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• Method of transforming 3D vectors with a matrix

  - by Drew Noakes

  I've been doing some reading on transforming Vector3 with matrices, and am tossing up digging deeper into the math and coding this myself versus using existing code. For whatever reason my school curriculum never included matrices, so I'm filling a gap in my knowledge. Thankfully I only need a few simple things, I think. Context is that I'm programming a robot for the RoboCup 3D league. I'm coding it in C# but it'll have to run on Mono. Ideally I wouldn't use any existing graphics libraries for this (WinForms/WPF/XNA) as all I really need is a neat subset of matrix transformations. Specifically, I need translation and x/y/z rotations, and a way of combining multiple transformations into a single matrix. This will then be applied to my own Vector3 type to produce the transformed Vector3. I've read different advice about this. For example, some model the transformation with a 4x3 matrix, others with a 4x4 matrix. Also, some examples show that you need a forth value for the vector's matrix of 1. What happens to this value when it's included in the output? [1 0 0 0] [x y z 1] * [0 1 0 0] = [a b c d] [0 0 1 0] [2 4 6 1] The parts I'm missing are: What sizes my matrices should be Compositing transformations by multiplying the transformation matrices together Transforming 3D vectors with the resulting matrix As I mostly just want to get this running, any psuedo-code would be great. Information about what matrix values perform what transformations is quite clearly defined on many pages, so need not be discussed here unless you're very keen :)

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• Great Circle & Rhumb line intersection

  - by Karl T

  I have a Latitude, Longitude, and a direction of travel in degrees true north. I would like to calculate if I will intersect a line defined by two more Lat/Lon points. I figure the two points defining the line would create my great circle and my location and azimuth would define my Rhumb line. I am only interested in intersections that will occur with a few hundred kilometers so I do not need every possible solution. I have no idea where to begin.

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• Distance from a point to a polygon

  - by clwen

  I am trying to determine the distance from a point to a polygon in 2D space. The point can be inside or outside the polygon; The polygon can be convex or concave. If the point is within the polygon or outside the polygon with a distance smaller than a user-defined constant d, the procedure should return True; False otherwise. I have found a similar question: Distance from a point to a polyhedron or to a polygon. However, the space is 2D in my case and the polygon can be concave, so it's somehow different from that one. I suppose there should be a method simpler than offsetting the polygon by d and determining it's inside or outside the polygon. Any algorithm, code, or hints for me to google around would be appreciated.

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• Convet from line points to shape points

  - by VOX

  I have an array of points that make up a line. However I need to draw the line with the width of n pixels. How can I transform that points for lines to points for polygon (or a shape) so I can directly draw it on canvas. I'm developing two program at the same time, one is j2me and another is .NET CF. j2me doesn't support drawing lines with width. Please take a look at the picture. link text

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• How to improve performance

  - by Ram

  Hi, In one of mine applications I am dealing with graphics objects. I am using open source GPC library to clip/merge two shapes. To improve accuracy I am sampling (adding multiple points between two edges) existing shapes. But before displaying back the merged shape I need to remove all the points between two edges. But I am not able to find an efficient algorithm that will remove all points between two edges which has same slope with minimum CPU utilization. Currently all points are of type PointF Any pointer on this will be a great help.

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• Fastest way of converting a quad to a triangle strip?

  - by Tina Brooks

  What is the fastest way of converting a quadrilateral (made up of foyr x,y points) to a triangle strip? I'm well aware of the general triangulation algorithms that exist, but I need a short, well optimized algorithm that deals with quadrilaterals only. My current algorithm does this, which works for most quads but still gets the points mixed up for some: #define fp(f) bounds.p##f /* Sort four points in ascending order by their Y values */ point_sort4_y(&fp(1), &fp(2), &fp(3), &fp(4)); /* Bottom two */ if (fminf(-fp(1).x, -fp(2).x) == -fp(2).x) { out_quad.p1 = fp(2); out_quad.p2 = fp(1); } else { out_quad.p1 = fp(1); out_quad.p2 = fp(2); } /* Top two */ if (fminf(-fp(3).x, -fp(4).x) == -fp(3).x) { out_quad.p3 = fp(3); out_quad.p4 = fp(4); } else { out_quad.p3 = fp(4); out_quad.p4 = fp(3); }

