A cone projection maps complex plane arcs to conic sections with fixed focus and directrix.
problem Mapping complex plane arcs to conic sections with fixed focus and directrix.
method Elementary spatial construction and reciprocal lens identity.
result The cone projection maps every line not through the origin onto an arc of a conic with focus at the origin and directrix the line itself.
The cone projection fR(z)=z/(1+∣z∣/R) maps lines to conic arcs with specific properties.
problem Mapping lines to conic arcs with specific properties.
method Using a reciprocal lens identity and radial homeomorphism.
result The Self-Directrix Theorem and Confocal--Codirectrix Theorem.
A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and inve…
In this article, we use the harmonic sequence associated to a weakly conformal harmonic map f:S→S6 in order to determine explicit examples of linearly full almost complex 2-spheres of S6 with at most two singularities. We prove that the singularity type of these almost complex 2-spheres has an extra symmetry a…
This paper continues the study of a class of compact convex hypersurfaces in Euclidean space Rn+1, n≥1, which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In R3 reflectors arise naturall…
The paper studies singularities on parallels of tangent developable surfaces of frontal curves.
problem Understanding singularities on parallels of tangent developable surfaces.
method Generalization of tangent developable surfaces and parallel deformations for frontal curves in arbitrary dimensions.
result Classification of generic singularities on parallels of tangent developable surfaces for frontal curves in 3 or 4 dimensional Euclidean spaces.
Study on biconservative surfaces in 4D hyperbolic space, providing extrinsic descriptions.
problem Characterizing biconservative surfaces in hyperbolic space.
method Local extrinsic description through normal flow and analysis of vector fields.
result Classification of non-CMC, PNMC biconservative surfaces in H4 into three cases. New approach to rotational Weingarten surfaces using geometric momentum.
problem Classifying and characterizing rotational Weingarten surfaces.
method Introducing geometric linear momentum of a plane curve to reduce Weingarten conditions to differential equations.
result Classification of non-degenerate quadric surfaces and elasticoids.
Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.
problem Understanding geometric properties of curves and surfaces in Riemannian spaces.
method Developing a theoretical framework to study curves and surfaces by their angle with a parallel transported vector field.
result Surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces.