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.
Navigating effectively in virtual environments at human scales is a difficult problem. However, it is even more difficult to navigate in large-scale virtual environments such as those simulating the physical Universe; the huge spatial range of astronomical simulations and the dominance of empty space make it hard for users to acquire reliable spatial knowledge of astronomical contexts. This paper introduces a careful combination of navigation and visualization techniques to resolve the unique problems of large-scale real-time exploration in terms of travel and wayfinding. For large-scale travel, spatial scaling techniques and constrained navigation manifold methods are adapted to the large spatial scales of the virtual Universe. We facilitate large-scale wayfinding and context awareness using visual cues such as power-of-10 reference cubes, continuous exponential zooming into points of interest, and a scalable world-in-miniature (WIM) map. These methods enable more effective exploration and assist with accurate context-model building, thus leading to improved understanding of virtual worlds in the context of large-scale astronomy.