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• Calculating 2D angles for 3D objects in perspective

  - by Will

  Imagine a photo, with the face of a building marked out. Its given that the face of the building is a rectangle, with 90 degree corners. However, because its a photo, perspective will be involved and the parallel edges of the face will converge on the horizon. With such a rectangle, is it possible to calculate the angle in 2D of the edges of a face that is 90 degrees to it? In the image below, the blue is the face marked on the photo, and I'm wondering how to calculate the 2D vector of the red lines of the other face:

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• Generate 2D cross-section polygon from 3D mesh

  - by nornagon

  I'm writing a game which uses 3D models to draw a scene (top-down orthographic projection), but a 2D physics engine to calculate response to collisions, etc. I have a few 3D assets for which I'd like to be able to automatically generate a hitbox by 'slicing' the 3D mesh with the X-Y plane and creating a polygon from the resultant edges. Google is failing me on this one (and not much helpful material on SO either). Suggestions?

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• Find point which sum of distances to set of other points is minimal

  - by Pawel Markowski

  I have one set (X) of points (not very big let's say 1-20 points) and the second (Y), much larger set of points. I need to choose some point from Y which sum of distances to all points from X is minimal. I came up with an idea that I would treat X as a vertices of a polygon and find centroid of this polygon, and then I will choose a point from Y nearest to the centroid. But I'm not sure whether centroid minimizes sum of its distances to the vertices of polygon, so I'm not sure whether this is a good way? Is there any algorithm for solving this problem? Points are defined by geographical coordinates.

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• Categorize the approximate shape of an array of Points in 3D Space

  - by user1295133

  I have a set of points in 3d space and I want to be able to categorize the shape that best fits them - cube, sphere, cylinder, planar (flat) etc. I've looked at supervised/machine learning but since I need first generate a large training data set that's not really suitable. My dream solution would be a java library with a wonderful magical function something like : public enum ShapeType { CUBE, SPHERE, CYLINDER, PLANAR } public ShapeType CategorizeShapeFromPoints( 3DPoint[] points ) However, any and all help will be appreciated. Thanks

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• Determining line orientation using vertex shaders

  - by Brett

  Hi, I want to be able to calculate the direction of a line to eye coordinates and store this value for every pixel on the line using a vertex and fragment shader. My idea was to calculate the direction gradient using atan2(Gy/Gx) after a modelview tranformation for each pair of vertices then quantize this value as a color intensity to pass to a fragment shader. How can I get access to the positions of pairs of vertices to achieve this or is there another method I should use? Thanks

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• Converting convex hull to binary mask

  - by Jonas

  I want to generate a binary mask that has ones for all pixels inside and zeros for all pixels outside a volume. The volume is defined by the convex hull around a set of 3D coordinates (<100; some of the coordinates are inside the volume). I can get the convex hull using CONVHULLN, but how do I convert that into a binary mask? In case there is no good way to go via the convex hull, do you have any other idea how I could create the binary mask?

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• Sort latitude and longitude coordinates into clockwise quadrangle

  - by Dave Jarvis

  Problem Users can provide up to four latitude and longitude coordinates, in any order. They do so with Google Maps. Using Google's Polygon API (v3), the coordinates they select should highlight the selected area between the four coordinates. Solutions and Searches http://stackoverflow.com/questions/242404/sort-four-points-in-clockwise-order Graham's scan seems too complicated for four coordinates Sort the coordinates into two arrays (one by latitude, the other longitude) ... then? Question How do you sort the coordinates in (counter-)clockwise order, using JavaScript? Code Here is what I have so far: // Ensures the markers are sorted: NW, NE, SE, SW function sortMarkers() { var ns = markers.slice( 0 ); var ew = markers.slice( 0 ); ew.sort( function( a, b ) { if( a.lat() < b.lat() ) { return -1; } else if( a.lat() > b.lat() ) { return 1; } return 0; }); ns.sort( function( a, b ) { if( a.lng() < b.lng() ) { return -1; } else if( a.lng() > b.lng() ) { return 1; } return 0; }); } What is a better approach? Thank you.

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• CGAL: Transformation Matrix for Rotation given two lines/vectors/directions

  - by Vulcan Eager

  How do I generate a transformation matrix for rotating points/others by the angle between two lines/vectors/directions in CGAL? 2D is what I need. 3D is what I love.

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• Rectangles Covering

  - by den bardadym

  I have N rectangles with sides parallel Ox and Oy. Exists another rectangele (model). I need create algorithm, which can tell: is model covered by N rectangles? and code him. I have some ideas. First I think need sort rectangles by left side (it can be done by O(n log n)). Then I think need use vertical sweeping line. Thanks.

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• How can I test if a point lies within a 3d shape with its surface defined by a point cloud?

  - by Ben

  Hi I have a collection of points which describe the surface of a shape that should be roughly spherical, and I need a method with which to determine if any other given point lies within this shape. I've previously been approximating the shape as an exact sphere, but this has proven too inaccurate and I need a more accurate method. Simplicity and speed is favourable over complete accuracy, a good approximation will suffice. I've come across techniques for converting a point cloud to a 3d mesh, but most things I have found have been very complicated, and I am looking for something as simple as possible. Any ideas? Many thanks, Ben.

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• Real life usage of the projective plane theory

  - by Elazar Leibovich

  I'm learning about the theory of the projective plane. Very generally speaking, it is an extension of the plane, which includes additional points which are defined as the intersection points of two parallel lines. In the projective plane, every two lines have an interesection point. Whether they're parallel or not. Every point in the projective plane can be represented by three numbers (you actually need less than that, but nevemind now). Is there any real life application which uses the projective plane? I can think that, for instance, a software which needs to find the intersections of a line, can benefit from always having an intersection point which might lead to simpler code, but is it really used?

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• Most "thorough" distribution of points around a circle

  - by hippietrail

  This question is intended to both abstract and focus one approach to my problem expressed at "Find the most colourful image in a collection of images". Imagine we have a set of circles, each has a number of points around its circumference. We want to find a metric that gives a higher rating to a circle with points distributed evenly around the circle. Circles with some points scattered through the full 360° are better but circles with far greater numbers of points in one area compared to a smaller number in another area are less good. The number of points is not limited. Two or more points may coincide. Coincidental points are still relevant. A circle with one point at 0° and one point at 180° is better than a circle with 100 points at 0° and 1000 points at 180°. A circle with one point every degree around the circle is very good. A circle with a point every half degree around the circle is better. In my other (colour based question) it was suggested that standard deviation would be useful but with caveat. Is this a good suggestion and does it cope with the closeness of 359° to 1°?

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• GIS: line_locate_point() in Python

  - by miracle2k

  I'm pretty much a beginner when it comes to GIS, but I think I understand the basics - it doesn't seem to hard. But: All these acronyms and different libraries, GEOS, GDAL, PROJ, PCL, Shaply, OpenGEO, OGR, OGC, OWS and what not, each seemingly depending on any number of others, is slightly overwhelming me. Here's what I would like to do: Given a number of points and a linestring, I want to determine the location on the line closest to a certain point. In other words, what PostGIS's line_locate_point() does: http://postgis.refractions.net/documentation/manual-1.3/ch06.html#line%5Flocate%5Fpoint Except I want do use plain Python. Which library or libraries should I have a look at generally for doing these kinds of spatial calculations in Python, and is there one that specifically supports a line_locate_point() equivalent?

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• Perturb vector by some angle

  - by Myx

  Hello: I have a unit vector in 3D space whose direction I wish to perturb by some angle within the range 0 to theta, with the position of the vector remaining the same. What is a way I can accomplish this? Thanks.

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• How can I group an array of rectangles into "Islands" of connected regions?

