#convex hull

[epochtutorials]

How To Make Convex Hull In Image Processing With C# - Epoch Abuse Convex hull in image processing is a morphological operation, where we encapsulate a shape or and object in an image into a convex shape.

[notreallyherehahaha]

In a kite-rhombus solid, or KRS, all faces are either kites or rhombi, and there are at least some of both of these quadrilateral-types as faces. I have found eight such polyhedra, all of which are formed by creating the convex hull of different Archimedean-Catalan base-dual compounds. Not all Archimedean-Catalan compounds produce kite-rhombus solids, but one of the eight that does is derived from the truncated dodecahedron, as explained below.

The next step is to create the compound of this solid and its dual, the triakis icosahedron. In the image below, this dual is the blue polyhedron.

The convex hull of this compound, below, I’m simply calling “the KRS derived from the truncated dodecahedron,” until and unless someone invents a better name for it.

The next KRS shown is derived, in the same manner, from the truncated tetrahedron.

Here is the KRS derived from the truncated cube.

The truncated icosahedron is the “seed” from which the next KRS shown is derived. This KRS is a “stretched” form of a zonohedron called the rhombic enneacontahedron.

Another of these kite-rhombus solids, shown below, is based on the truncated octahedron.

The next KRS shown is based on the rhombcuboctahedron.

Two of the Archimedeans are chiral, and they both produce chiral kite-rhombus solids. This one is derived from the snub cube.

Finally, to complete this set of eight, here is the KRS based on the snub dodecahedron.

You may be wondering what happens when this same process is applied to the other five Archimedean solids. The answer is that all-kite polyhedra are produced; they have no rhombic faces. Two are “stretched” forms of Catalan solids, and are derived from the cuboctahedron and the icosidodecahedron:

If this procedure is applied to the rhombicosidodecahedron, the result is an all-kite polyhedron with two different face-types, as seen below.

The two remaining Archimedean solids are the great rhombcuboctahedron and the great rhombicosidodecahedron, each of which produces a polyhedron with three different types of kites as faces.

The polyhedron-manipulation and image-production for this post was performed using Stella 4d: Polyhedron Navigator, which may be purchased or tried for free at http://www.software3d.com/Stella.php.

    Eight Kite-Rhombus Solids, Plus Five All-Kite Polyhedra — the Convex Hulls of the Thirteen Archimedean-Catalan Compounds In a kite-rhombus solid, or KRS, all faces are either kites or rhombi, and there are at least some of both of these quadrilateral-types as faces.

[peeterjoot]

ECE1505H Convex Optimization. Lecture 3: Matrix functions, SVD, and types of Sets. Taught by Prof. Stark Draper

ECE1505H Convex Optimization. Lecture 3: Matrix functions, SVD, and types of Sets. Taught by Prof. Stark Draper

[Click here for a PDF of this post with nicer formatting]

Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

These are notes for the UofT course ECE1505H, Convex Optimization, taught by Prof. Stark Draper.

Matrix inner product

Given real matrices \( X, Y \in \mathbb{R}^{m\times n} \), one possible matrix inner product definition is

\begin{equation}\label{eqn:convexOp…

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[sluicergames]

huller: a tool to generate a collision shape from an image

I wrote a small command line tool called huller to generate a convex collision shape from an image: https://github.com/clofresh/huller

The idea is to generate the collision shape outside of your game like this:

./huller path/to/sprite.png > sprite.lua

And then in your game you can do (assuming you’re using HC):

HC = require('HC') function love.load() spriteShapeData = require('sprite') sprite = { image = love.graphics.newImage('path/to/sprite.png'), shape = HC.polygon(unpack(spriteShapeData)), } end function love.draw() sprite.shape:draw() love.graphics.draw(sprite.image) end

Now you can do all the nice collision things with HC and have it (mostly) match what the image looks like.

[mmmlib]

Mechanics of form

[diegomontesinos]

Frames from my implementation of the algorithm to compute the Convex Hull 3D (polyhedron) applied to the mesh of the head from a friend.

[diegomontesinos]

Convex Hull (3D - polyhedron) of a set of 100 points in the space.

I implement the incremental algorithm to compute the convex hull of a set of 3D points and it’s ready to 3D print!

[diegomontesinos]

Animation of the incremental algorithm to construct the convex hull O(nk).