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  • signed angle between two 3d vectors with same origin within the same plane? recipe?

    - by Advanced Customer
    Was looking through the web for an answer but it seems like there is no clear recipe for it. What I need is a signed angle of rotation between two vectors Va and Vb lying within the same 3D plane and having the same origin knowing that: the plane contatining both vectors is an arbitrary and is not parallel to XY or any other of cardinal planes Vn - is a plane normal both vectors along with the normal have the same origin O = { 0, 0, 0 } Va - is a reference for measuring the left handed rotation at Vn The angle should be measured in such a way so if the plane would be XY plane the Va would stand for X axis unit vector of it. I guess I should perform a kind of coordinate space transformation by using the Va as the X-axis and the cross product of Vb and Vn as the Y-axis and then just using some 2d method like with atan2() or something. Any ideas? Formulas?

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  • What algorithm can I use to determine points within a semi-circle?

    - by khayman218
    I have a list of two-dimensional points and I want to obtain which of them fall within a semi-circle. Originally, the target shape was a rectangle aligned with the x and y axis. So the current algorithm sorts the pairs by their X coord and binary searches to the first one that could fall within the rectangle. Then it iterates over each point sequentially. It stops when it hits one that is beyond both the X and Y upper-bound of the target rectangle. This does not work for a semi-circle as you cannot determine an effective upper/lower x and y bounds for it. The semi-circle can have any orientation. Worst case, I will find the least value of a dimension (say x) in the semi-circle, binary search to the first point which is beyond it and then sequentially test the points until I get beyond the upper bound of that dimension. Basically testing an entire band's worth of points on the grid. The problem being this will end up checking many points which are not within the bounds.

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  • drawing circle without floating point calculation

    - by zaharpopov
    This is common interview question (according to some interview sites) but I can find no normal answers in Internet - some are wrong and some point to complex theory I expect not looked for in interview (like Bressenham algorithm). The question is simple: The circle equation is: x^2 + y^2 = R^2. Given R, draw 0,0-centered circle as best as possible without using any floating point (no trigo, square roots, and so on, only integers)

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  • Prims vs Polys: what are the pros and cons of each?

    - by Richard Inglis
    I've noticed that most 3d gaming/rendering environments represent solids as a mesh of (usually triangular) 3d polygons. However some examples, such as Second Life, or PovRay use solids built from a set of 3d primitives (cube, sphere, cone, torus etc) on which various operations can be performed to create more complex shapes. So my question is: why choose one method over the other for representing 3d data? I can see there might be benefits for complex ray-tracing operations to be able to describe a surface as a single mathematical function (like PovRay does), but SL surely isn't attempting anything so ambitious with their rendering engine. Equally, I can imagine it might be more bandwidth-efficient to serve descriptions of generalised solids instead of arbitrary meshes, but is it really worth the downside that SL suffers from (ie modelling stuff is really hard, and usually the results are ugly) - was this just a bad decision made early in SL's development that they're now stuck with? Or is it an artefact of what's easiest to implement in OpenGL?

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  • How to convert any text/font to its bezier path representation?

    - by yizzreel
    I have a bezier path library to draw complex bezier paths without problem. Now, I need to know how to read a text or font and extract its path information to draw it as a path instead of as text. I came across a C applicaiton, FontForge. It does exactly what I need, picks any font and extract its path information. But what I need to know is how it does it to add that feature to my drawing library.

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  • Algorithm: Determine shape of two sectors delineated by an arbitrary path, and then fill one.

    - by Arseniy Banayev
    NOTE: This is a challenging problem for anybody who likes logic problems, etc. Consider a rectangular two-dimensional grid of height H and width W. Every space on the grid has a value, either 0 1 or 2. Initially, every space on the grid is a 0, except for the spaces along each of the four edges, which are initially a 2. Then consider an arbitrary path of adjacent (horizontally or vertically) grid spaces. The path begins on a 2 and ends on a different 2. Every space along the path is a 1. The path divides the grid into two "sectors" of 0 spaces. There is an object that rests on an unspecified 0 space. The "sector" that does NOT contain the object must be filled completely with 2. Define an algorithm that determines the spaces that must become 2 from 0, given an array (list) of values (0, 1, or 2) that correspond to the values in the grid, going from top to bottom and then from left to right. In other words, the element at index 0 in the array contains the value of the top-left space in the grid (initially a 2). The element at index 1 contains the value of the space in the grid that is in the left column, second from the top, and so forth. The element at index H contains the value of the space in the grid that is in the top row but second from the left, and so forth. Once the algorithm finishes and the empty "sector" is filled completely with 2s, the SAME algorithm must be sufficient to do the same process again. The second (and on) time, the path is still drawn from a 2 to a different 2, across spaces of 0, but the "grid" is smaller because the 2s that are surrounded by other 2s cannot be touched by the path (since the path is along spaces of 0). I thank whomever is able to figure this out for me, very very much. This does not have to be in a particular programming language; in fact, pseudo-code or just English is sufficient. Thanks again! If you have any questions, just leave a comment and I'll specify what needs to be specified.

