This Month's Feature Column

Moving Remy in Harmony: Pixar's Use of Harmonic Functions

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The rat in the cage

To make an engaging animated film, you need characters, like Remy from Pixar Animation Studio's Ratatouille, that are lovable with loads of personality

© Disney/Pixar

and the ability to move them in ways that are both believable and interesting.

© Disney/Pixar

While there are many mathematical issues that show up in computer animation, this article will describe some new mathematical techniques being tested at Pixar for use in upcoming films. In particular, we'll see how harmonic functions give life to animated characters by enabling their movement and expression of emotion.

Animated characters are usually constructed from a grid of points with a means to interpolate between the grid points and produce a natural, smooth appearance. On the character shown below can be seen some of the 8019 points forming the curvilinear rectangular grid on the character's body:

© Disney/Pixar

When we want to make the character move, we then need to move each of the 8019 points in the grid. This sounds like a daunting task: not only does the location of each point need to be modified, but the motion of nearby points needs to be coordinated to create the illusion of a realistic solid body.

Consequently, animators may place the character inside a "cage," a coarser grid of control points containing the character. For instance, the character above is shown contained in a cage of 112 points.

© Disney/Pixar

To move the character, we would like to move the relatively small number of control points defining the cage and have the points inside respond in some natural way. For instance, to make the character's arm bend, we could just move the relatively small number of points on the cage as shown below.

© Disney/Pixar	© Disney/Pixar

To give a more familiar example, the surface of Remy's body is defined by 9775 points while his cage has 325 control points.

© Disney/Pixar

In fact, we may give greater expressive control by placing smaller cages of control points around special features, such as Woody's eye:

© Disney/Pixar

This raises an important question that must be addressed: when one of the control points on the cage moves, how should the interior points move in response? This article will describe two new ways to approach this question.

David Austin
Grand Valley State University
austind at gvsu.edu