Given a point Q other than the incenter I of a reference triangle, we give a simple conic construction of a homothety mapping I into Q so that the image of the intouch triangle is perspective with the reference triangle. This is a generalization of the Kariya theorem in the case Q = I that the homothety can be arbitrary. The ratio of the homothety (the Kariya factor) is a unique nonzero finite ...

Bézier surfaces are an important design tool in Computer Aided Design. They are parameterized surfaces where the parameterization can be represented as a homogeneous polynomial in barycentric coordinates. Usually, Wachspress coordinates are used to obtain tensor product Bézier surfaces over rectangular domains. Recently, Floater introduced mean value coordinates as an alternative to Wachspress ...

We introduce two direct quadrature methods based on linear rational interpolation for solving general Volterra integral equations of the second kind. The first, deduced by a direct application of linear barycentric rational quadrature given in former work, is shown to converge at the same rate, but is costly on long integration intervals. The second, based on a composite version of the rational...

The antiprism triangulation provides a natural way to subdivide simplicial complex \(\Delta \), similar barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as the of chains multi-pointed faces from point view, by successively applying construction, or balanced stellar subdivisions, on geometric view. This paper studie...

In order to find the optimal control of continuous state-space and time reinforcement learning (RL) problems, we approximate the value function (VF) with a particular class of functions called the barycentric interpolators. We establish sufficient conditions under which a RL algorithm converges to the optimal VF, even when we use approximate models of the state dynamics and the reinforcement fu...

This paper describes a combined method of facial reconstruction and morphing between two heads, showing the extensive usage of feature points detected from pictures. We first present an efficient method to generate a 3D head for animation from picture data and then a simple method to do 3Dshape interpolation and 2D morphing based on triangulation. The basic idea is to generate an individualized...

Bernstein polynomials are a classical tool in Computer Aided Design to create smooth maps with a high degree of local control. They are used for the construction of Bézier surfaces, free-form deformations, and many other applications. However, classical Bernstein polynomials are only defined for simplices and parallelepipeds. These can in general not directly capture the shape of arbitrary obje...

Barycentric plotting, achieved by placing gaussian kernels in distant corners of the feature space and projecting multidimensional output of neural network on a plane, provides information about the process of training and certain features of the network. Additional visual guides added to the plot show tendencies and irregularities in the training process.

= convex hull of {a0, . . . , an}. The points a0, . . . , an are geometrically independent if a1 − a0, . . . , an − a0 is a linearly independent set over R. Note that this is independent of the order of a0, . . . , an. In this case, we say that the simplex a0 . . . an is n -dimensional, or an n -simplex. Given a point Pn i=1 λiai belonging to an n-simplex, we say it has barycentric coordinates ...

Hele-Shaw flow describes the slow flow of a viscous liquid between two parallel plates separated by a small gap. In some configurations such a flow generates instabilities known as Saffman-Taylor fingers, which form intricate visual patterns. While these patterns have been an inspiration for artists, as well as thoroughly analyzed by mathematicians, efficiently simulating them remains challengi...