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• Mathematics for Computer Science Students

  - by Ender

  To cut a long story short, I am a CS student that has received no formal Post-16 Maths education for years. Right now even my Algebra is extremely rusty and I have a couple of months to shape up my skills. I've got a couple of video lectures in my bookmarks, consisting of: Pre-Calculus Algebra Calculus Probability Introduction to Statistics Differential Equations Linear Algebra My aim as of today is to be able to read the CLRS book Introduction to Algorithms and be able to follow the Mathematical notation in that, as well as being able to confidently read and back-up any arguments written in Mathematical notation. Aside from these video lectures, can anyone recommend any good books to help teach someone wishing to go from a low-foundation level to a more advanced level of Mathematics? Just as a note, I've taken a first-year module in Analytical Modelling, so I understand some of the basic concepts of Discrete Mathematics. EDIT: Just a note to those that are looking to learn Linear Algebra using the Video Lectures I have posted up. Peteris Krumins' Blog contains a run-through of these lecture notes as well as his own commentary and lecture notes, an invaluable resource for those looking to follow the lectures too.

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• Recommendations with hierarchical data on non-relational databases?

  - by Luki

  I'm developing an web application that uses a non-relational database as a backend (django-nonrel + AppEngine). I need to store some hierarchical data (projects/subproject_1/subproject_N/tasks), and I'm wondering which pattern should I use. For now I thought of: Adjacency List (store the item's parent id) Nested sets (store left and right values for the item) In my case, the depth of nesting for a normal user will not exceed 4-5 levels. Also, on the UI, I would like to have a pagination for the items on the first level, to avoid to load too many items at the first page load. From what I understand so far, nested sets are great when the hierarchy is used more for displaying. Adjacency lists are great when editing on the tree is done often. In my case I guess I need the displaying more than the editing (when using nested sets, even if the display would work great, the above pagination could complicate things on editing). Do you have any thoughts and advice, based on your experience with the non-relational databases?

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• If some standards apply when "it depends" then should I stick with custom approaches?

  - by Travis J

  If I have an unconventional approach which works better than the industry standard, should I just stick with it even though in principal it violates those standards? What I am talking about is referential integrity for relational database management systems. The standard for enforcing referential integrity is to CASCADE delete. In practice, this is just not going to work all the time. In my current case, it does not. The alternative suggested is to either change the reference to NULL, DEFAULT, or just to take NO ACTION - usually in the form of a "soft delete". I am all about enforcing referential integrity. Love it. However, sometimes it just does not fully apply to use all the standards in practice. My approach has been to slightly abandon a small part of one of those practices which is the part about leaving "hanging references" around. Oops. The trade off is plentiful in this situation I believe. Instead of having deprecated data in the production database, a splattering of "soft delete" logic all across my controllers (and views sometimes depending on how far down the chain the soft delete occurred), and the prospect of queries taking longer and longer - instead of all that - I now have a recycle bin and centralized logic. The only tradeoff is that I must explicitly manage the possibility of "hanging references" which can be done through generics with one class. Any thoughts?

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• Algorithm for rating books: Relative perception

  - by suneet

  So I am developing this application for rating books (think like IMDB for books) using relational database. Problem statement : Let's say book "A" deserves 8.5 in absolute sense. In case if A is the best book I have ever seen, I'll most probably rate it 9.5 whereas for someone else, it might be just an average book, so he/they will rate it less (say around 8). Let's assume 4 such guys rate it 8. If there are 10 guys who are like me (who haven't ever read great literature) and they all rate it 9.5-10. This will effectively make it's cumulative rating greater than 9 (9.5*10 + 8*4) / 14 = 9.1 whereas we needed the result to be 8.5 ... How can I take care of(normalize) this bias due to incorrect perception of individuals. MyProposedSolution : Here's one of the ways how I think it could be solved. We can have a variable Lit_coefficient which tells us how much knowledge a user has about literature. If I rate "A"(the book) 9.5 and person "X" rates it 8, then he must have read books much better than "A" and thus his Lit_coefficient should be higher. And then we can normalize the ratings according to the Lit_coefficient of user. Could there be a better algorithm/solution for the same?

