A pseudo-haptic knot diagram interface

Abstract

To make progress in understanding knot theory, we will need to interact with the projected representations of mathematical knots which are of course continuous in 3D but significantly interrupted in the projective images. One way to achieve such a goal would be to design an interactive system that allows us to sketch 2D knot diagrams by taking advantage of a collision-sensing controller and explore their underlying smooth structures through a continuous motion. Recent advances of interaction techniques have been made that allow progress to be made in this direction. Pseudo-haptics that simulates haptic effects using pure visual feedback can be used to develop such an interactive system. This paper outlines one such pseudo-haptic knot diagram interface. Our interface derives from the familiar pencil-and-paper process of drawing 2D knot diagrams and provides haptic-like sensations to facilitate the creation and exploration of knot diagrams. A centerpiece of the interaction model simulates a "physically" reactive mouse cursor, which is exploited to resolve the apparent conflict between the continuous structure of the actual smooth knot and the visual discontinuities in the knot diagram representation. Another value in exploiting pseudo-haptics is that an acceleration (or deceleration) of the mouse cursor (or surface locator) can be used to indicate the slope of the curve (or surface) of whom the projective image is being explored. By exploiting these additional visual cues, we proceed to a full-featured extension to a pseudo-haptic 4D visualization system that simulates the continuous navigation on 4D objects and allows us to sense the bumps and holes in the fourth dimension. Preliminary tests of the software show that main features of the interface overcome some expected perceptual limitations in our interaction with 2D knot diagrams of 3D knots and 3D projective images of 4D mathematical objects.