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• Calculating up-vector to avoid gimbal lock using euler angles

  - by jessejuicer

  I wish to orbit a camera around a sphere, yet the problem is that when the camera rotates so that it is at the north pole (and pointing down) or the south pole (and pointing up) of the sphere the camera doesn't handle itself very well. It spins rapidly until arriving 180 degrees in the opposite direction. I believe this is known as gimbal lock. I understand you can avoid this problem using quaternions. But I also read in another forum that it's possible to avoid this easily using euler angles as well. Which I would prefer to do. It was said that all you need to do is "calculate a proper up-vector every frame, and that avoids the problem entirely." Well, I tried aligning the up-vector with the vertical axis of the camera whenever the camera changed orientation, but this didn't seem to work. Meaning that the up-vector followed exactly the orientation of the camera's y-axis (or it's up vector), instead of using a constant up-vector aligned to the up-vector of the world (0, 1, 0). How exactly do I go about calculating a proper up-vector as my camera orientation changes to avoid the gimbal lock problem mentioned above?

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• Cross-reference between delphi records

  - by Paul-Jan

  Let's say I have a record TQuaternion and a record TVector. Quaternions have some methods with TVector parameters. On the other hand, TVector supports some operations that have TQuaternion parameters. Knowing that Delphi (Win32) does not allow for forward record declarations, how do I solve this elegantly? Using classes is not really an option here, because I really want to use operator overloading for this rare case where it actually makes good sense. For now I simply moved these particular methods out of the records and into separate functions, the good old-fashioned way. Better suggestions are most welcome.

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• How to interpolate rotations?

  - by uhuu

  I have two vectors describing rotations; a start rotation A and a target rotation B. How would I best go about interpolating A by a factor F to approach B? Using a simple lerp on the vectors fails to work when more than one dimension needs to be interpolated (i.e. produces undesirable rotations). Maybe building quaternions from the rotation vectors and using slerp is the way to go. But how, then, could I extract a vector describing the new rotation from the resulting quaternion? Thanks in advance.

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• How do I implement a quaternion based camera?

  - by kudor gyozo

  I looked at several tutorials about this and when I thought I understood I tried to implement a quaternion based camera. The problem is it doesn't work correctly, after rotating for approx. 10 degrees it jumps back to -10 degrees. I have no idea what's wrong. I'm using openTK and it already has a quaternion class. I'm a noob at opengl, I'm doing this just for fun, and don't really understand quaternions, so probably I'm doing something stupid here. Here is some code: (Actually almost all the code except the methods that load and draw a vbo (it is taken from an OpenTK sample that demonstrates vbo-s)) I load a cube into a vbo and initialize the quaternion for the camera protected override void OnLoad(EventArgs e) { base.OnLoad(e); cameraPos = new Vector3(0, 0, 7); cameraRot = Quaternion.FromAxisAngle(new Vector3(0,0,-1), 0); GL.ClearColor(System.Drawing.Color.MidnightBlue); GL.Enable(EnableCap.DepthTest); vbo = LoadVBO(CubeVertices, CubeElements); } I load a perspective projection here. This is loaded at the beginning and every time I resize the window. protected override void OnResize(EventArgs e) { base.OnResize(e); GL.Viewport(0, 0, Width, Height); float aspect_ratio = Width / (float)Height; Matrix4 perpective = Matrix4.CreatePerspectiveFieldOfView(MathHelper.PiOver4, aspect_ratio, 1, 64); GL.MatrixMode(MatrixMode.Projection); GL.LoadMatrix(ref perpective); } Here I get the last rotation value and create a new quaternion that represents only the last rotation and multiply it with the camera quaternion. After this I transform this into axis-angle so that opengl can use it. (This is how I understood it from several online quaternion tutorials) protected override void OnRenderFrame(FrameEventArgs e) { base.OnRenderFrame(e); GL.Clear(ClearBufferMask.ColorBufferBit | ClearBufferMask.DepthBufferBit); double speed = 1; double rx = 0, ry = 0; if (Keyboard[Key.A]) { ry = -speed * e.Time; } if (Keyboard[Key.D]) { ry = +speed * e.Time; } if (Keyboard[Key.W]) { rx = +speed * e.Time; } if (Keyboard[Key.S]) { rx = -speed * e.Time; } Quaternion tmpQuat = Quaternion.FromAxisAngle(new Vector3(0,1,0), (float)ry); cameraRot = tmpQuat * cameraRot; cameraRot.Normalize(); GL.MatrixMode(MatrixMode.Modelview); GL.LoadIdentity(); Vector3 axis; float angle; cameraRot.ToAxisAngle(out axis, out angle); GL.Rotate(angle, axis); GL.Translate(-cameraPos); Draw(vbo); SwapBuffers(); } Here are 2 images to explain better: I rotate a while and from this: it jumps into this Any help is appreciated. Update1: I add these to a streamwriter that writes into a file: sw.WriteLine("camerarot: X:{0} Y:{1} Z:{2} W:{3} L:{4}", cameraRot.X, cameraRot.Y, cameraRot.Z, cameraRot.W, cameraRot.Length); sw.WriteLine("ry: {0}", ry); The log is available here: http://www.pasteall.org/26133/text. At line 770 the cube jumps from right to left, when camerarot.Y changes signs. I don't know if this is normal. Update2 Here is the complete project.

