Alicia Cantón

On Harmonic Functions on Trees

We study the asymptotic behaviour of harmonic and p-harmonic functions (1<p<∞) on trees, obtaining estimates about the Hausdorff dimension of ‘radial’ limits.

Geometric characteristics of conics in Bézier form

Synthetic derivation of closed formulae of the geometric characteristic of a conic given in Bezier form in terms of its control polygon, (P; Q; R) and weights, (1; w; 1g)

Gromov hyperbolicity of planar graphs

We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ℝ2 with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of ℝ2 such that every tile is a triangle and a partial answer to this question is given. A weaker version of this conjecture stating that every tessellation graph of ℝ2 with rectangular tiles is non-hyperbolic is given and partially answered. If this conjecture were true, many tessellation graphs of ℝ2 with tiles which are parallelograms would be non-hyperbolic.

Non-degenerate developable triangular bézier patches

In this talk we show a construction for characterising developable surfaces in the form of Bezier triangular patches. It is shown that constructions used for rectangular patches are not useful, since they provide degenerate triangular patches. Explicit constructions of non-degenerate developable triangular patches are provided.

Interpolation of a spline developable surface between a curve and two rulings

In this paper we address the problem of interpolating a spline developable patch bounded by a given spline curve and the first and the last rulings of the developable surface. To complete the boundary of the patch, a second spline curve is to be given. Up to now this interpolation problem could be solved, but without the possibility of choosing both endpoints for the rulings. We circumvent such difficulty by resorting to degree elevation of the developable surface. This is useful for solving not only this problem, but also other problems dealing with triangular developable patches.

Implicit Equations of Non-degenerate Rational Bezier Quadric Triangles

In this paper we review the derivation of implicit equations for non-degenerate quadric patches in rational Bezier triangular form. These are the case of Steiner surfaces of degree two. We derive the bilinear forms for such quadrics in a coordinate-free fashion in terms of their control net and their list of weights in a suitable form. Our construction relies on projective geometry and is grounded on the pencil of quadrics circumscribed to a tetrahedron formed by vertices of the control net and an additional point which is required for the Steiner surface to be a non-degenerate quadric.

Geometric elements and classification of quadrics in rational Bézier form

In this paper we classify and derive closed formulas for geometric elements of quadrics in rational Bezier triangular form (such as the center, the conic at infinity, the vertex and the axis of paraboloids and the principal planes), using just the control vertices and the weights for the quadric patch. The results are extended also to quadric tensor product patches. Our results rely on using techniques from projective algebraic geometry to find suitable bilinear forms for the quadric in a coordinate-free fashion, considering a pencil of quadrics that are tangent to the given quadric along a conic. Most of the information about the quadric is encoded in one coefficient, involving the weights of the patch, which allows us to tell apart oval from ruled quadrics. This coefficient is also relevant to determine the affine type of the quadric. Spheres and quadrics of revolution are characterized within this framework. Closed formulas for geometric elements of quadrics in rational Bezier form in terms of their weights and control points, using algebraic projective geometry.The results are derived for Bezier triangles, but are applicable to tensor product patches.Closed, coordinate-free formulas for implicit equations for quadrics in terms of their weights and control points.Affine classification of quadrics in Bezier form.

Isoperimetric inequalities in graphs and surfaces

Let M be the set of metric spaces that are either graphs with bounded degree or Riemannian manifolds with bounded geometry. Kanai proved the quasi-isometric stability of several geometric properties (in particular, of isoperimetric inequalities) for the spaces in M. Kanai proves directly these results for graphs with bounded degree; in order to prove the general case, he uses a graph (an e-net) associated to a Riemannian manifold with bounded geometry. This paper studies the stability of isoperimetric inequalities under quasi-isometries between non-exceptional Riemann surfaces (endowed with their Poincare metrics). The present work proves the stability of the linear isoperimetric inequality for planar surfaces (genus zero surfaces) without the condition on bounded geometry. It is also shown the stability of any non-linear isoperimetric inequality.