Implicit surface Implicit surface torus (R=40, a=15). Implicit surface of genus 2. Implicit non-algebraic surface (wineglass). An implicit surface is the set of zeros of a function of three variables. The graph of a function is usually described by an equation and is called an explicit representation. , where the x-, y- and z-coordinates of surface points are represented by three functions depending on common parameters . is given: (implicit), (parametric). Examples: For a plane, a sphere, and a torus there exist simple parametric representations. The implicit function theorem describes conditions under which an equation can be solved (at least implicitly) for x, y or z. If is polynomial in x, y and z, the surface is called algebraic. Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Formulas[edit] Throughout the following considerations the implicit surface is represented by an equation where function are Tangent plane and normal vector[edit] at
DGPF - Deutsche Gesellschaft für Photogrammetrie, Fernerkundung und Geoinformation e.V. - Home Isosurface Surface representing points of constant value within a volume The term isoline is also sometimes used for domains of more than 3 dimensions.[1] Applications[edit] Isosurfaces are normally displayed using computer graphics, and are used as data visualization methods in computational fluid dynamics (CFD), allowing engineers to study features of a fluid flow (gas or liquid) around objects, such as aircraft wings. Numerous other disciplines that are interested in three-dimensional data often use isosurfaces to obtain information about pharmacology, chemistry, geophysics and meteorology. Implementation algorithms[edit] Marching cubes[edit] Asymptotic decider[edit] The asymptotic decider algorithm was developed as an extension to marching cubes in order to resolve the possibility of ambiguity in it. Marching tetrahedra[edit] The marching tetrahedra algorithm was developed as an extension to marching cubes in order to solve an ambiguity in that algorithm and to create higher quality output surface.
Visualisation 3D d’une fonction réelle de deux variables (de ℝ² dans ℝ) avec Python -matplotlib | by Joséphine Picot | Medium PS : Si vous avez déjà suivi le tutoriel “Visualisation des cercles de niveau d’une fonction réelle”, vous pouvez passer directement à la partie “Visualisation” de ce tutoriel car la préparation des données est la même. Tout d’abord quelques librairies sont nécessaires : installation pip install numpypip install matplotlib importation import numpy as npimport matplotlib.pyplot as pltimport mpl_toolkitsfrom mpl_toolkits.mplot3d import Axes3D Si vous avez une erreur lors de l’importation de mpl_toolkit, essayez d’exécuter la commande suivante : ! Étapes de préparation Soit une fonction f de ℝ² dans ℝ. f(x, y) = 2*x² — x*y + 2*y² On définit la fonction f en python : def f(x, y): return 2*x**2 - x*y + 2*y**2 On choisit un intervalle de valeurs dans l’ensemble des réels ℝ pour x et y (je prendrai pour exemple [-100, 100] avec 100 valeurs pour chaque variable). x = np.linspace(-100, 100, 100)y = np.linspace(-100, 100, 100) X, Y = np.meshgrid(x, y) Z = f(X,Y) Visualisation interactive %matplotlib notebook
Polygonising a scalar field (Marching Cubes) Also known as: "3D Contouring", "Marching Cubes", "Surface Reconstruction" Written by Paul Bourke May 1994 Based on tables by Cory Gene Bloyd along with additional example source code marchingsource.cppAn alternative table by Geoffrey Heller.rchandra.zip: C++ classes contributed by Raghavendra Chandrashekara.OpenGL source code, sample volume: cell.gz (old)volexample.zip: An example showing how to call polygonise including a sample MRI dataset.Improved (2018) Qt/OpenGL example courtesy Dr. This document describes an algorithm for creating a polygonal surface representation of an isosurface of a 3D scalar field. There are many applications for this type of technique, two very common ones are: Reconstruction of a surface from medical volumetric datasets. The fundamental problem is to form a facet approximation to an isosurface through a scalar field sampled on a rectangular 3D grid. The indexing convention for vertices and edges used in the algorithm are shown below Another example Source code
Direction Fields - Geometry Central This section describes routines for computing n-direction fields on a surface. An n-direction field on a surface assigns n evenly-spaced unit tangent vectors to each point on the surface. For example, a 1-direction field is an ordinary direction field, a 2-direction field is a line field, and a 4-direction field is a cross field. Most of these routines only depend on the intrinsic geometry of a surface (via the IntrinsicGeometryInterface). Therefore, you can run them on abstract geometric domains as well as traditional surfaces in 3D. #include "geometrycentral/surface/direction_fields.h" Smoothest Direction Fields These routines compute the smoothest possible n-direction field on the input surface. The two routines are almost identical. Example Smoothest Boundary-Aligned Direction Fields This routine works like the previous ones, except it imposes Dirichlet boundary conditions to force the generated direction field to be aligned with the mesh’s boundary. Curvature-Aligned Direction Fields
Implicit surfaces Also known as "Metaballs", "Blobbies", "Soft objects" Written by Paul Bourke June 1997 Introduction Most computer based 3D geometric modelling is done with basic primitives such as lines, planes, boxes, etc. Many smooth and deformable objects are difficult or inefficient to represent with such building blocks, even if primitives such as spheres or Bezier/spline surfaces are used. The following summarises three common methods for creating so called "implicit surfaces". Example Consider the field function D(r) = 1/r2 and a number of control points in 3D space. r is the distance of a point in space to a particular control point. For example if there is a single control point different colour levels will result in spheres of different radius. If the control structure is a line or a plane then the distance r is normally taken to be the closest distance to nay point on the line or plane. Blobby Molecules "b" is related to the standard deviation of the curve, "a" to the height. Meta Balls Notes D.
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Marching Cubes: Curve Wrapping & More Metaballs - Grasshopper UPDATE: 30-Jan-2014 140130_marching_cubes.gh I have added breps as an input for wrapping geometry (it also can take lines now), as in the above definition. It has been updated for some efficiencies...now you can taper a curve at both ends if you choose, you only need to feed one radius, although you can feed as many as you like (it acts like the longest list component). Some of the work posted lately by Nick Tyrer has gotten me thinking about marching cubes again...I had done some stuff with marching tetrahedra and cubes a ways back, and with some new inspiration (and a little time I could carve out today) I figured I'd take a stab at trying to make it more flexible and robust. The short of it is that the inputs can take any combination of points and curves, along with variable radii of influence for each geometry object. The marching cube stuff is derived from the amazing Paul Bourke's work. G = A list of base Geometry, which can be any combination of curves or points