We denote that set by A(z,w,k) and (following [1]) call it the Apollonius circle of constant k associated to the points z and w. The set A(z,w,k) is a circle for all values of k other than 1 when it is a line. In this paper, we consider z,w ∈ U, show that if z = w, then necessarily A(z,w, √ (1−|w|2)/(1−|z|2)) meets the unit circle twice, consider the arc on the unit circle with those endpoints,...

The paper studies triangle centres and central lines of a suitable reference triangle. It derives formulas for the ratios of the distances between triangle centres along various central lines, including lines on which the Apollonius point, the centre of the Apollonius circle and the midpoint of the join of the two Brocard points lie. It identifies Eulerian lines, some already known, and some ne...

A self-organizing cooperation strategy for multiple unmanned surface vessels (USVs) to pursue intelligent evaders in the case of a dynamic obstacle vessel is proposed. Firstly, cooperative hunting proposed form an Apollonius circle. According escape strategies under different encirclement states, pursuers are divided into pursuit group and ambush group. The drives area completes together with I...

Conflict detection is the process by which potential collisions between aircraft are identified. Developing efficient detection methods is an engineering priority since the prevention of a mid-air collision is a major safety constraint for all airspace design. Proximity between two aircraft each flying with instantaneous uniform straight motion is re-examined. The Apollonius theorem for constru...

acceleration, 29, 46–48, 69, 93, 139 of gravity, 93, 152, 161, 264 accelerometers, 109, 114 active constraints, 214 actual flight time approximation ballistic guidance, 165–167 actual state vectors, 2 adherence property (of the supremum), 24, 196 adjoint vectors, 14, 37–39, 46, 137, 142–143, 147, 167–169, 183, 263 equation, 38 state transition matrix, 38 aerospace mission, 6, 289 aileron deflec...

In the first picture in Figure 1 there are three mutually tangent circles packed in a large circle on the outside, with four curvilinear triangles in-between. Such a configuration of circles is called a Descartes configuration and is the starting point for a bounded Apollonian circle packing (ACP). One may also consider an unbounded Apollonian circle packing in which the original Descartes conf...

UNLABELLED We present Voroprot, an interactive cross-platform software tool that provides a unique set of capabilities for exploring geometric features of protein structure. Voroprot allows the construction and visualization of the Apollonius diagram (also known as the additively weighted Voronoi diagram), the Apollonius graph, protein alpha shapes, interatomic contact surfaces, solvent accessi...