As-Killing-as-possible vector fields for planar deformation


Cartoon animation, image warping, and several other tasks in two-dimensional computer graphics reduce to the formulation of a reasonable model for planar deformation. A deformation is a map from a given shape to a new one, and its quality is determined by the type of distortion it introduces. In many applications, a desirable map is as isometric as possible. Finding such deformations, however, is a nonlinear problem, and most of the existing solutions approach it by minimizing a nonlinear energy. Such methods are not guaranteed to converge to a global optimum and often suffer from robustness issues. We propose a new approach based on approximate Killing vector fields (AKVFs), first introduced in shape processing. AKVFs generate near-isometric deformations, which can be motivated as direction fields minimizing an “as-rigid-as-possible” (ARAP) energy to first order. We first solve for an AKVF on the domain given user constraints via a linear optimization problem and then use this AKVF as the initial velocity field of the deformation. In this way, we transfer the inherent nonlinearity of the deformation problem to finding trajectories for each point of the domain having the given initial velocities. We show that a specific class of trajectories — the set of logarithmic spirals — is especially suited for this task both in practice and through its relationship to linear holomorphic vector fields. We demonstrate the effectiveness of our method for planar deformation by comparing it with existing state-of-the-art deformation methods.

Download publication

Related Resources

See what’s new.



Blocks-to-CAD: A Cross-Application Bridge from Minecraft to 3D Modeling

Learning a new software application can be a challenge, requiring the…



AvatAR: An Immersive Analysis Environment for Human Motion Data Combining Interactive 3D Avatars and Trajectories

Analysis of human motion data can reveal valuable insights about the…



Toward a Multi-Level and Multi-Paradigm Platform for Building Occupant Simulation

In recent years, simulation has been used to investigate the mutual…



Supporting Creative Concept Generation by Engineering Students with Biomimetic Design

Biomimetic design uses ideas from biology as inspiration for design,…

Get in touch

Something pique your interest? Get in touch if you’d like to learn more about Autodesk Research, our projects, people, and potential collaboration opportunities.

Contact us