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The rules in a shape grammar apply in terms of embedding to take advantage of the parts that emerge visually in the appearance of shapes. While the shapes are kept unanalyzed as a computation moves forward, part-structures for shapes can be…
The efficiency of modern computer graphics allows us to explore collections of space curves simultaneously with "drag-to-rotate" interfaces. This inspires us to replace "scatterplots of points" with "scatterplots of curves" to…
The directed landscape is a prominent model of random geometry which is believed to be the universal scaling limit of all planar random geometries in the Kardar-Parisi-Zhang universality class. It comes equipped with a few different natural…
Measuring comodules are defined and shown to provide a useful generalization of the set of maps between modules with a broad range of applications. Three applications are described. Connections on bundles are described in terms of measuring…
Solving a long-standing open question in convex geometry, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, we use an easy curvature definition imitating the idea of…
We propose a novel approach for performing convolution of signals on curved surfaces and show its utility in a variety of geometric deep learning applications. Key to our construction is the notion of directional functions defined on the…
In Finsler geometry, we use calculus to study the geometry of regular inner metric spaces. In this note I will briefly discuss various curvatures and their geometric meanings from the metric geometry point of view, without going into the…
In this paper we construct the action describing dynamics of the particle moving in curved spacetime, with a non-trivial momentum space geometry. Curved momentum space is the core feature of theories where relative locality effects are…
We propose a deep representation of appearance, i. e., the relation of color, surface orientation, viewer position, material and illumination. Previous approaches have useddeep learning to extract classic appearance representationsrelating…
To unify general relativity and quantum theory is hard in part because they are formulated in two very different mathematical languages, differential geometry and functional analysis. A natural candidate for bridging this language gap, at…
3D perception of object shapes from RGB image input is fundamental towards semantic scene understanding, grounding image-based perception in our spatially 3-dimensional real-world environments. To achieve a mapping between image views of…
Despite high-dimensionality of images, the sets of images of 3D objects have long been hypothesized to form low-dimensional manifolds. What is the nature of such manifolds? How do they differ across objects and object classes? Answering…
Implicit surface representations, such as signed-distance functions, combined with deep learning have led to impressive models which can represent detailed shapes of objects with arbitrary topology. Since a continuous function is learned,…
In this paper we define a numerical shape invariant of a continuous map called shape dimension of a map, which generalizes the shape dimension of a topological space. Some basic properties and applications of this invariant are given. The…
World models compress rich sensory streams into compact latent codes that anticipate future observations. We let separate agents acquire such models from distinct viewpoints of the same environment without any parameter sharing or…
This article establishes several remarkably simple identities relating certain metric invariants of level curves of real and complex functions. In particular, we relate lengths of level curves to their curvature and to the gradient field of…
Subsurface projection has become indispensable in studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two…
The increasingly common applications of machine-learning schemes to atomic-scale simulations have triggered efforts to better understand the mathematical properties of the mapping between the Cartesian coordinates of the atoms and the…
Using an interface inserted in a background mesh is an alternative way of constructing a complex geometrical shape with a relative low meshing efforts. However, this process may require special treatment of elements cut by the interface.…
It is hard to imagine curved spacetimes of General Relativity. A simple but powerful way how to achieve this is visualizing them via embedding diagrams of both ordinary geometry and optical reference geometry. They facilitate to gain an…