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This article is a survey of research areas in which motion plays a pivotal role. The aim of the article is to review current approaches to modeling motion together with related data structures and algorithms, and to summarize the challenges that lie ahead in producing a more unified theory of motion representation that would be useful across several disciplines.

We present a computational kinematic theory of higher pairs with multiple contacts, including simultaneous contacts, intermittent contacts, and changing contacts. The theory systematizes single- and multiple-contact kinematic analysis by mapping it into geometric computation in configuration space. It derives the contact conditions, contact functions, and relations between contacts from the shapes and degrees of freedom of the parts. It helps identify common design flaws, such as undercutting, interference, and jamming, that cannot be systematically identified with current methods. We describe a program for the most common pairs: planar higher pairs with two degrees of freedom.

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We introduce the occlusion camera: a non-pinhole camera with 3D distorted rays. Some of the rays sample surfaces that are occluded in the reference view, while the rest sample visible surfaces. The extra samples alleviate disocclusion errors. The silhouette curves are pushed back, so nearly visible samples become visible. A single occlusion camera covers the entire silhouette of an object, whereas many depth images are required to achieve the same effect. Like regular depth images, occlusion-camera images have a single layer thus the number of samples they contain is bounded by the image resolution, and connectivity is defined implicitly. We construct and use occlusion-camera images in hardware. An occlusion-camera image does not guarantee that all disocclusion errors are avoided. Objects with complex geometry are rendered using the union of the samples stored by a planar pinhole camera and an occlusion camera depth image.