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• glm matrix conversion for DirectX

  - by niktehpui

  For on of the coursework specification I need to work with DirectX, so I tried to implement a DirectX Renderer in my small cross-platform framework (to have it optionally available for Windows). Since I want to stick to my dependencies I want use glm for vector/matrix/quaternions math. The vectors seem to be fully compatible with DirectX, but the glm::mat4 is not working properly in DirectX Effects Framework. I assumed the reason is that DirectX uses row majors layouts and OpenGL column majors (although if I remember right internally in HLSL DX uses column major as well), so I transposed the matrix, but I still get no proper results compared to using XNA-Math. XNA-Version of the code (works): XMMATRIX world = XMMatrixIdentity(); XMMATRIX view = XMMatrixLookAtLH(XMVectorSet(5.0, 5.0, 5.0, 1.0f), XMVectorZero(), XMVectorSet(0.0f, 1.0f, 0.0f, 0.0f)); XMMATRIX proj = XMMatrixPerspectiveFovLH(0.25f*3.14f, 1.25f, 1.0f, 1000.0f); XMMATRIX worldViewProj = world*view*proj; m_fxWorldViewProj->SetMatrix(reinterpret_cast<float*>(&worldViewProj)); This works flawlessly and displays the expected colored cube. GLM-Version (does not work): glm::mat4 world(1.0f); glm::mat4 view = glm::lookAt(glm::vec3(5.0f, 5.0f, 5.0f), glm::vec3(0.0f, 0.0f, 0.0f), glm::vec3(0.0f, 1.0f, 0.0f)); glm::mat4 proj = glm::perspective(0.25f*3.14f, 1.25f, 1.0f, 1000.0f); glm::mat4 worldViewProj = glm::transpose(world*view*proj); m_fxWorldViewProj->SetMatrix(glm::value_ptr(worldViewProj)); Displays nothing, screen stays black. I really would like to stick to glm on all platforms.

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• Spatial Rotation in Gmod Expression2.

  - by Fascia

  I'm using expression2 to program behavior in Garry's mod (http://wiki.garrysmod.com/?title=Wire_Expression2) Okay so, to set the precedent. In Gmod I have a block and I am at a complete loss of how to get it to rotate around the 3 up, down and right vectors (Which are local. ie; if I pitch it 45 degrees the forward vector is 0.707, 0.707, 0). Essentially, From the 3 vectors I'd like to be able to get local Pitch/Roll/Yaw. By Local Pitch Roll Yaw I mean that they are completely independent of one another allowing true 3d rotation. So for example; if I place my craft so its nose is parallel to the floor the X,Y,Z would be 0,0,0. If I turn it parallel to the floor (World and Local Yaw) 90 degrees it's now 0, 0, 90. If I then pitch it (World Roll, Local Pitch) it 180 degrees it's now 180, 0, 90. I've already explored quaternions however I don't believe I should post my code here as I think I was re-inventing the wheel. I know I didn't explain that well but I believe the problem is pretty generic. Any help anyone could offer is greatly appreciated. Oh, I'd like to avoid gimblelock too. Essentially calculating the rotation around each of the crafts up/forward/right vectors using the up/forward/right vectors. To simply the question a generic implementation rather than one specific to Gmod is absolutely fine.

