A 2-dimensional point-line framework is a collection of points and lines in the plane which are linked by pairwise constraints that fix some angles between pairs of lines and also some point-line and point-point distances. It is rigid if every contin
uous motion of the points and lines which preserves the constraints results in a point-line framework which can be obtained from the initial framework by a translation or a rotation. We characterise when a generic point-line framework is rigid. Our characterisation gives rise to a polynomial algorithm for solving this decision problem.
A 2-dimensional framework is a straight line realisation of a graph in the Euclidean plane. It is radically solvable if the set of vertex coordinates is contained in a radical extension of the field of rationals extended by the squared edge lengths.
We show that the radical solvability of a generic framework depends only on its underlying graph and characterise which planar graphs give rise to radically solvable generic frameworks. We conjecture that our characterisation extends to all graphs.
TW Coronae Borealis (TW CrB) is a binary system likely to be active showing evidence of starspots and a hotspot. We calculated a new ephemeris based on all available timings from 1946 and find the period to be 0.58887492(2) days. We have revised the
average rate of change of period down from 1.54(16) x 10-7 days yr-1 to 6.66(14) x10-8 days yr-1. Based on light curve simulation analysis we conclude that the two stars are close to filling their Roche lobes, or possibly that one of the stars Roche lobe has been filled. The modelling also led to a hotspot and two starspots being identified. Conservative mass transfer is one of a number of possible mechanisms considered that could explain the change in period, but the mass transfer rate would be significantly lower than previous estimates. We found evidence that suggests the period was changing in a cyclical manner, but we do not have sufficient data to make a judgement on the mechanism causing this variation. The existence of a hotspot suggests mass transfer with a corresponding increase in the amplitude in the B band as compared with the R and V bands. The chromospheric activity implied by the starspots make this binary a very likely X-ray source.
Given a rigid realisation of a graph $G$ in ${mathbb R}^2$, it is an open problem to determine the maximum number of pairwise non-congruent realisations which have the same edge lengths as the given realisation. This problem can be restated as findin
g the number of solutions of a related system of quadratic equations and in this context it is natural to consider the number of solutions in ${mathbb C}^2$ rather that ${mathbb R}^2$. We show that the number of complex solutions, $c(G)$, is the same for all generic realisations of a rigid graph $G$, characterise the graphs $G$ for which $c(G)=1$, and show that the problem of determining $c(G)$ can be reduced to the case when $G$ is $3$-connected and has no non-trivial $3$-edge-cuts. We consider the effect of the Henneberg moves and the vertex-splitting operation on $c(G)$. We use our results to determine $c(G)$ exactly for two important families of graphs, and show that the graphs in both families have $c(G)$ pairwise equivalent generic real realisations. We also show that every planar isostatic graph on $n$ vertices has at least $2^{n-3}$ pairwise equivalent real realisations.
Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in R^d. These form the basis for a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are
obtained for the Jacobian of diverse framework systems, including constrained point-line systems that appear in CAD, body-pin frameworks, hybrid systems of distance constrained objects and infinite bar-joint frameworks. This leads to generalised forms of the Fowler-Guest character formula together with counting rules in terms of counts of symmetry-fixed elements. Necessary conditions for isostaticity are obtained for asymmetric frameworks, both when symmetries are present in subframeworks and when symmetries occur in partition-derived frameworks.
SimDialog is a visual editor for dialog in computer games. This paper presents the design of SimDialog, illustrating how script writers and non-programmers can easily create dialog for video games with complex branching structures and dynamic respons
e characteristics. The system creates dialog as a directed graph. This allows for play using the dialog with a state-based cause and effect system that controls selection of non-player character responses and can provide a basic scoring mechanism for games.