  - by Eric

  The problem I have an array of java.awt.Rectangles. For those who are not familiar with this class, the important piece of information is that they provide an .intersects(Rectangle b) function. I would like to write a function that takes this array of Rectangles, and breaks it up into groups of connected rectangles. Lets say for example, that these are my rectangles (constructor takes the arguments x, y, width,height): Rectangle[] rects = new Rectangle[] { new Rectangle(0, 0, 4, 2), //A new Rectangle(1, 1, 2, 4), //B new Rectangle(0, 4, 8, 2), //C new Rectangle(6, 0, 2, 2) //D } A quick drawing shows that A intersects B and B intersects C. D intersects nothing. A tediously drawn piece of ascii art does the job too: +-------+ +---+ ¦A+---+ ¦ ¦ D ¦ +-+---+-+ +---+ ¦ B ¦ +-+---+---------+ ¦ +---+ C ¦ +---------------+ Therefore, the output of my function should be: new Rectangle[][]{ new Rectangle[] {A,B,C}, new Rectangle[] {D} } The failed code This was my attempt at solving the problem: public List<Rectangle> getIntersections(ArrayList<Rectangle> list, Rectangle r) { List<Rectangle> intersections = new ArrayList<Rectangle>(); for(Rectangle rect : list) { if(r.intersects(rect)) { list.remove(rect); intersections.add(rect); intersections.addAll(getIntersections(list, rect)); } } return intersections; } public List<List<Rectangle>> mergeIntersectingRects(Rectangle... rectArray) { List<Rectangle> allRects = new ArrayList<Rectangle>(rectArray); List<List<Rectangle>> groups = new ArrayList<ArrayList<Rectangle>>(); for(Rectangle rect : allRects) { allRects.remove(rect); ArrayList<Rectangle> group = getIntersections(allRects, rect); group.add(rect); groups.add(group); } return groups; } Unfortunately, there seems to be an infinite recursion loop going on here. My uneducated guess would be that java does not like me doing this: for(Rectangle rect : allRects) { allRects.remove(rect); //... } Can anyone shed some light on the issue?

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• finding a point on an ellipse circumference which is inside a rectangle having center point, height

  - by Shlomi Assaf

  Hi all. I have a rectangle in .NET in which I draw an ellipse. I know the width, height and center point of that rectangle. Ofcourse the cetner point of the rectangle is also the center point of the ellipse. I know how to calculate a point on a circle, however I have no clue about an ellipse. I have those parameters and an angle, i need the point on the ellipse, can someone post the formula? I saw somewhere you need to calculate 2 points in which 2 raduises will go, the sum of the radiuses will be fixed and they will change in size accordingly. I dont know how to do that, I only have the rectange height, width and center point and ofcourse the angle I wish to find the point at. thanks for any help Shlomi

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• A simple algorithm for polygon intersection

  - by Elazar Leibovich

  I'm looking for a very simple algorithm for computing the polygon intersection/clipping. That is, given polygons P, Q, I wish to find polygon T which is contained in P and in Q, and I wish T to be maximal among all possible polygons. I don't mind the run time (I have a few very small polygons), I can also afford getting an approximation of the polygons' intersection (that is, a polygon with less points, but which is still contained in the polygons' intersection). But it is really important for me that the algorithm will be simple (cheaper testing) and preferably short (less code). edit: please note, I wish to obtain a polygon which represent the intersection. I don't need only a boolean answer to the question of whether the two polygons intersect.

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• Calculate the distance between a point and a line segment on an ellipsoid (or WGS84 coordinates)?

  - by Jochen

  Hi, I have a line segment AB between two WGS84 coordinates on the earth's surface (or, alternatively, on an ellipsoid), and need to calculate the distance between a point P and the point nearest to P that is on line segment AB. How can I do this? Any help is greatly appreciated. Regards, Jochen

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• Largest triangle from a set of points

  - by Faken

  I have a set of random points from which i want to find the largest triangle by area who's verticies are each on one of those points. So far I have figured out that the largest triangle's verticies will only lie on the outside points of the cloud of points (or the convex hull) so i have programmed a function to do just that (using Graham scan in nlogn time). However that's where I'm stuck. The only way I can figure out how to find the largest triangle from these points is to use brute force at n^3 time which is still acceptable in an average case as the convex hull algorithm usually kicks out the vast majority of points. However in a worst case scenario where points are on a circle, this method would fail miserably. Dose anyone know an algorithm to do this more efficiently? Note: I know that CGAL has this algorithm there but they do not go into any details on how its done. I don't want to use libraries, i want to learn this and program it myself (and also allow me to tweak it to exactly the way i want it to operate, just like the graham scan in which other implementations pick up collinear points that i don't want).

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