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• NET Math Libraries

  - by JoshReuben

  NET Mathematical Libraries   .NET Builder for Matlab The MathWorks Inc. - http://www.mathworks.com/products/netbuilder/ MATLAB Builder NE generates MATLAB based .NET and COM components royalty-free deployment creates the components by encrypting MATLAB functions and generating either a .NET or COM wrapper around them. .NET/Link for Mathematica www.wolfram.com a product that 2-way integrates Mathematica and Microsoft's .NET platform call .NET from Mathematica - use arbitrary .NET types directly from the Mathematica language. use and control the Mathematica kernel from a .NET program. turns Mathematica into a scripting shell to leverage the computational services of Mathematica. write custom front ends for Mathematica or use Mathematica as a computational engine for another program comes with full source code. Leverages MathLink - a Wolfram Research's protocol for sending data and commands back and forth between Mathematica and other programs. .NET/Link abstracts the low-level details of the MathLink C API. Extreme Optimization http://www.extremeoptimization.com/ a collection of general-purpose mathematical and statistical classes built for the.NET framework. It combines a math library, a vector and matrix library, and a statistics library in one package. download the trial of version 4.0 to try it out. Multi-core ready - Full support for Task Parallel Library features including cancellation. Broad base of algorithms covering a wide range of numerical techniques, including: linear algebra (BLAS and LAPACK routines), numerical analysis (integration and differentiation), equation solvers. Mathematics leverages parallelism using .NET 4.0's Task Parallel Library. Basic math: Complex numbers, 'special functions' like Gamma and Bessel functions, numerical differentiation. Solving equations: Solve equations in one variable, or solve systems of linear or nonlinear equations. Curve fitting: Linear and nonlinear curve fitting, cubic splines, polynomials, orthogonal polynomials. Optimization: find the minimum or maximum of a function in one or more variables, linear programming and mixed integer programming. Numerical integration: Compute integrals over finite or infinite intervals, over 2D and higher dimensional regions. Integrate systems of ordinary differential equations (ODE's). Fast Fourier Transforms: 1D and 2D FFT's using managed or fast native code (32 and 64 bit) BigInteger, BigRational, and BigFloat: Perform operations with arbitrary precision. Vector and Matrix Library Real and complex vectors and matrices. Single and double precision for elements. Structured matrix types: including triangular, symmetrical and band matrices. Sparse matrices. Matrix factorizations: LU decomposition, QR decomposition, singular value decomposition, Cholesky decomposition, eigenvalue decomposition. Portability and performance: Calculations can be done in 100% managed code, or in hand-optimized processor-specific native code (32 and 64 bit). Statistics Data manipulation: Sort and filter data, process missing values, remove outliers, etc. Supports .NET data binding. Statistical Models: Simple, multiple, nonlinear, logistic, Poisson regression. Generalized Linear Models. One and two-way ANOVA. Hypothesis Tests: 12 14 hypothesis tests, including the z-test, t-test, F-test, runs test, and more advanced tests, such as the Anderson-Darling test for normality, one and two-sample Kolmogorov-Smirnov test, and Levene's test for homogeneity of variances. Multivariate Statistics: K-means cluster analysis, hierarchical cluster analysis, principal component analysis (PCA), multivariate probability distributions. Statistical Distributions: 25 29 