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• At what point does caching become necessary for a web application?

  - by Zaemz

  I'm considering the architecture for a web application. It's going to be a single page application that updates itself whenever the user selects different information on several forms that are available that are on the page. I was thinking that it shouldn't be good to rely on the user's browser to correctly interpret the information and update the view, so I'll send the user's choices to the server, and then get the data, send it back to the browser, and update the view. There's a table with 10,000 or so rows in a MySQL database that's going to be accessed pretty often, like once every 5-30 seconds for each user. I'm expecting 200-300 concurrent users at one time. I've read that a well designed relational database with simple queries are nothing for a RDBMS to handle, really, but I would still like to keep things quick for the client. Should this even be a concern for me at the moment? At what point would it be helpful to start using a separate caching service like Memcached or Redis, or would it even be necessary? I know that MySQL caches popular queries and the results, would this suffice?

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• Embedded non-relational (nosql) data store

  - by Igor Brejc

  I'm thinking about using/implementing some kind of an embedded key-value (or document) store for my Windows desktop application. I want to be able to store various types of data (GPS tracks would be one example) and of course be able to query this data. The amount of data would be such that it couldn't all be loaded into memory at the same time. I'm thinking about using sqlite as a storage engine for a key-value store, something like y-serial, but written in .NET. I've also read about FriendFeed's usage of MySQL to store schema-less data, which is a good pointer on how to use RDBMS for non-relational data. sqlite seems to be a good option because of its simplicity, portability and library size. My question is whether there are any other options for an embedded non-relational store? It doesn't need to be distributable and it doesn't have to support transactions, but it does have to be accessible from .NET and it should have a small download size. UPDATE: I've found an article titled SQLite as a Key-Value Database which compares sqlite with Berkeley DB, which is an embedded key-value store library.

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• Points on lines where the two lines are the closest together

  - by James Bedford

  Hey guys, I'm trying to find the points on two lines where the two lines are the closest. I've implemented the following method (Points and Vectors are as you'd expect, and a Line consists of a Point on the line and a non-normalized direction Vector from that point): void CDClosestPointsOnTwoLines(Line line1, Line line2, Point* closestPoints) { closestPoints[0] = line1.pointOnLine; closestPoints[1] = line2.pointOnLine; Vector d1 = line1.direction; Vector d2 = line2.direction; float a = d1.dot(d1); float b = d1.dot(d2); float e = d2.dot(d2); float d = a*e - b*b; if (d != 0) // If the two lines are not parallel. { Vector r = Vector(line1.pointOnLine) - Vector(line2.pointOnLine); float c = d1.dot(r); float f = d2.dot(r); float s = (b*f - c*e) / d; float t = (a*f - b*c) / d; closestPoints[0] = line1.positionOnLine(s); closestPoints[1] = line2.positionOnLine(t); } else { printf("Lines were parallel.\n"); } } I'm using OpenGL to draw three lines that move around the world, the third of which should be the line that most closely connects the other two lines, the two end points of which are calculated using this function. The problem is that the first point of closestPoints after this function is called will lie on line1, but the second point won't lie on line2, let alone at the closest point on line2! I've checked over the function many times but I can't see where the mistake in my implementation is. I've checked my dot product function, scalar multiplication, subtraction, positionOnLine() etc. etc. So my assumption is that the problem is within this method implementation. If it helps to find the answer, this is function supposed to be an implementation of section 5.1.8 from 'Real-Time Collision Detection' by Christer Ericson. Many thanks for any help!

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• How to detect 2D line on line collision?

  - by Vish

  I'm a flash actionscript game developer who is a bit backward with mathematics, though I find physics both interesting and cool. For reference this is a similar game to the one I'm making: Untangled flash game I have made an untangled game almost to full completion of logic. But, when two lines intersect, I need those intersected or 'tangled' lines to show a different color; red. It would be really kind of you people if you could suggest an algorithm with/without math for detecting line segment collisions. I'm basically a person who likes to think 'visually' than 'arithmetically' :) P.S I'm trying to make a function as private function isIntersecting(A:Point, B:Point, C:Point, D:Point):Boolean Thanks in advance.

