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Intersection curve
About: Intersection curve is a research topic. Over the lifetime, 441 publications have been published within this topic receiving 7788 citations.
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TL;DR: In this paper, the topographies of two potential energy surfaces are examined in the vicinity of their intersection, and two main patterns are found: peaked and sloped topologies are discussed explicitly.
Abstract: The topographies of two potential energy surfaces are examined in the vicinity of their intersection. A brief account of the basic theory is given and the possible surface types are discussed explicitly. Two main patterns are found. One of these (‘‘peaked’’) has the character of a tilted double cone in that the lower (upper) surface decreases (increases) in all directions from the intersection which is a point where an infinite number, in fact, all orthogonal trajectories emanate. The other pattern (‘‘sloped’’) results when both surfaces are monotonically sloped and touch each other along the slope, with most orthogonal trajectories bypassing the intersection. When the latter pattern prevails, the intersection can lie on a steepest descent line which originates at a transition state and hence may qualify as a reaction path model. An intermediate pattern, involving a horizontal slope on both surfaces, is also possible. The topographical patterns also differ markedly with respect to the bunching of the steepest descent lines. In general, the latter tend to veer away from the intersection on the lower surface favoring bifurcations, but are funneled towards the intersection on the upper surface, making the vicinity of the intersection a region favoring radiationless transitions. The various cases are classified and illustrated through quantitative graphs of contours and orthogonal trajectories.
428 citations
TL;DR: An algorithm to test whether their intersection is empty, and if so to find a separating plane, and to construct their intersection polyhedron is developed, which runs in timeO (n log n), where n is the sum of the numbers of vertices of the two polyhedra.
311 citations
TL;DR: A numerical tracing procedure is given in which a third-order Taylor approximant is constructed for taking steps of variable length, and the points so found are improved by Newton iteration.
245 citations
TL;DR: An adaptive algorithm is developed for finding the intersection curve(s) of pairs of rectangular parametric patches which are continuously differentiable, controlled by a set of tolerances.
194 citations
TL;DR: In this paper, it was shown that the hyperbolicity of the complex of curves associated to a compact orientable surface is bounded by a logarithmic function of the complexity of the surface, for example the genus plus the number of boundary components.
Abstract: Abstract We give another proof of the result of Masur and Minsky that the complex of curves associated to a compact orientable surface is hyperbolic. Our proof is more combinatorial in nature and can be expressed mostly in terms of intersection numbers. We show that the hyperbolicity constant is bounded above by a logarithmic function of the complexity of the surface, for example the genus plus the number of boundary components. The geodesics in the complex can be described, up to bounded Hausdorff distance, by a simple relation of intersection numbers. We can also give a similar criterion for recognising if two geodesic segments are close.
190 citations