There is likely no solution to this problem that is significantly better than O(n^2) in a standard model of computation. The problem of finding three collinear points reduces to the problem of finding the line that goes through the most points, and finding three collinear points is 3SUM-hard, meaning that solving it in less than … Read more
The problem can be solved by finding a feasible point of a Linear Program. If you’re interested in the full details, as opposed to just plugging an LP into an existing solver, I’d recommend reading Chapter 11.4 in Boyd and Vandenberghe’s excellent book on convex optimization. Set A = (X[1] X[2] … X[n]), that is, … Read more
The key to solving this problem is first making an observation: the center of the largest circle that will fit inside an arbitrary polygon is the point that is: Inside the polygon; and Furthest from any point on the edges of the polygon. Why? Because every point on the edge of a circle is equidistant … Read more
Does each vertex have its own normals or does each triangle have its own normals? Like so often the answer is: “It depends”. Since a normal is defined as being the vector perpendicular to all vectors within a given plane (in N dimensions), you need a plane to calculate a normal. A vertex position is … Read more
If i remember correctly, the algorithm is to draw a horizontal line through your test point. Count how many lines of of the polygon you intersect to reach your point. If the answer is odd, you’re inside. If the answer is even, you’re outside. Edit: Yeah, what he said (Wikipedia):
You can make things a lot easier than the Gift-Wrapping Algorithm… that’s a good answer when you have a set of points w/o any particular boundary and need to find the convex hull. In contrast, consider the case where the polygon is not self-intersecting, and it consists of a set of points in a list … Read more
I have been facing this dilemma for some months now, and made exhaustive research. Algorithms Mainly there are 2 categories of algorithms: computation geometry, and implicit surfaces. Computation Geometry They fit the mesh on the existing points. Probably the most famous algorithm of this group is powercrust, because it is theoretically well-established – it guarantees … Read more
First, compute the center point. Then sort the points using whatever sorting algorithm you like, but use special comparison routine to determine whether one point is less than the other. You can check whether one point (a) is to the left or to the right of the other (b) in relation to the center by … Read more
I thought I might briefly mention my own polygon clipping and offsetting library – Clipper. While Clipper is primarily designed for polygon clipping operations, it does polygon offsetting too. The library is open source freeware written in Delphi, C++ and C#. It has a very unencumbered Boost license allowing it to be used in both … Read more
Some of the suggested methods will fail in the case of a non-convex polygon, such as a crescent. Here’s a simple one that will work with non-convex polygons (it’ll even work with a self-intersecting polygon like a figure-eight, telling you whether it’s mostly clockwise). Sum over the edges, (x2 − x1)(y2 + y1). If the … Read more