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• Detect line line collision in an Untangled game

  - by Vish

  Pardon me if this is a repeat question,but I've been searching google for vain since the past few days, I'm a flash actionscript game developer who is a bit backward with mathematics, though I find physics both interesting and cool. Similiar game : Untangled flash game I have made an untangled game almost to full completion of logic. But, when two lines intersect , I need those intersected or 'tangled' lines to show a different color; red. It would be really kind of you people if you could suggest an algorithm with / without math for detecting line segment collisions. I'm basically a person who likes to think 'visually' than 'arithmetically' :) P.S I'm trying to make a function as private function isIntersecting(A:Point, B:Point, C:Point, D:Point):Boolean Thanks in advance. Vishnu Ajit

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• Raycasting tutorial / vector math question

  - by mattboy

  I'm checking out this nice raycasting tutorial at http://lodev.org/cgtutor/raycasting.html and have a probably very simple math question. In the DDA algorithm I'm having trouble understanding the calcuation of the deltaDistX and deltaDistY variables, which are the distances that the ray has to travel from 1 x-side to the next x-side, or from 1 y-side to the next y-side, in the square grid that makes up the world map (see below screenshot). In the tutorial they are calculated as follows, but without much explanation: //length of ray from one x or y-side to next x or y-side double deltaDistX = sqrt(1 + (rayDirY * rayDirY) / (rayDirX * rayDirX)); double deltaDistY = sqrt(1 + (rayDirX * rayDirX) / (rayDirY * rayDirY)); rayDirY and rayDirX are the direction of a ray that has been cast. How do you get these formulas? It looks like pythagorean theorem is part of it, but somehow there's division involved here. Can anyone clue me in as to what mathematical knowledge I'm missing here, or "prove" the formula by showing how it's derived?

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• View matrix in opengl

  - by user5584

  Hi! Sorry for my clumsy question. But I don't know where I am wrong at creating view matrix. I have the following code: createMatrix(vec4f(xAxis.x, xAxis.y, xAxis.z, dot(xAxis,eye)), vec4f( yAxis.x_, yAxis.y_, yAxis.z_, dot(yAxis,eye)), vec4f(-zAxis.x_, -zAxis.y_, -zAxis.z_, -dot(zAxis,eye)), vec4f(0, 0, 0, 1)); //column1, column2,... I have tried to transpose it, but with no success. I have also tried to use gluLookAt(...) with success. At the reference page, I watched the remarks about the to-be-created matrix, and it seems the same as mine. Where I am wrong?

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• Transform between two 3d cartesian coordinate systems

  - by Pris

  I'd like to know how to get the rotation matrix for the transformation from one cartesian coordinate system (X,Y,Z) to another one (X',Y',Z'). Both systems are defined with three orthogonal vectors as one would expect. No scaling or translation occurs. I'm using OpenSceneGraph and it offers a Matrix convenience class, if it makes finding the matrix easier: http://www.openscenegraph.org/documentation/OpenSceneGraphReferenceDocs/a00403.html.

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• For normal mapping, why can we not simply add the tangent normal to the surface normal?

  - by sebf

  I am looking at implementing bump mapping (which in all implementations I have seen is really normal mapping), and so far all I have read says that to do this, we create a matrix to convert from world-space to tangent-space, in order to transform the lights and eye direction vectors into tangent space, so that the vectors from the normal map may be used directly in place of those passed through from the vertex shader. What I do not understand though, is why we cannot just use the normalised sum of the sampled-normal vector, and the surface-normal? (assuming we already transform and pass through the surface normal for the existing lighting functions) Take the diagram below; the normal is simply the deviation from the 'reference normal' for any given coordinate system, correct? And transforming the surface normal of a mapped surface from world space to tangent space makes it equivalent to the tangent space 'reference normal', no? If so, why do we transform all lighting vectors into tangent space, instead of simply transforming the sampled tangent once in the pixel shader?

