The difference (algorithmically) is: in quadtrees, the data reaching a node is split into a fixed (2^d), equal size cells, whereas in kdtrees, the data is split into two regions based on some data analysis (e.g. the median of some coordinate). Quadtrees do not scale well to high dimensions, due to the exponential dependency in … Read more
Pick a safe starting point. Can be e.g. the endpoint with maximum x. March along the line segment. Upon encountering any intersection, always turn left and march along this new segment. Upon encountering an endpoint, record it. Goto 2. Stop when you have returned to your starting point. Your list of recorded endpoints now makes … Read more
The Voronoi diagram is just the dual graph of the Delaunay triangulation. So, the edges of the Voronoi diagram are along the perpendicular bisectors of the edges of the Delaunay triangulation, so compute those lines. Then, compute the vertices of the Voronoi diagram by finding the intersections of adjacent edges. Finally, the edges are then … Read more
The shape you have is one of so called “Goldberg polyhedra”, is also a geodesic polyhedra. The (rather elegant) algorithm to generate this (and many many more) can be succinctly encoded in something called a Conway Polyhedron Notation. The construction is easy to follow step by step, you can click the images below to get … Read more
Here is a sketch of an O(n^2 log n) solution. First, the preliminaries shared with other answers. When we have a line passing through some rectangles, we can translate it to any of the two sides until it passes through a corner of some rectangle. After that, we fix that corner as the center of … Read more
I solved something similar for a local online judge once using simulated annealing. That was the official solution as well and the program got AC. The only difference was that the point I had to find did not have to be part of the N given points. This was my C++ code, and N could … Read more
From my “Geometry” class: public struct Line { public static Line Empty; private PointF p1; private PointF p2; public Line(PointF p1, PointF p2) { this.p1 = p1; this.p2 = p2; } public PointF P1 { get { return p1; } set { p1 = value; } } public PointF P2 { get { return p2; … Read more
The following resources survey some popular numerical methods for inverse kinematics problems: Samuel R. Buss. Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods Bill Baxter. Fast Numerical Methods for Inverse Kinematics Chris Welman. Inverse Kinematics and Geometric Constraints for Articulated Figure Manipulation Buss’s survey may be particularly interesting, because it … Read more
I have assumed the points are distinct, otherwise there might not even be such a line. If points are distinct, then such a line always exists and is possible to find using a deterministic O(nlogn) time algorithm. Say the points are P1, P2, …, P2n. Assume they are not all on the same line. If … Read more