ARAP表面建模：非线性能量优化与直观细节保持

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ARAP (As-Rigid-As-Possible) 表面建模是一种在计算机图形学领域中的重要方法，特别关注于表面变形和编辑任务的分析。其核心理念是通过观察表面的局部行为来定义和执行操作，强调保持局部变换的刚度。这种方法在设计中非常实用，因为它提供了一种非线性的、概念简单的能量形式，这个能量需要在满足特定建模约束的情况下被最小化。 ARAP建模的基本思想是将表面变形视为一个优化问题，通过最小化与原始形状的偏差，同时保持局部区域尽可能接近其原始状态。这种刚性约束使得算法能够产生细节保留且直观的变形，这对于诸如角色动画、产品设计、建筑设计等应用至关重要。它不同于传统的光滑表面处理技术，后者可能在处理细节或保持结构完整性时遇到挑战。该算法的核心在于开发了一个基于迭代的网格编辑策略，它易于实现且效率高，这使得ARAP成为实际建模场景中的理想选择。其优势在于能够在保持模型精度的同时，简化设计师的工作流程，提高了工作效率。由于其计算几何和对象建模的特性，ARAP技术主要归类于ACM CCSS标准下的I.3.5子类别——计算几何和对象建模，专注于几何算法、语言和系统。在实践中，ARAP建模可以应用于以下场景： 1. **角色动画**：在游戏角色的实时动画制作中，ARAP可以帮助保持角色肢体的自然运动和动态细节，如肌肉收缩和骨骼运动的逼真表现。 2. **工业设计**：在产品设计阶段，ARAP有助于创建精确而流畅的产品表面，如汽车外形、家具表面等，确保符合设计规范和美学要求。 3. **建筑设计**：在建筑模型中，ARAP可以帮助保持结构的稳定性和美学一致性，尤其是在处理复杂曲面和细节部分。 4. **计算机辅助制造(CAM)**：ARAP模型能提供稳定的过渡曲线，用于指导数控机床进行精细加工，确保零件的质量。总结来说，ARAP表面建模是一种强大的工具，它将概念上的简单性和实际应用的效率相结合，为计算机图形学中的表面变形和编辑任务提供了创新的解决方案。它的广泛应用表明了在当今的数字化环境中，对保持形状局部刚度的关注对于提高设计质量和工作效率的重要性。

To appear at the Eurographics Symposium on Geometry Processing (2007)

Alexander Belyaev, Michael Garland (Editors)

As-Rigid-As-Possible Surface Modeling

Olga Sorkine and Marc Alexa

TU Berlin, Germany

Abstract

Modeling tasks, such as surface deformation and editing, can be analyzed by observing the local behavior of the

surface. We argue that deﬁning a modeling operation by asking for rigidity of the local transformations is useful

in various settings. Such formulation leads to a non-linear, yet conceptually simple energy formulation, which is

to be minimized by the deformed surface under particular modeling constraints. We devise a simple iterative mesh

editing scheme based on this principle, that leads to detail-preserving and intuitive deformations. Our algorithm

is effective and notably easy to implement, making it attractive for practical modeling applications.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometr y

and Object Modeling – geometric algorithms, languages, and systems

1. Introduction

When we talk about shape, we usually refer to a property

that does not change with the orientation or position of an

object. In that sense, preserving shape means that an object

is only rotated or translated, but not scaled or sheared. In

the context of interactive shape modeling it is clear, how-

ever, that a shape has to to be stretched or sheared to satisfy

the modeling constraints placed by the user. Users intuitively

expect the deformation to preserve the shape of the object

locally, as happens with physical objects when a smooth,

large-scale deformation is applied to them. In other words,

small parts of the shape should change as rigidly as possible.

Our goal is to create a shape deformation framework that

is directly based upon the above principle. When local sur-

face deformations induced by modeling operations are close

to rigid, surface details tend to be preserved. This is a highly

important property for surface editing schemes that are

meant to be applied to complex, detailed surfaces, such as

those coming from scanning real 3D objects or from sophis-

ticated virtual sculpting tools. Recently, detail-preserving

surface editing techniques have been receiving much atten-

tion in geometric modeling research [Sor06,BS07], thanks to

the increasing proliferation of such detailed models, which

usually come in the form of irregular polyg onal meshes.

We propose the following conceptual model derived from

the principle of local rigidity: The surface of the object is

covered with small overlapping cells. An ideal deformation

seeks to keep the transformation for the surface in each cell

as rigid as possible. Overlap of the cells is necessary to avoid

surface stretching or shearing at the boundary of the cells.

Figure 1: Large deformation obtained by translating a

single vertex constraint (in yellow) using our as-rigid-as-

possible technique.

For this modeling framework to become practical we shall

deﬁne how rigidity is measured in each of the cells. A nat-

ural choice is to estimate the rigid transformation for each

cell based on corresponding points on the initial and the de-

formed surfaces, then apply this rigid transformation to the

original shape and measure the deviation to the deformed

shape. Note that estimating a linear transformation from the

corresponding surface points and then measuring its non-

orthogonality is not a good measure of rigidity: an optimal

approximate linear transformation of an arbitrary deforma-

tion could well be orthogonal. Consequently, in order to de-

ﬁne locally shape-preserving deformation, a direct optimiza-

tion of the rigid transformation should be performed instead.

Assuming we can measure deviation from rigidity in each

cell, setting up the modeling framework requires deciding

on the size and placement (or, equivalently, overlap) of

 The Eurographics Association 2007.

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