Non-technical summary Transforming towards global sustainability requires a dramatic acceleration of social change. Hence, there is growing interest in finding ‘positive tipping points’ at which small interventions can trigger self-reinforcing feedbacks that accelerate systemic Examples have recently been seen power generation and personal transport, but how we identify positive points yet to o...

We discuss a characterization of positively curved surfaces M with the property that at each point the tangent plane to M is not a support plane for the entire surface. A one parameter family of examples which have special relevance with respect to the characterization is also given. Each member of this family is a smooth embedded surface in R 3 that is topologically a disk, has everywhere posi...

Given A ∈ Ωn, the n-dimensional spectral unit ball, we show that B is a ”generalized” tangent vector at A to an entire curve in Ωn if and only if B is in the tangent cone CA to the isospectral variety at A. If B 6∈ CA, then the Kobayashi–Royden pseudometric is positive at (A;B). In the case of Ω3, the zero set of this metric is completely described.

Given a strictly convex plane curve γ and a point x in its exterior, there are two tangent segments to γ from x. The equitangent locus of γ is the set of points x for which the two tangent segments have equal lengths. An equitangent n-gon of γ is a circumscribed n-gon such that the tangent segments to γ from the vertices have equal lengths, see Figure 1. For example, if γ is a circle, the equit...

The inverse tangent function can be bounded by different inequalities, for example by Shafer’s inequality. In this publication, we propose a new sharp double inequality, consisting of a lower and an upper bound, for the inverse tangent function. In particular, we sharpen Shafer’s inequality and calculate the best corresponding constants. The maximum relative errors of the obtained bounds are ap...

Studying human behavior in the natural context of everyday visual tasks--including locomotor tasks such as driving--can reveal visual strategies or even suggest underlying visual mechanisms. This paper reviews empirical and theoretical work in the past 20 years (1994-2014) on the visual control of steering a vehicle along a winding path-one of the most comprehensively studied forms of visually ...

Car drivers can employ a number of strategies to negotiate curves. The tangent point strategy proposes the use of the angle between the tangent point of the inner lane markings and the car's current heading direction, which is proportional to the required steering angle. The gaze-sampling strategy proposes to fixate points on the future path and measure the curvature of optic flow vectors which...

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. ...