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• Typical Applications of Linear System Solver in Game Developemnt

  - by craftsman.don

  I am going to write a custom solver for linear system. I would like to survey the typical problems involved the linear system solving in games. So that I can custom optimization on these problems based on the shape of the matrix. currently I am focus on these problems: B-Spline editing (I use a linear solve to resolve the C0, C1, C2 continuity) Constraint in Simulation (especially Position-Constraint, cloth) Both of them are Banded Matrix. I want to hear about some other applications of a linear system in games. Thank you.

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• How can I calculate a vertex normal for a hard edge?

  - by K.G.

  Here is a picture of a lovely polygon: Circled is a vertex, and numbered are its adjacent faces. I have calculated the normals of those faces as such (not yet normalized, 0-indexed): Vertex 1 normal 0: 0.000000 0.000000 -0.250000 Vertex 1 normal 1: 0.000000 0.000000 -0.250000 Vertex 1 normal 2: -0.250000 0.000000 0.000000 Vertex 1 normal 3: -0.250000 0.000000 0.000000 Vertex 1 normal 4: 0.250000 0.000000 0.000000 What I'm wondering is, how can I determine, taken as given that I want this vertex to represent a hard edge, whether its normal should be the normal of 1/2 or 3/4? My plan after I glanced at the sketch I used to put this together was "Ha! I'll just use whichever two faces have the same normal!" and now I see that there are two sets of two faces for which this is true. Is there a rule I can apply based on the face winding, angle of the adjacent edges, moon phase, coin flip, to consistently choose a normal direction for this box? For the record, all of the other polygons I plan to use will have their normals dictated in Maya, but after encountering this problem, it made me really curious.

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• Make an object slide around an obstacle

  - by Isaiah

  I have path areas set up in a game I'm making for canvas/html5 and have got it working to keep the player within these areas. I have a function isOut(boundary, x, y) that returns true if the point is outside the boundary. What I do is check only the new position x/y separately with the corresponding old position x/y. Then if each one is out I assign them the past value from the frame before. The old positions are kept in a variable from a closure I made. like this: opos = [x,y];//old position npos = [x,y];//new position if(isOut(bound, npos[0], opos[1])){ npos[0] = opos[0]; //assign it the old x position } if(isOut(bound, opos[0], npos[1])){ npos[1] = opos[1]; //assign it the old y position } It looks nice and works good at certain angles, but if your boundary has diagonal regions it results in jittery motion. What's happening is the y pos exits the area while x doesn't and continues pushing the player to the side, once it has moved the player to the side a bit the player can move forward and then the y exits again and the whole process repeats. Anyone know how I may be able to achieve a smoother slide? I have access to the player's velocity vector, the angle, and the speed(when used with the angle). I can move the play with either angle/speed or x/yvelocities as I've built in backups to translate one to the other if either have been altered manually.

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• How can I test if a point lies between two parallel lines?

  - by Harold

  In the game I'm designing there is a blast that shoots out from an origin point towards the direction of the mouse. The width of this blast is always going to be the same. Along the bottom of the screen (what's currently) squares move about which should be effected by the blast that the player controls. Currently I am trying to work out a way to discover if the corners of these squares are within the blast's two bounding lines. I thought the best way to do this would be to rotate the corners of the square around an origin point as if the blast were completely horizontal and see if the Y values of the corners were less than or equal to the width of the blast which would mean that they lie within the effected region, but I can't work out

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• Moving objects colliding when using unalligned collision avoidance (steering)

  - by James Bedford

  I'm having trouble with unaligned collision avoidance for what I think is a rare case. I have set two objects to move towards each other but with a slight offset, so one of the objects is moving slightly upwards, and one of the objects is moving slightly downwards. In my unaligned collision avoidance steering algorithm I'm finding the points on the object's forward line and the other object's forward line where these two lines are the closest. If these closest points are within a collision avoidance distance, and if the distance between them is smaller than the two radii of the two object's bounding spheres, then the objects should steer away in the appropriate direction. The problem is that for my case, the closest points on the lines are calculated to be really far away from the actual collision point. This is because the two forward lines for each object are moving away from each other as the objects pass. The problem is that because of this, no steering takes place, and the two objects partially collide. Does anyone have any suggestions as to how I can correctly calculate the point of collision? Perhaps by somehow taking into account the size of the two objects?

