OpenGL Math - Projecting Screen space to World space coords

Question

Time for a little bit of math for the end of the day..

I need to project 4 points of the window size:

<0,0> <1024,768>

Into a world space coordinates so it will form a quadrilateral shape that will later be used for terrain culling - without GluUnproject

For test only, I use mouse coordinates - and try to project them onto the world coords

Laurilaurianne

isn't the window size irrelevant? the shape/aspect sure, but what you are after is the specific rectangular cone shape the camera view forms, right? – Gono

i found something cool collagefactory.blogspot.com/2010/03/… – Calisaya

Breton, the thing i am looking for is to construct a 'bounding box' from actual camera view that i will use later on to find which terrain cells are 'inside' so i can render them. So i need to know where is the top left, top right, bottom left and bottom right located in 3d world space so i can later on find the terrain cells that are within this range. – Laurilaurianne

Why do you use camera position in model matrix? This should be model position! If your model is at zero, then you dont need any matrix. For the inverse, it's easy to test. Multiply the result by the source, and it works if you obtain an identity matrix – Landbert

And you won't get a bounding box from camera corner point, but a frustum – Landbert

Answer 1

RESOLVED

Here's how to do it exactly, step by step.

1. Obtain your mouse coordinates within the client area
2. Get your Projection matrix and View matrix if no Model matrix required.
3. Multiply Projection * View
4. Inverse the results of multiplication

5. Construct a vector4 consisting of

   x = mouseposition.x within a range of window x

   • transform to values between -1 and 1

   y = mouseposition.y within a range of window y

   • transform to values between -1 and 1
   • remember to invert mouseposition.y if needed

   z = the depth value ( this can be obtained with glReadPixel)

   • you can manually go from -1 to 1 ( zNear, zFar )

   w = 1.0

6. Multiply the vector by inversed matrix created before

7. Divide result vector by it's w component after matrix multiplication ( perspective division )

       POINT mousePos;
       GetCursorPos(&mousePos);
       ScreenToClient( this->GetWindowHWND(), &mousePos );         
   
       CMatrix4x4 matProjection = m_pCamera->getViewMatrix() *  m_pCamera->getProjectionMatrix() ;
   
       CMatrix4x4 matInverse =  matProjection.inverse();
   
   
       float in[4];
       float winZ = 1.0;
   
   
       in[0]=(2.0f*((float)(mousePos.x-0)/(this->GetResolution().x-0)))-1.0f,
       in[1]=1.0f-(2.0f*((float)(mousePos.y-0)/(this->GetResolution().y-0)));
       in[2]=2.0* winZ -1.0;
       in[3]=1.0;          
   
       CVector4 vIn = CVector4(in[0],in[1],in[2],in[3]);
       pos = vIn * matInverse;
   
       pos.w = 1.0 / pos.w;
   
       pos.x *= pos.w;
       pos.y *= pos.w;
       pos.z *= pos.w;
   
       sprintf(strTitle,"%f %f %f / %f,%f,%f ",m_pCamera->m_vPosition.x,m_pCamera->m_vPosition.y,m_pCamera->m_vPosition.z,pos.x,pos.y,pos.z);
   
       SetWindowText(this->GetWindowHWND(),strTitle);
   

Laurilaurianne

You can just do pos.x /= pos.w – Umeh

why you calculate the projection by viewMatrix * projectMatrix instead of projectMatrix * viewMatrix? – Delapaz

Do not divide by w. – Scolecite

Answer 2

I had to make some adjustments to the answers provided here. But here's the code I ended up with (Note I'm using GLM, that could affect multiplication order). nearResult is the projected point on the near plane and farResult is the projected point on the far plane. I want to perform a ray cast to see what my mouse is hovering over so I convert them to a direction vector which will then originate from my camera's position.

vec3 getRayFromScreenSpace(const vec2 & pos)
{
    mat4 invMat= inverse(m_glData.getPerspective()*m_glData.getView());
    vec4 near = vec4((pos.x - Constants::m_halfScreenWidth) / Constants::m_halfScreenWidth, -1*(pos.y - Constants::m_halfScreenHeight) / Constants::m_halfScreenHeight, -1, 1.0);
    vec4 far = vec4((pos.x - Constants::m_halfScreenWidth) / Constants::m_halfScreenWidth, -1*(pos.y - Constants::m_halfScreenHeight) / Constants::m_halfScreenHeight, 1, 1.0);
    vec4 nearResult = invMat*near;
    vec4 farResult = invMat*far;
    nearResult /= nearResult.w;
    farResult /= farResult.w;
    vec3 dir = vec3(farResult - nearResult );
    return normalize(dir);
}

Amaris

This answer is long, but I was able to understand the math instead of just copy and paste, thanks – Syck

Answer 3

Multiply all your matrices. Then invert the result. Point after projection are always in the -1,1. So the four corner screen points are -1,-1; -1,1; 1,-1;1,1. But you still need to choose th z value. If you are in OpenGL, z is between -1 and 1. For directx, the range is 0 to 1. Finally take your points and transform them with the matrix

Landbert

This is what i do actually, I do a MVP matrix multiplying projection*(View * model) Then i inverse that, have a look above ( updated first post ) - maybe there's something i'm missing within the Z value..? – Laurilaurianne

@Laurilaurianne At the moment your z is -3, I'm sure this isn't intended, as z has to be in [-1,1] (maybe you just want -1?). The rest seems reasonable, but don't forget to invert source.y if your window coordinate system has y going from top to bottom. And I hope VectorMatrixMultiply implicitly assumes a 1 for the w component, and not a 0. – Lemoine

Christian, thanks i am now patching what you have pointed out, and also i have updated the main post with CMatrix4x4 operator* method, so you can have a look if it's ok - as this class is not mine, and i don't know if i can trust it in this situation. – Laurilaurianne

Your * miss the translation part. – Landbert

crazyjul, fixed by implementing new mathlib. – Laurilaurianne

Answer 4

If you have access to the glu libraries, use gluUnProject(winX, winY, winZ, model, projection, viewport, &objX, &objY, &objZ);

winX and winY will be the corners of your screen in pixels. winZ is a number in [0,1] which will specify where between zNear and zFar (clipping planes) the points should fall. objX-Z will hold the results. The middle variables are the relevant matrices. They can be queried if needed.

Lantana

Yeah sorry for that, i haven't mentioned i don't want to use gluunproject - i need to operate on matrices only. – Laurilaurianne

@Laurilaurianne But the linked documentation explains the implementation of gluUnProject, and as it's extremely easy, you can just write it yourself. See crazyjul's answer. – Lemoine