## geometry

### Puzzle: Find largest rectangle (maximal rectangle problem)

What's the most efficient algorithm to find the rectangle with the largest area which will fit in the empty space? Let's say the screen looks like this ('#' represents filled area): .................... ..............###### ##.................. .................### .................### #####............... #####............... #####......

### How do I reverse-project 2D points into 3D?

I have 4 2D points in screen-space, and I need to reverse-project them back into 3D space. I know that each of the 4 points is a corner of a 3D-rotated rigid rectangle, and I know the size of the rectangle. How can I get 3D coordinates from this? I am not using any particular API, and I do not have an existing projection matrix. I'm jus...

### Query points epsilon-close to a cut plane in point cloud using the GPU.

Hello everybody, I am trying to solve the current problem using GPU capabilities: "given a point cloud P and an oriented plane described by a point and a normal (Pp, Np) return the points in the cloud which lye at a distance equal or less than EPSILON from the plane". Talking with a colleague of mine I converged toward the following so...

### What's the best way to calculate a 3D (or n-D) centroid?

As part of a project at work I have to calculate the centroid of a set of points in 3D space. Right now I'm doing it in a way that seems simple but naive -- by taking the average of each set of points, as in: centroid = average(x), average(y), average(z) where x, y and z are arrays of floating-point numbers. I seem to recall that ther...

### Is there an efficient algorithm to generate a 2D concave hull?

Having a set of (2D) points from a GIS file (a city map), I need to generate the polygon that defines the 'contour' for that map (its boundary). Its input parameters would be the points set and a 'maximum edge length'. It would then output the corresponding (probably non-convex) polygon. The best solution I found so far was to generate ...

### How do I derive a Voronoi diagram given its point set and its Delaunay triangulation?

I'm working on a game where I create a random map of provinces (a la Risk or Diplomacy). To create that map, I'm first generating a series of semi-random points, then figuring the Delaunay triangulations of those points. With that done, I am now looking to create a Voronoi diagram of the points to serve as a starting point for the pr...

### How do you calculate the axis-aligned bounding box of an ellipse?

If the major axis of the ellipse is vertical or horizontal, it's easy to calculate the bounding box, but what about when the ellipse is rotated? The only way I can think of so far is to calculate all the points around the perimeter and find the max/min x and y values. It seems like there should be a simpler way. If there's a function (...

### Algorithm for hit test in non-overlapping rectangles

I have a collection of non-overlapping rectangles that cover an enclosing rectangle. What is the best way to find the containing rectangle for a mouse click? The obvious answer is to have an array of rectangles and to search them in sequence, making the search O(n). Is there some way to order them by position so that the algorithm is le...

### How to test if a line segment intersects an axis-aligned rectange in 2D?

How to test if a line segment intersects an axis-aligned rectange in 2D? The segment is defined with its two ends: p1, p2. The rectangle is defined with top-left and bottom-right points. ...

### Oriented Bounding Box vs Oriented Bounding Box intersection test (c/c++)

I'd like an optimized implementation for this test. It need not conform exactly to this prototype. bool OOBBIntersectOOBB( float center0, float halfExtents0, // or some other bounding description float rotation0, // quaternion would also be fine float center1, float halfExtents1, float rotation1); ...

### Calculate the Hilbert value of a point for use in a Hilbert R-Tree?

I have an application where a Hilbert R-Tree (wikipedia) (citeseer) would seem to be an appropriate data structure. Specifically, it requires reasonably fast spatial queries over a data set that will experience a lot of updates. However, as far as I can see, none of the descriptions of the algorithms for this data structure even mentio...

### Bezier clipping

I'm trying to find/make an algorithm to compute the intersection (a new filled object) of two arbitrary filled 2D objects. The objects are defined using either lines or cubic beziers and may have holes or self-intersect. I'm aware of several existing algorithms doing the same with polygons, listed here. However, I'd like to support bezie...

### How do I overlap widgets with the pack geometry manager in Python/Tkinter?

I want to put a canvas with an image in my window, but then I want to pack() widgets over top of it so the canvas acts as a background. Is it possible to have two states for the pack() manager, one for one set of widgets and another for another set? ...

### Algorithm to detect intersection of two rectangles?

I'm looking for an algorithm to detect if two rectangles intersect (one at an arbitrary angle, the other with only vertical/horizontal lines). Testing if a corner of one is in the other ALMOST works. It fails if the rectangles form a cross-like shape. It seems like a good idea to avoid using slopes of the lines, which would require sp...

### Fast plane rotation algorithm?

I am working on an application that detects the most prominent rectangle in an image, then seeks to rotate it so that the bottom left of the rectangle rests at the origin, similar to how IUPR's OSCAR system works. However, once the most prominent rectangle is detected, I am unsure how to take into account the depth component or z-axis, ...

### What is a good source for geometric algorithms?

I am looking for any good sources for geometric algorithms specifically; The simple stuff like when two lines cross and so on is easy enough (and easy to find), but I would like to find somewhere with algorithms for the more tricky things, such as finding the shape formed by expanding a given polygon by some amount; fast algorithms for...

### How do I find the center of a number of geographic points?

If I have a series of points as longitude and latitude, how would I calculate the center of all of those points? ...

### Reverse-projection 2D points into 3D

Suppose we have a 3d Space with a plane on it with an arbitary equation : ax+by+cz+d=0 now suppose that we pick 3 random points on that plane: (x0,y0,z0) (x1,y1,z1) (x1,y1,z1) now i have a different point of view(camera) for this plane. i mean i have a different camera that will look at this plane from a different point of view. From th...

### Code or formula for intersection of two parabolas in any rotation

I am working on a geometry problem that requires finding the intersection of two parabolic arcs in any rotation. I was able to intesect a line and a parabolic arc by rotating the plane to align the arc with an axis, but two parabolas cannot both align with an axis. I am working on deriving the formulas, but I would like to know if ther...

### How can I convert a JTS-Geometry into an AWT-Shape?

Is it possible to convert a com.vividsolutions.jts.geom.Geometry (or a subclass of it) into a class that implements java.awt.Shape? Which library/method can I use to achieve that goal? ...