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• Understanding dot notation

  - by Starkers

  Here's my interpretation of dot notation: a = [2,6] b = [1,4] c = [0,8] a . b . c = (2*6)+(1*4)+(0*8) = 12 + 4 + 0 = 16 What is the significance of 16? Apparently it's a scalar. Am I right in thinking that a scalar is the number we times a unit vector by to get a vector that has a scaled up magnitude but the same direction as the unit vector? So again, what is the relevance of 16? When is it used? It's not the magnitude of all the vectors added up. The magnitude of all of them is calculated as follows: sqrt( ax * ax + ay * ay ) + sqrt( bx * bx + by * by ) + sqrt( cx * cx + cy * cy) sqrt( 2 * 2 + 6 * 6 ) + sqrt( 1 * 1 + 4 * 4 ) + sqrt( 0 * 0 + 8 * 8) sqrt( 4 + 36 ) + sqrt( 1 + 16 ) + sqrt( 0 + 64) sqrt( 40 ) + sqrt( 17 ) + sqrt( 64) 6.3 + 4.1 + 8 10.4 + 8 18.4 So I don't really get this diagram: Attempting with sensible numbers: a = [1,0] b = [4,3] a . b = (1*0) + (4*3) = 0 + 12 = 12 So what exactly is a . b describing here? The magnitude of that vector? Because that isn't right: the 'a.b' vector = [4,0] sqrt( x*x + y*y ) sqrt( 4*4 + 0*0 ) sqrt( 16 + 0 ) 4 So what is 12 describing?

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• Lerping to a center point while in motion

  - by Fibericon

  I have an enemy that initially flies in a circular motion, while facing away from the center point. This is how I achieve that: position.Y = (float)(Math.Cos(timeAlive * MathHelper.PiOver4) * radius + origin.Y); position.X = (float)(Math.Sin(timeAlive * MathHelper.PiOver4) * radius + origin.X); if (timeAlive < 5) { angle = (float)Math.Atan((0 - position.X) / (0 - position.Y)); if (0 < position.Y) RotationMatrix = Matrix.CreateRotationX(MathHelper.PiOver2) * Matrix.CreateRotationZ(-1 * angle); else RotationMatrix = Matrix.CreateRotationX(MathHelper.PiOver2) * Matrix.CreateRotationZ(MathHelper.Pi - angle); } That part works just fine. After five seconds of this, I want the enemy to turn inward, facing the center point. However, I've been trying to lerp to that point, since I don't want it to simply jump to the new rotation. Here's my code for trying to do that: else { float newAngle = -1 * (float)Math.Atan((0 - position.X) / (0 - position.Y)); angle = MathHelper.Lerp(angle, newAngle, (float)gameTime.ElapsedGameTime.Milliseconds / 1000); if (0 < position.Y) RotationMatrix = Matrix.CreateRotationX(MathHelper.PiOver2) * Matrix.CreateRotationZ(MathHelper.Pi - angle); else RotationMatrix = Matrix.CreateRotationX(MathHelper.PiOver2) * Matrix.CreateRotationZ(-1 * angle); } That doesn't work so fine. It seems like it's going to at first, but then it just sort of skips around. How can I achieve what I want here?

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• How to get xy coordinates along a given path

  - by netbrain

  Say i have two points (x,y), (0,0) and (10,10). Now i wan´t to get coordinates along the line by stepping through values of x and y. I thought i solved it with the following functions: fy = startY + (x - startX) * ((destY-startY)/(destX-startX)); fx = (y + startY) / ((destY-startY)/(destX-startX)) + startX; taken from http://en.wikipedia.org/wiki/Linear_interpolation However, it seems that im getting a problem when destX and startX is the same value, so you get division by zero. Is there a better way of getting coordinates along a line when knowing the start and endpoint of the line?