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  • smallest perimiter rectangle with given integer area and integer sides

    - by remuladgryta
    Given an integer area A, how can one find integer sides w and h of a rectangle such that w*h = A and w+h is as small as possible? I'd rather the algorithm be simple than efficient (although within reasonable efficiency). What would be the best way to accomplish this? Finding out the prime factors of A, then combining them in some way that tries to balance w and h? Finding the two squares with integer sides with areas closest to A and then somehow interpolating between them? Any other method i'm not thinking of?

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  • Find the centroid of a polygon with weighted vertices

    - by Calle Kabo
    Hi, I know how to find the centroid (center of mass) of a regular polygon. This assumes that every part of the polygon weighs the same. But how do I calculate the centroid of a weightless polygon (made from aerogel perhaps :), where each vertex has a weight? Simplified illustration of what I mean using straight line: 5kg-----------------5kg ^center of gravity 10kg---------------5kg ^center of gravity offset du to weight of vertices Of course, I know how to calculate the center of gravity on a straight line with weighted vertices, but how do I do it on a polygon with weighted vertices? Thanks for your time!

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  • Efficiently remove points with same slope

    - 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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  • 3D line plane intersection, with simple plane

    - by clamp
    hello, i have two points in 3D space which have X-coordinates with different signum. so one of them lies definitely on one side of the X-plane and one on the other. now i want to find the intersection of this plane and the line made up by the two points in the most simple and optimized way. i know how to do general line plane intersection, but since in this case the plane is just the x-plane, i think there should be some shortcuts i can take. thanks!

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  • .NET Ascertaining mouse is on line drawn between two arbitrary points

    - by johnc
    I have an arrow drawn between two objects on a Winform. What would be the simplest way to determine that my mouse is currently hovering over, or near, this line. I have considered testing whether the mouse point intersects a square defined and extrapolated by the two points, however this would only be feasible if the two points had very similar x or y values. I am thinking, also, this problem is probably more in the realms of linear algebra rather than simple trigonometry, and whilst I do remember the simpler aspects of matrices, this problem is beyond my knowledge of linear algebra. On the other hand, if a .NET library can cope with the function, even better.

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  • Coloring close points

    - by ooboo
    I have a dense set of points in the plane. I want them colored so that points that are close to each other have the same color, and a different color if they're far away. For simplicity assume that there are, say, 5 different colors to choose from. Turns out I've not the slightest idea how to do that .. I'm using Tkinter with Python, by the way

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  • Two parallel line segments intersection

    - by Judarkness
    I know there are many algorithms to verify whether two line segments are intersected. But once they encountered parallel condition, they just tell the user a big "No" and pretend there is no overlap, share end point, or end point collusion. I know I can can calculate the distance between 2 lines segments. If the distance is 0, check the end points located in the other line segments or not. And this means I have to use a lot of if else and && || conditions. This is not difficult, but my question is "Is there a trick( or mathematics) method to calculate this special parallel case?"

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  • Find the set of largest contiguous rectangles to cover multiple areas

