We show that every c-planar clustered graph has a straight-line c-planar drawing in which each cluster is represented by an axis-parallel rectangle, thus solving a problem posed by Eades, Feng, Lin, and Nagamochi [Algorithmica, 2006 ].

We give a linear-time algorithm to decide whether a graph has a planar LL-drawing, i.e. a planar drawing on two parallel lines. This has previously been known only for trees. We utilize this result to obtain planar drawings on three lines for a generalization of bipartite graphs, also in linear time.

Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar st-graphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(logn) running time using n= logn processors in the EREW PRAM mod...

A microrobotic leg actuated with amplifying mechanisms is developed in this letter to mimic the motion of legs natural organisms on small scale. The device has a planar structure and fabricated bulk micromachining. Two electrothermal actuators are used drive back forth workspace. actuator transmitted elastically using flexible arms. mechanism symmetrical similar standard parallel manipulators b...

In this paper we consider planar polygons with parallel opposite sides. This type of polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For this class of polygons, we define discrete versions of the area evolute, central symmetry set, equidistants and area parallels and show that they behave quite simila...

Parallel manipulators which are singular with respect to the Schönflies motion group X(a) are called Schönflies-singular, or more precisely X(a)-singular, where a denotes the direction of the rotary axis. A special class of such manipulators are architecturally singular ones because they are singular with respect to any Schönflies group. Another remarkable set of Schönfliessingular planar paral...

In this paper, 3-PPR planar parallel manipulators with ∆or U-shape base are compared with respect to their workspace size and kinematic sensitivity to joint clearances. First, the singularities and workspace of a general 3-PPR planar parallel manipulator are analyzed. Then, an error prediction model applicable to both serial and parallel manipulators is developed. As a result, two nonconvex qua...

This paper studies the kinematic geometry of general 3-RPR planar parallel robots with actuated base joints. These robots, while largely overlooked, have simple direct kinematics and large singularity-free workspace. Furthermore, their kinematic geometry is the same as that of a newly developed parallel robot with SCARA-type motions. Starting from the direct and inverse kinematic model, the exp...