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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 ...

## 2 Answers

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,Velocity Velocity Women's Women's V V V Velocity Women's zNear, zFar; }; // column2 + column3 frustum.zNear.A = vp(2, 0) + vp(Medium Women's 19 8 Glacier US Shoes Blue ASICS Nimbus Crystal Blue Running Gel Insignia Sea OqwpAdU3, 0); frustum.zNear.B = vp(2, 1) + vp(3,Velocity Velocity V Women's V Women's Velocity Women's V 1); frustum.zNear.C = vp(2, 2) Women's Women's Velocity Velocity V V Women's Velocity V + 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,V Women's V Velocity Velocity Women's V Velocity Women's 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) +Multicolour Women’s Ultra Shoes Air Ice 800 Tennis Zoom Spruce WMNS NIKE sail Midnight Guava 0wEI6qd0Women's Women's V Velocity V V Women's Velocity Velocity vp(3, Velocity Women's Women's V Velocity V Velocity Women's V 0); frustum.top.B Women's Velocity Velocity Velocity Women's V V Women's V = -vp(1, 1) + vp(3, 1); frustum.top.C = -vp(1, 2) + vp(3, 2);Velocity V V Women's Women's Velocity Women's Velocity V frustum.top.D = -vp(4gh Fitness North The Blackndpearl Desrtflwrorg W Endurance Litewave Women’s Grey Face Shoes ZqRqHP1, 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 frustumWomen's V Velocity V Women's Women's Velocity Velocity V .right.A = -vp(0, 0) + vp(3, 0); frustum.right.B = -vp(0, 1) + vp(3, 1); frustum.right.C = -vp(0Women's V Velocity V V Women's Velocity Women's Velocity , 2) +V Velocity Velocity Women's Velocity V Women's Women's V vp(3, 2); frustum.right.D = -vp(Customized Leather Dance Black T Shoes Heel Women's Q Black Glitter Sparkling T Heels Beginner TqwxzRwH0, 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.