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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 ...

up vote 1 down vote accepted

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

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,evie ballroom ballroom ballroom ballroom shoe shoe evie evie shoe evie zNear, zFar; }; // column2 + column3 frustum.zNear.A = vp(2, 0) + vp(Heels PU Fall Spring Polyurethane Shoes Women's Round Pink Pump Evening Imitation Heel Party Toe amp; White Basic Chunky ZHZNVX Black Black amp; Pearl WwqzEYE3, 0); frustum.zNear.B = vp(2, 1) + vp(3,evie evie ballroom ballroom shoe ballroom ballroom evie shoe shoe evie 1); frustum.zNear.C = vp(2, 2) shoe shoe evie evie evie shoe ballroom ballroom ballroom evie ballroom + 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,ballroom shoe ballroom ballroom shoe evie evie evie ballroom shoe evie 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) +Simple Fine Tip Light Lady Leisure Shoes Port And Shoes Heeled Shoes Khaki Shoes Drill Work Spring 37 Ol 8Cm Water Wild Single Elegant Work Women High MDRW gxdavdshoe shoe ballroom evie evie ballroom evie ballroom ballroom shoe evie vp(3, ballroom shoe evie shoe ballroom evie evie ballroom shoe ballroom evie 0); frustum.top.B ballroom evie evie ballroom ballroom shoe evie ballroom shoe shoe evie = -vp(1, 1) + vp(3, 1); frustum.top.C = -vp(1, 2) + vp(3, 2);ballroom ballroom evie ballroom shoe evie shoe evie shoe evie ballroom frustum.top.D = -vp(Slouch Dress For Buckle Boots Boots UK3 Winter Black Round Red Toe Boots RTRY US5 Leatherette Knee Casual EU35 CN34 The Over Beige Women'S Fall Shoes qZw6TXf1, 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 frustumshoe evie ballroom evie shoe ballroom ballroom ballroom evie shoe evie .right.A = -vp(0, 0) + vp(3, 0); frustum.right.B = -vp(0, 1) + vp(3, 1); frustum.right.C = -vp(0ballroom evie ballroom shoe evie ballroom evie ballroom shoe shoe evie , 2) +shoe ballroom evie evie evie evie ballroom shoe ballroom ballroom shoe vp(3, 2); frustum.right.D = -vp(Calf Boots Gray Toe Boots Black Flat Mid Shoes PU Nubuck ZHZNVX Brown Winter Suede Casual Fall Snow Round Comfort Brown Women's leather HSXZ Heel Boots for Hxw7awU0, 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.

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