(a) y′ = 1 + xy (b) y′ = −2xy (c) y′ = 1− 2xy Answer: If (b) were correct, then, when x = 0, we would have that y′ = 0. However, the slope of the tangent line is clearly positive when x = 0 in the picture, so (b) cannot be correct. If (a) were correct, then y′ would be positive for all positive values of x. However, the slope of the tangent line is clearly negative for most positive x, so (a) c...

Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the att...

A classic problem in geometric modelling is curve interpolation to data points. Some of the existing interpolation schemes only require point data, whereas others, require higher order information, such as tangents or curvature values, in the data points. Since measured data usually lack this information, estimation of these quantities becomes necessary. Several tangent estimation methods for p...

For nearly 20 years, looking at the tangent point on the road edge has been prominent in models of visual orientation in curve driving. It is the most common interpretation of the commonly observed pattern of car drivers looking through a bend, or at the apex of the curve. Indeed, in the visual science literature, visual orientation towards the inside of a bend has become known as "tangent poin...

A unit-vector field n : P → S2 on a convex polyhedron P ⊂ R3 satisfies tangent boundary conditions if, on each face of P , n takes values tangent to that face. Tangent unit-vector fields are necessarily discontinuous at the vertices of P . We consider fields which are continuous elsewhere. We derive a lower bound E P (h) for the infimum Dirichlet energy Einf P (h) for such tangent unit-vector f...

The main question we address is: What is the minimal information required to generate closed, nonintersecting planar boundaries? For this paper, we restrict “shape” to this meaning. More precisely, we examine whether the medial axis, together with dynamics, can serve as a language to design shapes and to effect shape changes, e.g., for modeling, to generate a morph sequence, etc. We represent t...

Operational speed is an index that represents drivers’ speeding behaviors and shows the comfort and safety levels they experience. Many models had been proposed to predict operational speed in tangent and curve segments of highways. Most of these works had used geometric, with few of them conducted using roadside variables to predict operational speed. Also, the operational speed study in multi...

Given a point A outside of a closed strictly convex plane curve γ, there are two tangent segments from A to γ, the left and the right ones, looking from point A. Problem: Does there exist a curve γ such that one can walk around it so that, at all moments, the right tangent segment is smaller than the left one? In other words, does there exist a pair of simple closed curves, γ and Γ, the former ...