continuous and discrete statistical distributions, including uniform, Poisson, normal, lognormal, Weibull and Gumbel (extreme value) distributions. Random numbers: Random variates from any distribution, 4 high-quality random number generators, low discrepancy sequences, shufflers. New in version 4.0 (November, 2010) Support for .NET Framework Version 4.0 and Visual Studio 2010 TPL Parallellized – multicore ready sparse linear program solver - can solve problems with more than 1 million variables. Mixed integer linear programming using a branch and bound algorithm. special functions: hypergeometric, Riemann zeta, elliptic integrals, Frensel functions, Dawson's integral. Full set of window functions for FFT's. Product  Price Update subscription Single Developer License $999  $399  Team License (3 developers) $1999  $799  Department License (8 developers) $3999  $1599  Site License (Unlimited developers in one physical location) $7999  $3199    NMath http://www.centerspace.net .NET math and statistics libraries matrix and vector classes random number generators Fast Fourier Transforms (FFTs) numerical integration linear programming linear regression curve and surface fitting optimization hypothesis tests analysis of variance (ANOVA) probability distributions principal component analysis cluster analysis built on the Intel Math Kernel Library (MKL), which contains highly-optimized, extensively-threaded versions of BLAS (Basic Linear Algebra Subroutines) and LAPACK (Linear Algebra PACKage). Product  Price Update subscription Single Developer License $1295 $388 Team License (5 developers) $5180 $1554   DotNumerics http://www.dotnumerics.com/NumericalLibraries/Default.aspx free DotNumerics is a website dedicated to numerical computing for .NET that includes a C# Numerical Library for .NET containing algorithms for Linear Algebra, Differential Equations and Optimization problems. The Linear Algebra library includes CSLapack, CSBlas and CSEispack, ports from Fortran to C# of LAPACK, BLAS and EISPACK, respectively. Linear Algebra (CSLapack, CSBlas and CSEispack). Systems of linear equations, eigenvalue problems, least-squares solutions of linear systems and singular value problems. Differential Equations. Initial-value problem for nonstiff and stiff ordinary differential equations ODEs (explicit Runge-Kutta, implicit Runge-Kutta, Gear's BDF and Adams-Moulton). Optimization. Unconstrained and bounded constrained optimization of multivariate functions (L-BFGS-B, Truncated Newton and Simplex methods).   Math.NET Numerics http://numerics.mathdotnet.com/ free an open source numerical library - includes special functions, linear algebra, probability models, random numbers, interpolation, integral transforms. A merger of dnAnalytics with Math.NET Iridium in addition to a purely managed implementation will also support native hardware optimization. constants & special functions complex type support real and complex, dense and sparse linear algebra (with LU, QR, eigenvalues, ... decompositions) non-uniform probability distributions, multivariate distributions, sample generation alternative uniform random number generators descriptive statistics, including order statistics various interpolation methods, including barycentric approaches and splines numerical function integration (quadrature) routines integral transforms, like fourier transform (FFT) with arbitrary lengths support, and hartley spectral-space aware sequence manipulation (signal processing) combinatorics, polynomials, quaternions, basic number theory. parallelized where appropriate, to leverage multi-core and multi-processor systems fully managed or (if available) using native libraries (Intel MKL, ACMS, CUDA, FFTW) provides a native facade for F# developers