    - by joelpt
    I'm working on a tool called Quickfort for the game Dwarf Fortress. Quickfort turns spreadsheets in csv/xls format into a series of commands for Dwarf Fortress to carry out in order to plot a "blueprint" within the game. I am currently trying to optimally solve an area-plotting problem for the 2.0 release of this tool. Consider the following "blueprint" which defines plotting commands for a 2-dimensional grid. Each cell in the grid should either be dug out ("d"), channeled ("c"), or left unplotted ("."). Any number of distinct plotting commands might be present in actual usage. . d . d c c d d d d c c . d d d . c d d d d d c . d . d d c To minimize the number of instructions that need to be sent to Dwarf Fortress, I would like to find the set of largest contiguous rectangles that can be formed to completely cover, or "plot", all of the plottable cells. To be valid, all of a given rectangle's cells must contain the same command. This is a faster approach than Quickfort 1.0 took: plotting every cell individually as a 1x1 rectangle. This video shows the performance difference between the two versions. For the above blueprint, the solution looks like this: . 9 . 0 3 2 8 1 1 1 3 2 . 1 1 1 . 2 7 1 1 1 4 2 . 6 . 5 4 2 Each same-numbered rectangle above denotes a contiguous rectangle. The largest rectangles take precedence over smaller rectangles that could also be formed in their areas. The order of the numbering/rectangles is unimportant. My current approach is iterative. In each iteration, I build a list of the largest rectangles that could be formed from each of the grid's plottable cells by extending in all 4 directions from the cell. After sorting the list largest first, I begin with the largest rectangle found, mark its underlying cells as "plotted", and record the rectangle in a list. Before plotting each rectangle, its underlying cells are checked to ensure they are not yet plotted (overlapping a previous plot). We then start again, finding the largest remaining rectangles that can be formed and plotting them until all cells have been plotted as part of some rectangle. I consider this approach slightly more optimized than a dumb brute-force search, but I am wasting a lot of cycles (re)calculating cells' largest rectangles and checking underlying cells' states. Currently, this rectangle-discovery routine takes the lion's share of the total runtime of the tool, especially for large blueprints. I have sacrificed some accuracy for the sake of speed by only considering rectangles from cells which appear to form a rectangle's corner (determined using some neighboring-cell heuristics which aren't always correct). As a result of this 'optimization', my current code doesn't actually generate the above solution correctly, but it's close enough. More broadly, I consider the goal of largest-rectangles-first to be a "good enough" approach for this application. However I observe that if the goal is instead to find the minimum set (fewest number) of rectangles to completely cover multiple areas, the solution would look like this instead: . 3 . 5 6 8 1 3 4 5 6 8 . 3 4 5 . 8 2 3 4 5 7 8 . 3 . 5 7 8 This second goal actually represents a more optimal solution to the problem, as fewer rectangles usually means fewer commands sent to Dwarf Fortress. However, this approach strikes me as closer to NP-Hard, based on my limited math knowledge. Watch the video if you'd like to better understand the overall strategy; I have not addressed other aspects of Quickfort's process, such as finding the shortest cursor-path that plots all rectangles. Possibly there is a solution to this problem that coherently combines these multiple strategies. Help of any form would be appreciated.

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  • circles and triangles problem

    - by Faken
    Hello everyone, I have an interesting problem here I've been trying to solve for the last little while: I have 3 circles on a 2D xy plane, each with the same known radius. I know the coordinates of each of the three centers (they are arbitrary and can be anywhere). What is the largest triangle that can be drawn such that each vertice of the triangle sits on a separate circle, what are the coordinates of those verticies? I've been looking at this problem for hours and asked a bunch of people but so far only one person has been able to suggest a plausible solution (though i have no way of proving it). The solution that we have come up with involves first creating a triangle about the three circle centers. Next we look at each circle individually and calculate the equation of a line that passes through the circle's center and is perpendicular to the opposite edge. We then calculate two intersection points of the circle. This is then done for the next two circles with a result of 6 points. We iterate over the 8 possible 3 point triangles that these 6 points create (the restriction is that each point of the big triangle must be on a separate circle) and find the maximum size. The results look reasonable (at least when drawn out on paper) and it passes the special case of when the centers of the circles all fall on a straight line (gives a known largest triangle). Unfortunate i have no way of proving this is correct or not. I'm wondering if anyone has encountered a problem similar to this and if so, how did you solve it? Note: I understand that this is mostly a math question and not programming, however it is going to be implemented in code and it must be optimized to run very fast and efficient. In fact, I already have the above solution in code and tested to be working, if you would like to take a look, please let me know, i chose not to post it because its all in vector form and pretty much impossible to figure out exactly what is going on (because it's been condensed to be more efficient). Lastly, yes this is for school work, though it is NOT a homework question/assignment/project. It's part of my graduate thesis (abet a very very small part, but still technically is part of it). Thanks for your help.

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  • Does Quartz2D test intersection of rect by line before drawing it.

    - by ddnv
    I'm drawing a big scheme that consist of a lot of lines. I do it in the drawRect: method of UIView. The scheme is larger than the layer of view and I check each line and draw it only if it intersects the visible rect. But at one moment I thought, should I do this? Maybe Quartz is already doing this test? So the question is: When I use function CGContextAddLineToPoint() does the Core Graphics tests this line for intersection with layer rect or it just draw it anyway?

