The paper presents a solution to calculate intersection between a plane and bi-cubic Bezier curved surface. The paper first constructs bounding box based on the control grid by the characteristics of the Bezier surface, and then makes use of the plane half-space properties to convert the problem that calculates the intersection line between the plane and the parameter curved surface into calcul...

A degree is saturated bounding if it can compute a saturated model for any complete, decidable theory whose types are all computable. All high and PA degrees are saturated bounding (following from results of Jockusch and MacIntyre-Marker.) We show that for every n, no lown c.e. degree is saturated bounding, extending the previous known result that 0 is not saturated bounding, by Millar.

We consider a capillary drop that contacts several planar bounding walls so as to produce singularities (vertices) in the boundary of its free surface. It is shown under various conditions that when the number of vertices is less than or equal to three, then the free surface must be a portion of a sphere. These results extend the classical theorem of H. Hopf on constant mean curvature immersion...

We study the relation between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M . In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Further we characterize the simply connected space forms...

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeomorphic to a cusp cross-section of a (generalize...

This is a continuation of the papers of Braun and Fløystad [BF] and Cook [C] bounding the degree of smooth surfaces not of general type in P 4. We will prove the following: Theorem 1. Let S be a smooth surface of degree d in P 4 not of general type. Then d ≤ 66. The new idea in this paper is to bound the degree of the sporadic zeros of a generic hyperplane section of the surface by considering ...

This paper describes and demonstrates a new and highly innovative technique that identifies an approximation of the entire bounding surface of the feasible objective region directly, including deep concavities, disconnected regions and the edges of interior holes in the feasible areas. The Pareto front is a subset of the surface of the objective boundary and can be extracted easily. Importantly...