Related papers: Interactive Design and Optics-Based Visualization …
Learning low-dimensional numerical representations from symbolic data, e.g., embedding the nodes of a graph into a geometric space, is an important concept in machine learning. While embedding into Euclidean space is common, recent…
Boundaries such as walls, windows, and doors are ubiquitous in the physical world, yet their potential in mixed reality (MR) remains underexplored. We present Unbounded, a Research through Design inquiry into object--boundary interaction…
A recent development, called isogeometric analysis, provides a unified approach for design, analysis and optimization of functional products in industry. Traditional volume rendering methods for inspecting the results from the numerical…
Transformation optics offers an unconventional approach to the control of electromagnetic fields. A transformation optical structure is designed by first applying a form-invariant coordinate transform to Maxwell's equations, in which part…
Non-isotropic geometries are of interest to low-dimensional topologists, physicists and cosmologists. However, they are challenging to comprehend and visualize. We present novel methods of computing real-time native geodesic rendering of…
We introduce the concept of non-uniform metamirrors (full-reflection metasurfaces) providing full control of reflected wave fronts independently from the two sides of the mirror. Metamirror is a single planar array of electrically small…
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom.com.
Screen-based Augmented Reality (AR) systems can be built as a window into the real world as often done in mobile AR applications or using the Magic Mirror metaphor, where users can see themselves with augmented graphics on a large display.…
Heterogeneous object modelling is an emerging area where geometric shapes are considered in concert with their internal physically-based attributes. This paper describes a novel theoretical and practical framework for modelling volumetric…
Students in introductory physics courses struggle to understand virtual image formation by a plane mirror and the proper construction of ray diagrams. This difficulty, if not sufficiently addressed, results in further problems throughout…
This paper proposes a novel algorithm of discovering the structure of a kaleidoscopic imaging system that consists of multiple planar mirrors and a camera. The kaleidoscopic imaging system can be recognized as the virtual multi-camera…
We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean 3-space. Since humans cannot visualise Euclidean spaces of higher dimensions, it appears to be impossible to give pictorial representations of higher-dimensional…
A method for displaying volumetric images, which exploits our binocular vision and does not require eyewear, is discussed. The display can be rendered as a matrix of pivoting micromirrors irradiated by a light beam; each micromirror focuses…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to…
Embeddings are ubiquitous in machine learning, appearing in recommender systems, NLP, and many other applications. Researchers and developers often need to explore the properties of a specific embedding, and one way to analyze embeddings is…
In the era of foundation models and Large Language Models (LLMs), Euclidean space is the de facto geometric setting of our machine learning architectures. However, recent literature has demonstrated that this choice comes with fundamental…
The double orthogonal projection of the 4-space onto two mutually perpendicular 3-spaces is a method of visualization of four-dimensional objects in a three-dimensional space. We present an interactive animation of the stereographic…
This paper considers an extension of origami geometry to the case of "folding" a three dimensional (3D) space along a plane. First, all possible incidence constraints between given points, lines and planes are analyzed by using the geometry…
This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and…