Orthogonal Polynomials on Planar Cubic Curves

Orthogonal polynomials in two variables on cubic curves are considered. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and demonstrate their usage for solving differential equations with singular solutions.

Constructing cubic curves with involutions

In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles’ Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter’s construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions. As an application of Schroeter’s construction we provide a new parametrisation of elliptic curves with torsion group $$\mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/8\mathbb {Z}$$ Z / 2 Z × Z / 8 Z and give some configurations with all their points on a cubic curve.

Conversion Between Cubic Bezier Curves and Catmull–Rom Splines

Splines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, Bézier and Catmull–Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic Bézier and Catmull–Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull–Rom or Bézier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.

Let’s Join: A Pavilion Inspired by the Weaire and Phelan Space Tessellation

The pavilion presented in this paper is an articulation of space based on the Weaire and Phelan tessellation. In this pavilion a second generative pattern is partially engraved and cut out of the polygonal surfaces that bind the polyhedron. This pattern is made up of cubic curves that aim to spatially engage with the tessellation. The design and prototyping process was implemented and controlled using parametric and procedural models. The use of these models made it possible to define the shape, orientation, and size of each element of the pavilion: from the polyhedron and the pattern’s curves, to smaller components like the shape of the panels’ joints, holes, and the countersinks that hold the screws. This research illustrates the geometric, formal, and procedural design that was used to shape the pavilion in its current spatial configuration, to draw the pattern cut and engraved into the polyhedron’s surface, and finally, to determine the shape and scale of the joints that hold the faces of the polyhedron together and characterize the internal space. Regarding the topic of Patterns and Spatial Organization, the paper investigates shape design methodologies, which are based on historical practices but updated using new technologies.

Cubic Curves in Engineering Geometry

In this paper are considered historically the first (the 60’s of the 20th century) computational methods for algebraic cubic curves constructing. The analysis of a general cubic curve equation r(t)=a3t3+a2t2+a1t+a0 has been carried out. As an example has been considered the simplest cubic curve r(t)=it3+jt2+kt. Based on the general cubic curve equation have been obtained equations of a cubic curve passing through two predetermined points and having predetermined tangents at these points. The equations have been presented both in Ferguson and Bézier forms. It has been shown that the cubic curve vector equation (for example, the standard equation of a Bezier curve) can be represented in a point form. Have been considered examples for constructing segments of cubic curves meeting the given boundary conditions. The generalized cubic curve equation, containing weight coefficients, has been obtained by the method of exit into four-dimensional space. Has been considered a vector parametric equation of a conical section, passing through two given points and touching predetermined straight lines at these points. The conical section is considered as a special case of a cubic curve. Curvature can be specified as an additional boundary condition. Has been considered the possibility for constructing a cubic curve with fixed positions of contacting planes at end points and given radii of curvature. Has been proposed an algorithm for constructing a plane cubic curve with a given curvature at the end points. Have been considered algorithms for constructing smooth compound Ferguson-Bezier curves. Smoothness conditions are imposed on a compound curve: 1) at any of its points, the curve must have a tangent (no fractures are allowed), 2) the curvature vector must be changed continuously from point to point (no discontinuous jump in the curvature vector is allowed neither in modulus no in direction). Have been proposed examples for constructing compound Ferguson-Bézier curves. Has been performed comparison of polynomial cubic spline with compound parametrically defined curves. Have been given examples for constructing cubic splines with fastened and free ends. The paper is for educational purposes, and intended for in-depth study of computer graphics basics.