Previous research on impossible figures focuses extensively on single view modeling and rendering. Existing computer games that employ impossible figures as navigation maze for gaming either use a fixed third-person view with axonometric projection to retain the figure's impossibility perception, or simply break the figure's impossibility upon view changes. In this paper, we present a new approach towards 3D gaming with impossible figures, delivering for the first time navigation in 3D mazes constructed from impossible figures. Such result cannot be achieved by previous research work in modeling impossible figures. To deliver seamless gaming navigation and interaction, we propose i) a set of guiding principles for bringing out subtle perceptions and ii) a novel computational approach to construct 3D structures from impossible figure images and then to dynamically construct the impossible-figure maze subjected to user's view. In the end, we demonstrate and discuss our method with a variety of generic maze types.

Although we can interactively rotate a 3D projected high-dimensional geometry and observe its dynamic changes, this traditional visualization method is limited and highly sensitive to the choice of viewing direction. Parallel-coordinates plots supplement this visualization scenario by providing statistical analysis of the geometry for distinct pairs of co-dimensions. Such analysis results in visual signatures that embed geometric structures such as symmetry, and thus allows us to overview the status of the missing dimensions while exploring the projected geometry. This paper presents a blue-noise sampling approach for efficient construction of continuous parallel-coordinates plots of high-dimensional geometric surfaces defined by mathematical equations. We employ the parallel-coordinates plots with the embedded visual signatures to assist the interactive exploration of high-dimensional geometries, typically for 2-manifold embedded in 4-space. While we interactively explore the 3D projected geometry, we can observe dynamic changes on its visual signature. Various geometric properties can also be identified and visualized. Moreover, we can interactively brush the plots, and see their counterparts in the 3D projection. Assorted geometric properties such as curvature can further be used to enhance the visual signature.

A 3D burr puzzle is a 3D model that consists of interlocking pieces with a single-key property. That is, when the puzzle is assembled, all the pieces are notched except one single key component which remains mobile. The intriguing property of the assembled burr puzzle is that it is stable, perfectly interlocked, without glue or screws, etc. Moreover, a burr puzzle consisting of a small number of pieces is still rather difficult to solve since the assembly must follow certain orders while the combinatorial complexity of the puzzle's piece arrangements is extremely high. In this paper, we generalize the 6-piece orthogonal burr puzzle (a knot) to design and model burr puzzles from 3D models. Given a 3D input model, we first interactively embed a network of knots into the 3D shape. Our method automatically optimizes and arranges the orientation of each knot, and modifies pieces of adjacent knots with an appropriate connection type. Then, following the geometry of the embedded pieces, the entire 3D model is partitioned by splitting the solid while respecting the assembly motion of embedded pieces. The main technical challenge is to enforce the single-key property and ensure the assembly/disassembly remains feasible, as the puzzle pieces in a network of knots are highly interlocked. Lastly, we also present an automated approach to generate the visualizations of the puzzle assembly process.

This paper introduces a method for optimizing the tiles of a quad-mesh. Given a quad-based surface, the goal is to generate a set of K quads whose instances can produce a tiled surface that approximates the input surface. A solution to the problem is a K-set tilable surface, which can lead to an effective cost reduction in the physical construction of the given surface. Rather than molding lots of different building blocks, a K-set tilable surface requires the construction of K prefabricated components only. To realize the K-set tilable surface, we use a cluster-optimize approach. First, we iteratively cluster and analyze: clusters of similar shapes are merged, while edge connections between the K quads on the target surface are analyzed to learn the induced flexibility of the K-set tilable surface. Then, we apply a non-linear optimization model with constraints that maintain the K quads connections and shapes, and show how quad-based surfaces are optimized into K-set tilable surfaces. Our algorithm is demonstrated on various surfaces, including some that mimic the exteriors of certain renowned building landmarks.