Intersection curve of two parametric surfaces in Euclidean n-space

The aim of this paper is to study the differential geometric properties of the intersection curve of two parametric surfaces in Euclidean n-space. For this aim, we first present the mth order derivative formula of a curve lying on a parametric surface. Then, we obtain curvatures and Frenet vectors of the transversal intersection curve of two parametric surfaces in Euclidean n-space. We also provide computer code produced in MATLAB to simplify determining the coefficients relative to Frenet frame of higher order derivatives of a curve.

Constructing Rigid-Foldable Generalized Miura-Ori Tessellations for Curved Surfaces

Origami has shown the potential to approximate three-dimensional curved surfaces by folding through designed crease patterns on flat materials. The Miura-ori tessellation is a widely used pattern in engineering and tiles the plane when partially folded. Based on constrained optimization, this article presents the construction of generalized Miura-ori patterns that can approximate three-dimensional parametric surfaces of varying curvatures while preserving the inherent properties of the standard Miura-ori, including developability, flat foldability, and rigid foldability. An initial configuration is constructed by tiling the target surface with triangulated Miura-like unit cells and used as the initial guess for the optimization. For approximation of a single target surface, a portion of the vertexes on the one side is attached to the target surface; for fitting of two target surfaces, a portion of vertexes on the other side is also attached to the second target surface. The parametric coordinates are adopted as the unknown variables for the vertexes on the target surfaces, while the Cartesian coordinates are the unknowns for the other vertexes. The constructed generalized Miura-ori tessellations can be rigidly folded from the flat state to the target state with a single degree-of-freedom.

Modelamiento de Superficies en Coordenadas Esféricas a Través de GeoGebra

Este artículo tiene como objetivo presentar una herramienta que permita ayudar la enseñanza y el aprendizaje de graficar y describir superficies en coordenadas esféricas, a través del software libre GeoGebra, para lo cual se va representar superficies convencionales y no convencionales, así como sólidos que están limitados por un número finito de superficies por medio de parametrizaciones basadas en coordenadas esféricas y luego presentarlas en la vista gráfica 3D de GeoGebra usando el comando Superficie, de esta forma se tiene una inmejorable visualizar, percepción, manipulación y comprensión del esbozo de superficies en el espacio en un ambiente dinámico y amigable. Finalizamos con un caso de aplicación de modelado del planetario más grande del mundo aplicando superficies paramétricas basadas en las coordenadas esféricas. Palabras-clave: Modelamiento. Superficies. Parametrización. Coordenadas Esféricas. GeoGebra. Abstract This article aims to present a tool that aids the teaching and learning of graphing and describing surfaces in spherical coordinates, through the free GeoGebra software, for which specific and non-specific surfaces will be represented, as well as solids that are limited by a finite number of surfaces by means of parameterizations based on spherical coordinates and then present them in the GeoGebra 3D graphic view using the Surface command, in this way you have an unbeatable visualization, perception, manipulation and understanding of the outline of surfaces in the space in a dynamic and friendly environment. We conclude with a case of modeling application of the largest planetarium in the world applying parametric surfaces based on spherical coordinates Keywords: Modeling. Surfaces. Parameterization. Spherical Coordinates. GeoGebra.

Design Procedure of a Topologically Optimized Scooter Frame Part

This article describes the design procedure of a topologically optimized scooter frame part. It is the rear heel of the frame, one of the four main parts of a scooter made with stainless steel 3D printing. The first part of the article deals with the design area definition and the determination of load cases for topology calculation. The second part describes the process of the topology optimization itself and the creation of the volume body based on the calculation results. Finally, the final control using an FEM (Finite Element Method) analysis and optimization of created Computer-Aided Design (CAD) data is shown. Part of the article is also a review of partial iterations and resulting versions of the designed part. Symmetry was used to define boundary conditions, which led to computing time savings, as well as during the CAD model creation, where non-parametric surfaces were mirrored to shorten the design time.

Global Phase Portrait and Bifurcation Diagram for a Multi-Parametric Linear System

The goal of this article is to conduct a global dynamics study of a linear multiparameter system (real parameters (a,b,c) in R^3); for this, we take the different changes that these parameters present. First, we find the different parametric surfaces in which the space is divided, where the stability of the critical point is defined; we then create a bifurcation diagram to classify the different bifurcations that appear in the system. Finally, we determine and classify the critical points at infinity, considering the canonical shape of the Poincaré sphere, and thus, obtain a global phase portrait of the multiparametric linear system.