Related papers: CurvedLand: An Applet for Illustrating Curved Geom…

Embedding diagrams have been used extensively to visualize the properties of curved space in Relativity. We introduce a new kind of embedding diagram based on the {\it extrinsic} curvature (instead of the intrinsic curvature). Such an…

In this paper we study maps (curved flats) into symmetric spaces which are tangent at each point to a flat of the symmetric space. Important examples of such maps arise from isometric immersions of space forms into space forms via their…

In this article, we briefly describe various tools and approaches that algebraic geometry has to offer to straighten bent objects. Throughout this article we will consider a specific example of a bent or curved piece of paper which in our…

The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…

Does text have an intrinsic curvature? Language is increasingly modeled in curved geometries - hyperbolic spaces for hierarchy, mixed-curvature manifolds for compositional structure - yet a basic scientific question remains unresolved: what…

For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane…

The quantum-phase-field concept of matter is revisited with special emphasis on the introverted view of space. Extroverted space surrounds physical objects, while introverted space lies in between physical objects. Space between objects…

We introduce a guide to help deep learning practitioners understand and manipulate convolutional neural network architectures. The guide clarifies the relationship between various properties (input shape, kernel shape, zero padding, strides…

Mapping complex input data into suitable lower dimensional manifolds is a common procedure in machine learning. This step is beneficial mainly for two reasons: (1) it reduces the data dimensionality and (2) it provides a new data…

In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential…

We propose a framework for 2D shape analysis using positive definite kernels defined on Kendall's shape manifold. Different representations of 2D shapes are known to generate different nonlinear spaces. Due to the nonlinearity of these…

We introduce an alternative formalization of curved spaces in which the concept of a pointwise affine space, as defined here, replaces that of a manifold. New or modified definitions of familiar notions from differential geometry such as…

Hilbert's two-dimensional space-filling curve is appreciated for its good locality-preserving properties and easy implementation for many applications. However, Hilbert did not describe how to generalize his construction to higher…

Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…

We give a language for geometry which makes curves and number fields look alike.

I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski Systems) living on it, giving the local directions of space and time. Relative to these systems, special…

Brain representations of curvature may be formed on the basis of either vision or touch. Experimental and theoretical work by the author and her colleagues has shown that the processing underlying such representations directly depends on…

In computer vision and medical imaging, the problem of matching structures finds numerous applications from automatic annotation to data reconstruction. The data however, while corresponding to the same anatomy, are often very different in…

The passionate plea for the use of scientific colour maps misses some aspects in the visual presentation of scientific data. While a linear colour map based on scientific human colour perception is useful for the presentation of some…

A map is an abstract visual representation of a region, taken from a given space, usually designed for final human consumption. Traditional cartography focuses on the mapping of Euclidean spaces by using some distance metric. In this paper…