We show that there exist series-parallel graphs requiring Ω(n2 √ ) area in any straight-line or poly-line grid drawing. Such a result is achieved in two steps. First, we show that, in any straight-line or poly-line drawing of K2,n, one side of the bounding box has length Ω(n), thus answering two questions posed by Biedl et al. Second, we show a family of series-parallel graphs requiring Ω(2 √ )...

We introduce the notion of generalized Delaunay triangulation of a planar straight-line graph G = (V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm...

We show that there exist series-parallel graphs requiring Ω(n2 √ ) area in any straight-line or poly-line grid drawing. Such a result is achieved in two steps. First, we show that, in any straight-line or poly-line drawing of K2,n, one side of the bounding box has length Ω(n), thus answering two questions posed by Biedl et al. Second, we show a family of series-parallel graphs requiring Ω(2 √ )...

In the main result of this paper we show that if the Julia set of a meromorphic function f contains a free analytic Jordan arc then it must in fact be a straight line, circle, segment of a straight line or an arc of a circle. If f is transcendental then the Julia set is unbounded and so consists of one or two straight line segments. We construct examples of functions whose Julia sets are of thi...

Abstract 'Fhe straight line length available for post preparations in pulps of 957 human teeth was evaluated. Analysis of radiographs in both buccolingual and mesiodistal directions showed that only 2.7% of the teeth studied were straight to the apex in both views. Average straight line lengths were calculated for the different types of teeth and used to offer guidelines for the length of pcjst...

Every planar graph has a crossings-free drawing in the plane. Formally, a straight-line drawing of a planar graph G is one where each vertex is placed at a point in the plane and each edge is represented by a straight-line segment between the two corresponding points such that no two edges cross each other, except possibly at their common endpoints. A straight-line grid drawing of G is a straig...

I.4: “If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the remaining angles respectively, namely those which the equal sides subtend.” I.9: “To bisect a given rectilinear angle.” I.10: “To bisect a given finite straight line.” I.11: “...

A morph between two straight-line planar drawings of the same graph is a continuous transformation from the first to the second drawing such that planarity is preserved at all times. Each step of the morph moves each vertex at constant speed along a straight line. Although the existence of a morph between any two drawings was established several decades ago, only recently it has been proved tha...

Achilles tendon length has been measured using a straight-line model. However, this model is associated with a greater measurement error compared with a curved-line model. Therefore, we examined the influence of neglecting the curved path of the Achilles tendon on its length change at various ranges of motion. Ten male subjects participated in this study. First, the location of the Achilles ten...