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• Trouble with Collada bones

  - by KyleT

  I have a Collada file with a rigged mesh. I've read the node tags in the library_visual_scenes tag and extracted the matrix for each node and stored everything in a hierarchical bone structure. My Matrix container is "row major", so I'd store the first float of a matrix tag in the 1st row, 1st column, the second in the 1st row, 2nd column, etc. From what I gather this is the Bind Pose Matrix. After that I went through the tag and extracted the float array in the source tag of the skin tag of the controller for the mesh. I stored each matrix from this float array in their corresponding Bone as the Inverse Bind Matrix. I also extracted the bind-shape-matrix and stored it. Now I'd like to draw the skeleton with OpenGL to see if everything is working correctly before I go about skinning. I iterate once over my bones and multiply a bone's Bind Pose Matrix by it's parents and store that. After that I iterate again over the bones and multiply the result of the previous matrix multiplication by the Inverse Bind Matrix and then by the Bind Shape Matrix. The results look something like this: [0.2, 9.2, 5.8, 1.2 ] [4.6, -3.3, -0.2, -0.1 ] [-1.8, 0.2, -4.2, -3.9 ] [0, 0, 0, 1 ] I've had to go to various sources to get the little understanding of Collada I have and books about 3d transform matricies can get pretty intense. I've hit a brick wall and if you could please read through this and see if there is something I'm doing wrong, and how I'd go about getting an X,Y,Z to draw a point for each of these joints once I've calculated the final transform, I'd really appreciate it.

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• Matrices: Arrays or separate member variables?

  - by bjz

  I'm teaching myself 3D maths and in the process building my own rudimentary engine (of sorts). I was wondering what would be the best way to structure my matrix class. There are a few options: Separate member variables: struct Mat4 { float m11, m12, m13, m14, m21, m22, m23, m24, m31, m32, m33, m34, m41, m42, m43, m44; // methods } A multi-dimensional array: struct Mat4 { float[4][4] m; // methods } An array of vectors struct Mat4 { Vec4[4] m; // methods } I'm guessing there would be positives and negatives to each. From 3D Math Primer for Graphics and Game Development, 2nd Edition p.155: Matrices use 1-based indices, so the first row and column are numbered 1. For example, a12 (read “a one two,” not “a twelve”) is the element in the first row, second column. Notice that this is different from programming languages such as C++ and Java, which use 0-based array indices. A matrix does not have a column 0 or row 0. This difference in indexing can cause some confusion if matrices are stored using an actual array data type. For this reason, it’s common for classes that store small, fixed size matrices of the type used for geometric purposes to give each element its own named member variable, such as float a11, instead of using the language’s native array support with something like float elem[3][3]. So that's one vote for method one. Is this really the accepted way to do things? It seems rather unwieldy if the only benefit would be sticking with the conventional math notation.

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• Portal View/Projection Matrix near plane

  - by melak47

  For RenderToTexture/Camera based portal rendering, the basics seems simple enough. However, with a free camera, most of the time it is going to be looking at such portals at an angle: Now a regular near clipping plane will not always work here, it will either intersect with the wall the portal is sitting on, or possibly with objects in front of the wall. The desired near clipping plane would be aligned like the portal, producing a view volume more like this: or this in 3D: So here is my question: How does one construct or "truncate" a view/projection matrix to achieve such an off-camera-normal (near) clipping plane?

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• Mouse pointer position to screen space

  - by Ylisar

  If I have a mouse pointer position in pixels of canvas, I can easily convert it to the -1..1 range for both X & Y by lerping by dividing with canvas dimensions. However, the problem is what I should put in Z & W if I want my screen space position to be on the near plane? The step afterwards would be for me to multiply by the inverse of view-projection to take me to world space, where I easily can construct a ray from the cameras world space position.

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