Planar Shape Interpolation with Bounded Distortion 

Renjie Chen, Ofir Weber, Daniel Keren and Mirela Ben-Chen 

Two intermediate frames interpolating between the source shape (left) and the target (right). Note the natural interpolation of both large and small scale features (the tail and the horns), seamless handling of large rotations, and small distortion in the intermediate frames.


Planar shape interpolation is widely used in computer graphics applications. Despite a wealth of interpolation methods, there is currently no approach that produces shapes with a bounded amount of distortion with respect to the input. As a result, existing interpolation methods may produce shapes that are significantly different than the input and can suffer from fold-overs and other visual artifacts, making them less useful in many practical scenarios. We introduce a novel shape interpolation scheme designed specifically to produce results with a bounded amount of conformal (angular) distortion. Our method is based on an elegant continuous mathematical formulation and provides several appealing properties such as existence and uniqueness of the solution as well as smoothness in space and time domains.

We further present a discretization and an efficient practical algorithm to compute the interpolant and demonstrate its usability and good convergence behavior on a wide variety of input shapes. The method is simple to implement and understand. We compare our method to state-of-the-art interpolation methods and demonstrate its superiority in various cases.


shape interpolation, bounded distortion, conformal mapping, conformal distortion, quasi-conformal, triangle meshes


PDF download



This figure demonstrates how our method behaves for conformal and Teichmüller maps. Blending 6 input poses (top) to generate novel poses (bottom).
In the first and second rows, the color coding shows the correspondence between the source and the target.The plot shows the angular distortion.




Matlab version
(The bundled CETM code (Conformal Equivalence of Triangle Meshes [Springborn et. al. 2008]) is kindly provided by the authors.)

C++ version
(3rd party implementation)


This work is supported by the support of Aly Kaufmann postdoctoral fellowship at the Technion, ISF grant 699/12, ISF equipment grant, Marie Curie CIG 303511 and the state of Lower-Saxony and the Volkswagen Foundation, Hannover, Germany.


@article {Chen:SIGGRAPH2013,
    title = {Planar Shape Interpolation with Bounded Distortion},
    author = {Renjie Chen and Ofir Weber and Daniel Keren and Mirela Ben-Chen},
    journal = {ACM Transactions on Graphics (Proceedings of SIGGRAPH)},
    volume = {32},
    number = {4},
    pages = {},
    year = {2013}