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• MD5 vertex skinning problem extending to multi-jointed skeleton (GPU Skinning)

  - by Soapy

  Currently I'm trying to implement GPU skinning in my project. So far I have achieved single joint translation and rotation, and multi-jointed translation. The problem arises when I try to rotate a multi-jointed skeleton. The image above shows the current progress. The left image shows how the model should deform. The middle image shows how it deforms in my project. The right shows a better deform (still not right) inverting a certain value, which I will explain below. The way I get my animation data is by exporting it to the MD5 format (MD5mesh for mesh data and MD5anim for animation data). When I come to parse the animation data, for each frame, I check if the bone has a parent, if not, the data is passed in as is from the MD5anim file. If it does have a parent, I transform the bones position by the parents orientation, and the add this with the parents translation. Then the parent and child orientations get concatenated. This is covered at this website. if (Parent < 0){ ... // Save this data without editing it } else { Math3::vec3 rpos; Math3::quat pq = Parent.Quaternion; Math3::quat pqi(pq); pqi.InvertUnitQuat(); pqi.Normalise(); Math3::quat::RotateVector3(rpos, pq, jv); Math3::vec3 npos(rpos + Parent.Pos); this->Translation = npos; Math3::quat nq = pq * jq; nq.Normalise(); this->Quaternion = nq; } And to achieve the image to the right, all I need to do is to change Math3::quat::RotateVector3(rpos, pq, jv); to Math3::quat::RotateVector3(rpos, pqi, jv);, why is that? And this is my skinning shader. SkinningShader.vert #version 330 core smooth out vec2 vVaryingTexCoords; smooth out vec3 vVaryingNormals; smooth out vec4 vWeightColor; uniform mat4 MV; uniform mat4 MVP; uniform mat4 Pallete[55]; uniform mat4 invBindPose[55]; layout(location = 0) in vec3 vPos; layout(location = 1) in vec2 vTexCoords; layout(location = 2) in vec3 vNormals; layout(location = 3) in int vSkeleton[4]; layout(location = 4) in vec3 vWeight; void main() { vec4 wpos = vec4(vPos, 1.0); vec4 norm = vec4(vNormals, 0.0); vec4 weight = vec4(vWeight, (1.0f-(vWeight[0] + vWeight[1] + vWeight[2]))); normalize(weight); mat4 BoneTransform; for(int i = 0; i < 4; i++) { if(vSkeleton[i] != -1) { if(i == 0) { // These are interchangable for some reason // BoneTransform = ((invBindPose[vSkeleton[i]] * Pallete[vSkeleton[i]]) * weight[i]); BoneTransform = ((Pallete[vSkeleton[i]] * invBindPose[vSkeleton[i]]) * weight[i]); } else { // These are interchangable for some reason // BoneTransform += ((invBindPose[vSkeleton[i]] * Pallete[vSkeleton[i]]) * weight[i]); BoneTransform += ((Pallete[vSkeleton[i]] * invBindPose[vSkeleton[i]]) * weight[i]); } } } wpos = BoneTransform * wpos; vWeightColor = weight; vVaryingTexCoords = vTexCoords; vVaryingNormals = normalize(vec3(vec4(vNormals, 0.0) * MV)); gl_Position = wpos * MVP; } The Pallete matrices are the matrices calculated using the above code (a rotation and translation matrix get created from the translation and quaternion). The invBindPose matrices are simply the inverted matrices created from the joints in the MD5mesh file. Update 1 I looked at GLM to compare the values I get with my own implementation. They turn out to be exactly the same. So now i'm checking if there's a problem with matrix creation... Update 2 Looked at GLM again to compare matrix creation using quaternions. Turns out that's not the problem either.

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• Why differs floating-point precision in C# when separated by parantheses and when separated by state

  - by Andreas Larsen

  I am aware of how floating point precision works in the regular cases, but I stumbled on an odd situation in my C# code. Why aren't result1 and result2 the exact same floating point value here? const float A; // Arbitrary value const float B; // Arbitrary value float result1 = (A*B)*dt; float result2 = (A*B); result2 *= dt; From this page I figured float arithmetic was left-associative and that this means values are evaluated and calculated in a left-to-right manner. The full source code involves XNA's Quaternions. I don't think it's relevant what my constants are and what the VectorHelper.AddPitchRollYaw() does. The test passes just fine if I calculate the delta pitch/roll/yaw angles in the same manner, but as the code is below it does not pass: X Expected: 0.275153548f But was: 0.275153786f [TestFixture] internal class QuaternionPrecisionTest { [Test] public void Test() { JoystickInput input; input.Pitch = 0.312312432f; input.Roll = 0.512312432f; input.Yaw = 0.912312432f; const float dt = 0.017001f; float pitchRate = input.Pitch * PhysicsConstants.MaxPitchRate; float rollRate = input.Roll * PhysicsConstants.MaxRollRate; float yawRate = input.Yaw * PhysicsConstants.MaxYawRate; Quaternion orient1 = Quaternion.Identity; Quaternion orient2 = Quaternion.Identity; for (int i = 0; i < 10000; i++) { float deltaPitch = (input.Pitch * PhysicsConstants.MaxPitchRate) * dt; float deltaRoll = (input.Roll * PhysicsConstants.MaxRollRate) * dt; float deltaYaw = (input.Yaw * PhysicsConstants.MaxYawRate) * dt; // Add deltas of pitch, roll and yaw to the rotation matrix orient1 = VectorHelper.AddPitchRollYaw( orient1, deltaPitch, deltaRoll, deltaYaw); deltaPitch = pitchRate * dt; deltaRoll = rollRate * dt; deltaYaw = yawRate * dt; orient2 = VectorHelper.AddPitchRollYaw( orient2, deltaPitch, deltaRoll, deltaYaw); } Assert.AreEqual(orient1.X, orient2.X, "X"); Assert.AreEqual(orient1.Y, orient2.Y, "Y"); Assert.AreEqual(orient1.Z, orient2.Z, "Z"); Assert.AreEqual(orient1.W, orient2.W, "W"); } } Granted, the error is small and only presents itself after a large number of iterations, but it has caused me some great headackes.

