Related papers: Stereographic Projections on Some Quadric Surfaces
We give the parameters of any evaluation code on a smooth quadric surface. For hyperbolic quadrics the approach uses elementary results on product codes and the parameters of codes on elliptic quadrics are obtained by detecting a BCH…
We construct a number of sculptures, each based on a geometric design native to the three-dimensional sphere. Using stereographic projection we transfer the design from the three-sphere to ordinary Euclidean space. All of the sculptures are…
This paper aims to develop the mathematical representation of a surface generated by elliptical arcs joining the sides of a regular polygon to a point lying vertically upward on the central axis of the polygon. The volume of the…
The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge…
We study the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over the unit sphere, use the formula to derive convexity…
We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions…
The surface reconstruction problem from sets of planar parallel slices representing cross sections through 3D objects is presented. The final result of surface reconstruction is always based on the correct estimation of the structure of the…
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the…
We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular,…
In this paper, orthogonal projection along a geodesic to the chosen k-plane is introduced using edge and Gram matrix of an n-simplex in hyperbolic or spherical n-space. The distance from a point to k-plane is obtained by the orthogonal…
We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify…
The study of quadric surfaces of revolution is a cornerstone of classical Euclidean geometry, but its extension to the three-dimensional sphere $\mathbb{S}^3$ has not been sufficiently explored. This article addresses this important gap by…
I apply the algebraic framework developed in arXiv:1101.4542 to study geometry of elliptic spaces in 1, 2, and 3 dimensions. The background material on projectivised Clifford algebras and their application to Cayley-Klein geometries is…
The projection onto the intersection of sets generally does not allow for a closed form even when the individual projection operators have explicit descriptions. In this work, we systematically analyze the projection onto the intersection…
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…
We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general…
We survey different classification results for surfaces with parallel mean curvature immersed into some Riemannian homogeneous four-manifolds, including real and complex space forms, and product spaces. We provide a common framework for…
The squircle is an intermediate shape between the square and the circle. In this paper, we examine and discuss equations for different types of squircles. We then build upon these 2D shapes to come-up with various 3D surfaces based on…
In this paper, we study the geometry of trisections on certain rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to construct trisections and related plane curves with interesting properties…
In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…