. These properties are highlighted though examples. An industry application with respect to topology driven structures through deformation with 3D shape correspondence of varying nature further highlights the strength of these models.

## Introduction

The deformation of shapes corresponds as a basic problem in geometric modeling (González Hidalgo et al. 2013). Several methods have been developed to address rigid alignments, isometric shape articulations, part stretching nonrigid shape deformations, etc. Considering the inherent matching shapes dissimilarity, the search for correspondence becomes complex with respect to formulation of the problem computation cost. In order to deal with shapes varying in structure as well as geometry, the current techniques are basically data or knowledge based. They make use of various supervised learning or unsupervised co-analysis for collections of shape. In terms of design, these techniques give coarse as well as discrete correspondences which leaves structural discrepancies considering matched parts or their ensembles unsettled. Some examples worth mentioning are available in (Chaudhuri 2018).

The deformation process can be represented through hierarchy modeling which effectively represents object geometrically. Though this simple graphical primitives along with operators are combined together towards the formation of complicated objects. In the hierarchy of leaf nodes, there are hardware commands from where vectors are drawn, individual pixels’ colors change, and operations are performed on list of segmented polygons. With respect to proper algorithms, the users have strong intuitive feeling for manipulation of results. They can also think for every operation and can structure the objects they are looking for. These deformations enforce the user to consider an object from special topology that can be bent, twisted, tapered, compressed, expanded, or transformed towards a final construct. They have highly intuitive nature. They have visualized operations which simulate some kind of manufacturing processes towards objects’ fabrication. These deformations have also been considered towards traditional CAD/CAM modeling as well as surface patches. This decreases the data storage requirements, which simulate flexible objects geometrically. This makes it possible towards accurate modeling of physical properties for various elastic materials considering several partial differential equations. However, simple mathematical deformation techniques also exist. These simple techniques have reduced requirements in terms of computation. They model objects effectively.

A fair amount of work has been done on matching discrete structures, deformation driven correspondence, structure aware deformation, topology matching and blending, as well as co-analysis. This is in lines with deformations of shapes where the 3D geometrical structures are actively used. With this motivation, in this article we present some basic concepts related to shape deformation which are supported with illustrative examples. An industry application revolving around a deformation driven technique considering varying topology with shape correspondence in 3D is highlighted. This article is structured as follows. In proceeding section, the shape deformations overview is given. This is followed by an industry application in shape deformations. Finally, the conclusions are given.

## Shape Deformations: Overview

The shape deformation can be defined as any object’s shape change which happens because of a force applied externally or temperature change. For shape change of an object, the deformation energy is transferred through work done. For temperature change, the deformation energy is transferred through heat. The first instance is attributed towards tensile and compressive forces as well as shear, bending, or torsion. The second instance is determined through temperature and is the structural defects mobility where the movement is thermally activated. The deformation also results from strain. The deformation is also associated with stability of an elastic material through incremental deformations. It is based on superposed small deformation considering an equilibrium solution. When deformation takes place, the internal forces of inter-molecular nature take place that opposes the external forces. If external force is not adequate, the forces are enough towards fully resisting said force. This also allows objects towards new equilibrium and return to actual state as the load withdraws. The greater external forces will go towards object’s full deformation as well as its structural distortion.

Let us consider *a*_{1}, *a*_{2}, and *a*_{3} as the components of 3D vector \( \overline{a} \) in three directions.

The local deformation changes tangent space for solid. The differential vectors for solid’s substance are circulated and skewed accordingly. There is an integration of these vectors considering the position globally.

There are separate chain links considering differential vectors that may stretch as well as rotate. The deformation’s local specification is given by matrix function which may be circulating as well as skewing. The end link’s position considering chain is vector sum for previous links and is discussed here.

*a*

_{i};

*i*= 1, 2, 3 such that:

*i*th column for \( \overline{\overline{JM}} \) is achieved through partial derivative for \( \overline{H}\left(\overline{a}\right) \) considering

*a*

_{i}. The object’s surface is represented through parametric function for variables

*p*and

*q*such that:

Any tangent vector considering the surface is reached through linear combinations of partial derivatives for \( \overline{a} \) considering *p* and *q*. The normal vector direction can be obtained through cross product for linearly independent tangent vectors considering surface.

Since only normal vector’s direction is significant, it is not required to calculate determinant’s value. The jacobian’s determinant looks towards local volume ratio for each transformation point considering deformed as well as undeformed regions. Scaling is one of the simple examples of shape deformation (https://abaqus-docs.mit.edu/2017/English/SIMACAECAERefMap/simacae-t-customdeffactorframe.htm). Other notable examples include global tapering along any axis, global axial twists, global linear bends along any axis (https://abaqus-docs.mit.edu/2017/English/SIMACAEGSARefMap/simagsa-t-displayingandcustomizingadeformedshapeplot.htm), etc. We present few of these examples here.

## Global Tapering Along Any Axis

## Global Axial Twists

## Global Linear Bends Along Any Axis

- (a)
The undeformed input shape’s conversion into its tangent vectors considering differentiation

- (b)
The tangent vectors’ transformation through tangent transformation considering tangent vectors with respect to object deformed

- (c)
The integration of new tangent vectors towards new position vectors for deformed curve, surface, or solid spaces

The local towards global operations changes local tangent vectors and Jacobian matrix in terms of position vectors globally. The space’s absolute position for deformed object is specified within arbitrary constant vector integration. This gives complete deformation description which can be directly coupled towards output considering elasticity equations or finite element analysis for entities of deformable nature describing general shapes’ composition. To further augment this discussion, two significant applications include space curves as well as 3D surfaces and solids transformations available in (Chaudhuri 2018).

## Industry Application

This deformation model considers geometric as well as topological operations like part split, duplication, merging, etc. which leads towards fine-grained and piecewise results. The key aspect here is the deformation energy which affects geometric distortion. It also encourages preservation of structures as well as allows changes in topology. It accomplishes this by connecting shape parts through structural rods that behave in a similar manner as virtual springs. It allows encoding for energies coming from geometric, structural, and topological shapes. The optimal shape correspondence is reached considering the pruned beam search.

## Conclusion

In this article, we have presented some of the important conceptual aspects related to shape deformation models. The shape deformations find immense applications in computer graphics and image processing. Some of the basic properties of shape deformations are initially highlighted followed by illustrative examples. An industry application considering topology driven structures through deformation with 3D shape correspondence of varying nature justifies the significance of the shape deformation models.

## References

- Alhashim, I., Xu, K., Zhuang, Y., Cao, J., Simari, P., Zhang, H.: Deformation driven topology varying 3D shape correspondence. ACM Trans. Graph.
**34**(6), 236 (2015)CrossRefGoogle Scholar - Chaudhuri, A.: Some Investigations with Shape Deformation Models. Technical report, TR–8919. Samsung R&D Institute Delhi India (2018)Google Scholar
- González Hidalgo, M., Torres, A.M., Gómez, J.V. (eds.): Deformation Models: Tracking, Animation and Applications. Lecture Notes in Computer Vision and Biomechanics, vol. 7. Springer, New York (2013)Google Scholar