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• java quaternion 3D rotation implementation

  - by MRM

  I made a method to rotate a list of points using quaternions, but all i get back as output is the same list i gave to rotate on. Maybe i did not understood corectly the math for 3d rotations or my code is not implemented the right way, could you give me a hand? This is the method i use: public static ArrayList<Float> rotation3D(ArrayList<Float> points, double angle, int x, int y, int z) { ArrayList<Float> newpoints = points; for (int i=0;i<points.size();i+=3) { float x_old = points.get(i).floatValue(); float y_old = points.get(i+1).floatValue(); float z_old = points.get(i+2).floatValue(); double[] initial = {1,0,0,0}; double[] total = new double[4]; double[] local = new double[4]; //components for local quaternion //w local[0] = Math.cos(0.5 * angle); //x local[1] = x * Math.sin(0.5 * angle); //y local[2] = y * Math.sin(0.5 * angle); //z local[3] = z * Math.sin(0.5 * angle); //components for final quaternion Q1*Q2 //w = w1w2 - x1x2 - y1y2 - z1z2 total[0] = local[0] * initial[0] - local[1] * initial[1] - local[2] * initial[2] - local[3] * initial[3]; //x = w1x2 + x1w2 + y1z2 - z1y2 total[1] = local[0] * initial[1] + local[1] * initial[0] + local[2] * initial[3] - local[3] * initial[2]; //y = w1y2 - x1z2 + y1w2 + z1x2 total[2] = local[0] * initial[2] - local[1] * initial[3] + local[2] * initial[0] + local[3] * initial[1]; //z = w1z2 + x1y2 - y1x2 + z1w2 total[3] = local[0] * initial[3] + local[1] * initial[2] - local[2] * initial[1] + local[3] * initial[0]; //new x,y,z of the 3d point using rotation matrix made from the final quaternion float x_new = (float)((1 - 2 * total[2] * total[2] - 2 * total[3] * total[3]) * x_old + (2 * total[1] * total[2] - 2 * total[0] * total[3]) * y_old + (2 * total[1] * total[3] + 2 * total[0] * total[2]) * z_old); float y_new = (float) ((2 * total[1] * total[2] + 2 * total[0] * total[3]) * x_old + (1 - 2 * total[1] * total[1] - 2 * total[3] * total[3]) * y_old + (2 * total[2] * total[3] + 2 * total[0] * total[1]) * z_old); float z_new = (float) ((2 * total[1] * total[3] - 2 * total[0] * total[2]) * x_old + (2 * total[2] * total[3] - 2 * total[0] * total[1]) * y_old + (1 - 2 * total[1] * total[1] - 2 * total[2] * total[2]) * z_old); newpoints.set(i, x_new); newpoints.set(i+1, y_new); newpoints.set(i+2, z_new); } return newpoints; } For rotation3D(points, 50, 0, 1, 0) where points is: 0.0, 0.0, -9.0; 0.0, 0.0, -11.0; 20.0, 0.0, -11.0; 20.0, 0.0, -9.0; i get back the same list.

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