Finding the normals of the planes of a View Frustum

Question

Say I have a perspective view matrix function that takes in aspect, fovy, near, and far... Transforming the view into a frustum. Typical OpenGL stuff, right. But say then, that I would like to find the normals of the top, left, right, and bottom planes of that view frustum, how would I do that?

Edit: I forgot, the camera has a vector position, and a vector direction ...

Answer 1

It becomes much clearer if you draw it from a top-down perspective:

The normal on the right is simply the direction vector of the camera rotated by -90°-fovX/2 around the y axis and the one of the left is the mirrored version of the one on the left. Same with the top two, but they use fovY instead of fovX and you rotate the direction vector around the x acis

Answer 2

Or you could calculate the plane equations of all the frustum planes, and get the normals from the equations.

A plane equation has the form:

Ax + By + Cz + D = 0

(A, B, C) represents the plane normal.

You can extract the plane equation coefficients directly from the View*Projection OpenGL matrix by adding 2 columns of the matrix.

This method is described here: http://www.cs.otago.ac.nz/postgrads/alexis/planeExtraction.pdf

Letting vp = View*Projection;

Here is some code I use;

struct Plane
{
    float A, B, C, D;
};

struct Frustum
{
    Plane top, bottom, right, left, zNear, zFar;
};

// column2 + column3
frustum.zNear.A = vp(2, 0) + vp(3, 0);
frustum.zNear.B = vp(2, 1) + vp(3, 1);
frustum.zNear.C = vp(2, 2) + vp(3, 2);
frustum.zNear.D = vp(2, 3) + vp(3, 3);

// column3 - column2
frustum.zFar.A = -vp(2, 0) + vp(3, 0);
frustum.zFar.B = -vp(2, 1) + vp(3, 1);
frustum.zFar.C = -vp(2, 2) + vp(3, 2);
frustum.zFar.D = -vp(2, 3) + vp(3, 3);

// column1 + column3
frustum.bottom.A = vp(1, 0) + vp(3, 0);
frustum.bottom.B = vp(1, 1) + vp(3, 1);
frustum.bottom.C = vp(1, 2) + vp(3, 2);
frustum.bottom.D = vp(1, 3) + vp(3, 3);

// column3 - column1 
frustum.top.A = -vp(1, 0) + vp(3, 0);
frustum.top.B = -vp(1, 1) + vp(3, 1);
frustum.top.C = -vp(1, 2) + vp(3, 2);
frustum.top.D = -vp(1, 3) + vp(3, 3);

// column0 + column3
frustum.left.A = vp(0, 0) + vp(3, 0);
frustum.left.B = vp(0, 1) + vp(3, 1);
frustum.left.C = vp(0, 2) + vp(3, 2);
frustum.left.D = vp(0, 3) + vp(3, 3);

// column3 - column0
frustum.right.A = -vp(0, 0) + vp(3, 0);
frustum.right.B = -vp(0, 1) + vp(3, 1);
frustum.right.C = -vp(0, 2) + vp(3, 2);
frustum.right.D = -vp(0, 3) + vp(3, 3);

Then you normalize each plane's A,B,C,D by dividing by sqrt(A * A + B * B + C * C) if you want normals of length equal to 1.