Pseudohaptic interaction with knot diagrams

Abstract

To make progress in understanding knot theory, we need to interact with the projected representations of mathematical knots, which are continuous in three dimensions (3-D) but significantly interrupted in the projective images. One way to achieve such a goal is to design an interactive system that allows us to sketch two-dimensional (2-D) 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 in this direction. Pseudohaptics that simulate haptic effects using pure visual feedback can be used to develop such an interactive system. We outline one such pseudohaptic knot diagram interface. Our interface derives from the familiar pencil-and-paper process of drawing 2-D 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 pseudohaptics 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 which the projective image is being explored. By exploiting these additional visual cues, we proceed to a full-featured extension to a pseudohaptic four-dimensional (4-D) visualization system that simulates the continuous navigation on 4-D 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 2-D knot diagrams of 3-D knots and 3-D projective images of 4-D mathematical objects.