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  • Reverse-projection 2D points into 3D

    - by ehsan baghaki
    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 that camera point of view these points have different locations. for example (x0,y0,z0) will be (x0',y0') and (x1,y1,z1) will be (x1',y1') and (x2,y2,z2) will be (x2',y2') from the new camera point of view. So here is my a little hard question! I want to pick a point for example (X,Y) from the new camera point of view and tell where it will be on that plane. All i know is that 3 points and their locations on 3d space and their projection locations on the new camera view. Do you know the coefficients of the plane-equation and the camera positions (along with the projection), or do you only have the six points? - Nils i know the location of first 3 points. therefore we can calculate the coefficients of the plane. so we know exactly where the plane is from (0,0,0) point of view. and then we have the camera that can only see the points! So the only thing that camera sees is 3 points and also it knows their locations in 3d space (and for sure their locations on 2d camera view plane). and after all i want to look at camera view, pick a point (for example (x1,y1)) and tell where is that point on that plane. (for sure this (X,Y,Z) point should fit on the plane equation). Also i know nothing about the camera location.

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  • Determining if a coordinate is on a line

    - by TGCBraun
    I´m coding a little app that allows the user to draw multiple shapes and then remove or resize them. It´s working perfectly on rectangles and ovals, but I´m having issues with lines. Here´s a method that I wrote to find if the clicked spot on the screen is part of a specific line: public boolean containsLocation(int x, int y) { int m = (getY2() - getY()) / (getX2() - getX()); int b = getY() - (m * getX()); if (y == (m * x) + b) { return true; } return false; I´m using the famous y = mx + b formula and replacing y and x to find if the clicked spot is part of the line. The problem is when I click on the screen to remove the line, it only works if I click on the very fist coordinate (x,y) where the line starts. Nothing happens when I click anywhere else along the line. Can anyone shed a light on what I´m doing wrong? Thanks a lot.

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  • Cone-Line Segment Intersection 2D

    - by nophnoph
    Hi everyone, I would like to know is there any way to determine if a cone is intersecting with a (finite)line segment. The cone actually a circle located at P(x,y) with theta degree field of view and radius r. The simple visualization can be found here (sorry i can't post the picture here) I'm trying to do it in C# but I don't have any idea how to that, so for now this is what I'm doing : Check if the line segment is intersecting with the circle If the line segment is intersecting with the circle then I check every single point in the line segment using a function I found here. But I don't think this is the best way to do it. Does anyone have an idea? For additional info, I need this function to make some kind of simple vision simulator. Thanks in advance :)

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  • Generating a beveled edge for a 2D polygon

    - by Metaphile
    I'm trying to programmatically generate beveled edges for geometric polygons. For example, given an array of 4 vertices defining a square, I want to generate something like this. But computing the vertices of the inner shape is baffling me. Simply creating a copy of the original shape and then scaling it down will not produce the desired result most of the time. My algorithm so far involves analyzing adjacent edges (triples of vertices; e.g., the bottom-left, top-left, and top-right vertices of a square). From there, I need to find the angle between them, and then create a vertex somewhere along that angle, depending on how deep I want the bevel to be. And because I don't have much of a math background, that's where I'm stuck. How do I find that center angle? Or is there a much simpler way of attacking this problem?

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  • Uniforme distance between points

    - by Reonarudo
    Hello, How could I, having a path defined by several points that are not in a uniform distance from each other, redefine along the same path the same number of points but with a uniform distance. I'm trying to do this in Objective-C with NSArrays of CGPoints but so far I haven't had any luck with this. Thank you for any help. EDIT I was wondering if it would help to reduce the number of points, like when detecting if 3 points are collinear we could remove the middle one, but I'm not sure that would help.

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  • Detecting coincident subset of two coincident line segments

    - by Jared Updike
    This question is related to: How do I determine the intersection point of two lines in GDI+? (great explanation of algebra but no code) How do you detect where two line segments intersect? (accepted answer doesn't actually work) But note that an interesting sub-problem is completely glossed over in most solutions which just return null for the coincident case even though there are three sub-cases: coincident but do not overlap touching just points and coincident overlap/coincident line sub-segment For example we could design a C# function like this: public static PointF[] Intersection(PointF a1, PointF a2, PointF b1, PointF b2) where (a1,a2) is one line segment and (b1,b2) is another. This function would need to cover all the weird cases that most implementations or explanations gloss over. In order to account for the weirdness of coincident lines, the function could return an array of PointF's: zero result points (or null) if the lines are parallel or do not intersect (infinite lines intersect but line segments are disjoint, or lines are parallel) one result point (containing the intersection location) if they do intersect or if they are coincident at one point two result points (for the overlapping part of the line segments) if the two lines are coincident

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