Aweptimum / DeWallua

Implementation of the DeWall Algorithm in Lua

DeWallua

An implementation of the DeWall algorithm in pure lua, found here: http://vcg.isti.cnr.it/publications/papers/dewall.pdf

TODO:

1. Clean!

Installation:

**Note: Don't actually use it right now as-is. It has a lot of debugging statements that need to be deleted/commented out.

Clone the DeWallua folder into your project and require it with:

local DeWall = require('DeWallua')

Usage:

At the moment, the API offers constrained and unconstrained.

unconstrained takes two arguments. The first is the list of points to triangulate (in {x = x_val, y=y_val} format). The second (optional) argument is a list of faces to insert into the triangulation. The faces' endpoints must be present in points and must be specified as a list of index-pairs like so:

faces = {{1,2}, {5,6}}

constrained takes a single argument: a list of ccw-ordered points that make up a convex/concave poylgon. It will triangulate the polygon staying within its bounds.

Both functions return a list of simplices in {x1,y1,x2,y2,x3,y3} format and leave their arguments intact.

An example:

local vertices = {
	{x =  0, y = 0}, 
	{x =  0, y = 1}, 
	{x = -1, y = 1}, 
	{x = -1, y = 2}, 
	{x = -2, y = 2}, 
	{x = -2, y = 0}, 
}

local simplices = DeWall.unconstrained( vertices )
tprint( simplices )
-- Prints these indexes as their equivalent x,y points:
-- {{1,3,2}, {3,2,4}, {1,3,6}, {3,6,4}, {5,3,4}}

local simplices = DeWall.constrained( vertices )
tprint( simplices )
-- Prints these indexes as their equivalent x,y points:
-- {{1,3,2}, {1,3,6}, {3,6,4}, {5,3,4}}

-- Notice there's one less triangle now -
-- the triangle made by points {3,2,4} lies outside the shape 

How it Works

An overview of the algorithm:

1. Compute the convex hull of the given points, P (or use the verts themselves)
2. Compute the cutting plane, a (take the average of all numbers along an axis)
3. Divide the input points, P, along a into two subsets: P1 and P2
4. Choose the point in P1 and P2 closest to the cutting plane, p1 and p2
5. Choose the point in P, p3, that makes triangle p1-p2-p3 with the smallest circumcircle
6. Add all edges not present into their corresponding Active-Face-List, AFL, (else remove the duplicate)
7. For each face in the AFL that intersects the cutting plane, use it to build a new minimum triangle
8. Continue until no more edges remain intersecting a
9. Repeat the process recursively, subdividing P1 and P2 further until all AFL's are empty.

Make_Simplex()

This function is the real heart of the algorithm (and might be doing too much) Here's what it does:

1. Given a face (f), check if it isn't already present in the hull of the given points (skip if so)
2. Iterate through points in the counter (skip if the current point is in the given face)
3. Test each point, i, for two things:
   • Is i in the correct halfspace defined by f
   • Is i constrained to the hull? (Does connecting f to i create self-intersections?)
4. Calculate the circumcircle of the triangle formed by f and i
5. If the circumradius is smaller than the current radius, i becomes the new candidate point
6. Return the simplex formed by f and the candidate point that yields the minimum circumradius