Computer graphics algorithms | Mesh generation
Marching tetrahedra is an algorithm in the field of computer graphics to render implicit surfaces. It clarifies a minor ambiguity problem of the marching cubes algorithm with some cube configurations. It was originally introduced in 1991. While the original marching cubes algorithm was protected by a software patent, marching tetrahedrons offered an alternative algorithm that did not require a patent license. More than 20 years have passed from the patent filing date (June 5, 1985), and the marching cubes algorithm can now be used freely. Optionally, the minor improvements of marching tetrahedrons may be used to correct the aforementioned ambiguity in some configurations. In marching tetrahedra, each cube is split into six irregular tetrahedra by cutting the cube in half three times, cutting diagonally through each of the three pairs of opposing faces. In this way, the tetrahedra all share one of the main diagonals of the cube. Instead of the twelve edges of the cube, we now have nineteen edges: the original twelve, six face diagonals, and the main diagonal. Just like in marching cubes, the intersections of these edges with the isosurface are approximated by linearly interpolating the values at the grid points. Adjacent cubes share all edges in the connecting face, including the same diagonal. This is an important property to prevent cracks in the rendered surface, because interpolation of the two distinct diagonals of a face usually gives slightly different intersection points. An added benefit is that up to five computed intersection points can be reused when handling the neighbor cube. This includes the computed surface normals and other graphics attributes at the intersection points. Each tetrahedron has sixteen possible configurations, falling into three classes: no intersection, intersection in one triangle and intersection in two (adjacent) triangles. It is straightforward to enumerate all sixteen configurations and map them to vertex index lists defining the appropriate triangle strips. (Wikipedia).
Stanford artist collaborates with physics department for 'Drawing with Tetrahedra'
Physics faculty members and graduate students use tetrahedra to create a less-than-perfect structure that explores the connection between shape and sound. For more information, see: http://news.stanford.edu/news/2014/march/tetra-physics-vivaldi-040214.html
From playlist Stanford Highlights
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/q0PF.
From playlist 3D printing
How to construct a Tetrahedron
How the greeks constructed the first platonic solid: the regular tetrahedron. Source: Euclids Elements Book 13, Proposition 13. In geometry, a tetrahedron also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. Th
From playlist Platonic Solids
Cardboard Tetrahedron Pyramid Perfect Circle Solar How to make a pyramid out of cardboard
How to make a pyramid out of cardboard. A tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex.
From playlist HOME OF GREENPOWERSCIENCE SOLAR DIY PROJECTS
Using a set of points determine if the figure is a parallelogram using the midpoint formula
👉 Learn how to determine the figure given four points. A quadrilateral is a polygon with four sides. Some of the types of quadrilaterals are: parallelogram, square, rectangle, rhombus, kite, trapezoid, etc. Each of the types of quadrilateral has its properties. Given four points that repr
From playlist Quadrilaterals on a Coordinate Plane
Raytracing and raymarching simulations of non-euclidean geometries - Henry Segerman
Workshop on Topology: Identifying Order in Complex Systems Topic: Raytracing and raymarching simulations of non-euclidean geometries Speaker: Henry Segerman Affiliation: Oklahoma State University Date: December 4, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
We explain how to make fractal images from cohomology classes in hyperbolic three-manifolds. You can try out the web app for yourself at: https://henryseg.github.io/cohomology_fractals Cohomology fractal zoom: https://youtu.be/-g1wNbC9AxI Non-euclidean virtual reality using ray-marching: h
From playlist GPU shaders
Periodic Foams and Manifolds - Frank Lutz
Frank Lutz Technische Universitat Berlin March 2, 2011 WORKSHOP ON TOPOLOGY: IDENTIFYING ORDER IN COMPLEX SYSTEMS For more videos, visit http://video.ias.edu
From playlist Mathematics
Veering Dehn surgery - Saul Schleimer
Geometric Structures on 3-manifolds Topic: Veering Dehn surgery Speaker: Saul Schleimer Date: Tuesday, April 12 (Joint with Henry Segerman.) It is a theorem of Moise that every three-manifold admits a triangulation, and thus infinitely many. Thus, it can be difficult to learn anything
From playlist Mathematics
Catherine Meusburger: Turaev-Viro State sum models with defects
Talk by Catherine Meusburger in Global Noncommutative Geometry Seminar (Europe) http://www.noncommutativegeometry.nl/ncgseminar/ on March 17, 2021
From playlist Global Noncommutative Geometry Seminar (Europe)
Determining if a set of points makes a parallelogram or not
👉 Learn how to determine the figure given four points. A quadrilateral is a polygon with four sides. Some of the types of quadrilaterals are: parallelogram, square, rectangle, rhombus, kite, trapezoid, etc. Each of the types of quadrilateral has its properties. Given four points that repr
From playlist Quadrilaterals on a Coordinate Plane
The Signature and Natural Slope of Hyperbolic Knots - Marc Lackenby
DeepMind Workshop Topic: The Signature and Natural Slope of Hyperbolic Knots Speaker: Marc Lackenby Affiliation: University of Oxford Date: March 30, 2022 Andras Juhasz has explained in his talk how machine learning was used to discover a previously unknown relationship between invariant
From playlist DeepMind Workshop
Scott Kim - Motley Dissections - G4G13 April 2018
This talk discusses motley dissections — polygons cut into polygons and polyhedra cut into polyhedra such that no two pieces every completely share an edge or a face. The most famous motley dissection is the squared square. My contribution extends this to triangled triangles, pentagoned pe
From playlist G4G13 Videos
Henry Adams (5/1/21): Bridging applied and quantitative topology
I will survey emerging connections between applied topology and quantitative topology. Vietoris-Rips complexes were invented by Vietoris in order to define a (co)homology theory for metric spaces, and by Rips for use in geometric group theory. More recently, they have found applications in
From playlist TDA: Tutte Institute & Western University - 2021
Determine if a set of points is a parallelogram using the distance formula
👉 Learn how to determine the figure given four points. A quadrilateral is a polygon with four sides. Some of the types of quadrilaterals are: parallelogram, square, rectangle, rhombus, kite, trapezoid, etc. Each of the types of quadrilateral has its properties. Given four points that repr
From playlist Quadrilaterals on a Coordinate Plane
Types of Silicates Part 1: Orthosilicates, Disilicates, and Cyclosilicates
With seven classes of minerals down there is just one to go, and it is the most important class. Silicates! There are so many silicates we will need two tutorials to get through them all. Here we will discuss orthosilicates, disilicates, and cyclosilicates. Script by Jared Matteucci and B
From playlist Geology
Okay, you've got me - this is Pythagorus' theorum.... Really easy though! In a right angled triangle the square of the two smaller sides added together is the same as the square of the diagonal. To donate to the tecmath channel:https://paypal.me/tecmath To support tecmath on Patreon
From playlist Trigonometry
Hyperbolic Knot Theory (Lecture - 2) by Abhijit Champanerkar
PROGRAM KNOTS THROUGH WEB (ONLINE) ORGANIZERS: Rama Mishra, Madeti Prabhakar, and Mahender Singh DATE & TIME: 24 August 2020 to 28 August 2020 VENUE: Online Due to the ongoing COVID-19 pandemic, the original program has been canceled. However, the meeting will be conducted through onl
From playlist Knots Through Web (Online)