433 research outputs found
Morphing Planar Graph Drawings Optimally
We provide an algorithm for computing a planar morph between any two planar
straight-line drawings of any -vertex plane graph in morphing steps,
thus improving upon the previously best known upper bound. Further, we
prove that our algorithm is optimal, that is, we show that there exist two
planar straight-line drawings and of an -vertex plane
graph such that any planar morph between and requires
morphing steps
Morphing of Triangular Meshes in Shape Space
We present a novel approach to morph between two isometric poses of the same
non-rigid object given as triangular meshes. We model the morphs as linear
interpolations in a suitable shape space . For triangulated 3D
polygons, we prove that interpolating linearly in this shape space corresponds
to the most isometric morph in . We then extend this shape space
to arbitrary triangulations in 3D using a heuristic approach and show the
practical use of the approach using experiments. Furthermore, we discuss a
modified shape space that is useful for isometric skeleton morphing. All of the
newly presented approaches solve the morphing problem without the need to solve
a minimization problem.Comment: Improved experimental result
Optimal Morphs of Convex Drawings
We give an algorithm to compute a morph between any two convex drawings of
the same plane graph. The morph preserves the convexity of the drawing at any
time instant and moves each vertex along a piecewise linear curve with linear
complexity. The linear bound is asymptotically optimal in the worst case.Comment: To appear in SoCG 201
Pole Dancing: 3D Morphs for Tree Drawings
We study the question whether a crossing-free 3D morph between two
straight-line drawings of an -vertex tree can be constructed consisting of a
small number of linear morphing steps. We look both at the case in which the
two given drawings are two-dimensional and at the one in which they are
three-dimensional. In the former setting we prove that a crossing-free 3D morph
always exists with steps, while for the latter steps
are always sufficient and sometimes necessary.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Convexity-Increasing Morphs of Planar Graphs
We study the problem of convexifying drawings of planar graphs. Given any
planar straight-line drawing of an internally 3-connected graph, we show how to
morph the drawing to one with strictly convex faces while maintaining planarity
at all times. Our morph is convexity-increasing, meaning that once an angle is
convex, it remains convex. We give an efficient algorithm that constructs such
a morph as a composition of a linear number of steps where each step either
moves vertices along horizontal lines or moves vertices along vertical lines.
Moreover, we show that a linear number of steps is worst-case optimal.
To obtain our result, we use a well-known technique by Hong and Nagamochi for
finding redrawings with convex faces while preserving y-coordinates. Using a
variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and
Nagamochi's result which comes with a better running time. This is of
independent interest, as Hong and Nagamochi's technique serves as a building
block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201
Morphing Planar Graph Drawings with Unidirectional Moves
Alamdari et al. showed that given two straight-line planar drawings of a
graph, there is a morph between them that preserves planarity and consists of a
polynomial number of steps where each step is a \emph{linear morph} that moves
each vertex at constant speed along a straight line. An important step in their
proof consists of converting a \emph{pseudo-morph} (in which contractions are
allowed) to a true morph. Here we introduce the notion of \emph{unidirectional
morphing} step, where the vertices move along lines that all have the same
direction. Our main result is to show that any planarity preserving
pseudo-morph consisting of unidirectional steps and contraction of low degree
vertices can be turned into a true morph without increasing the number of
steps. Using this, we strengthen Alamdari et al.'s result to use only
unidirectional morphs, and in the process we simplify the proof.Comment: 13 pages, 9 figure
Motion Planning for Variable Topology Trusses: Reconfiguration and Locomotion
Truss robots are highly redundant parallel robotic systems that can be
applied in a variety of scenarios. The variable topology truss (VTT) is a class
of modular truss robots. As self-reconfigurable modular robots, a VTT is
composed of many edge modules that can be rearranged into various structures
depending on the task. These robots change their shape by not only controlling
joint positions as with fixed morphology robots, but also reconfiguring the
connectivity between truss members in order to change their topology. The
motion planning problem for VTT robots is difficult due to their varying
morphology, high dimensionality, the high likelihood for self-collision, and
complex motion constraints. In this paper, a new motion planning framework to
dramatically alter the structure of a VTT is presented. It can also be used to
solve locomotion tasks that are much more efficient compared with previous
work. Several test scenarios are used to show its effectiveness. Supplementary
materials are available at https://www.modlabupenn.org/vtt-motion-planning/.Comment: 20 pages, 36 figure
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