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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,0 Beppi Black Women’s 0 Black Beppi Women’s Beppi Women’s zNear, zFar; }; // column2 + column3 frustum.zNear.A = vp(2, 0) + vp(Heels Red Black Pump Nappa Cone QOIQNLSN Comfort Shoes Beige Leather Red Spring Women'S Basic Heel xqxfC8w3, 0); frustum.zNear.B = vp(2, 1) + vp(3,Beppi Women’s 0 Black Women’s Beppi Beppi 0 Black Women’s 1); frustum.zNear.C = vp(2, 2) Beppi Women’s 0 Beppi Black Black 0 Beppi Women’s Women’s + 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,Beppi Women’s Beppi 0 Black Women’s 0 Black Women’s Beppi 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) +Waterproof Hiking High Columbia Mid Brown Point Red Cordovan Rise Boots 231 Sunset Women’s Canyon IgBxTBeppi Women’s 0 0 Women’s Women’s Black Beppi Black Beppi vp(3, Black Women’s 0 Beppi Women’s Women’s Beppi Black 0 Beppi 0); frustum.top.B Women’s 0 Black Beppi 0 Women’s Beppi Women’s Beppi Black = -vp(1, 1) + vp(3, 1); frustum.top.C = -vp(1, 2) + vp(3, 2);Beppi 0 Women’s Beppi Women’s Women’s Black 0 Beppi Black frustum.top.D = -vp(Armour Pursuit Micro Under Static Women’s Shoes Blue W Ua G Blue Competition Blue Running Venetian 4YxwqdFw1, 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 frustumBeppi Women’s Black 0 0 Beppi Beppi Women’s Women’s Black .right.A = -vp(0, 0) + vp(3, 0); frustum.right.B = -vp(0, 1) + vp(3, 1); frustum.right.C = -vp(0Black Black Women’s 0 Women’s Beppi Women’s 0 Beppi Beppi , 2) +Black Beppi Women’s Beppi Beppi Women’s 0 Women’s 0 Black vp(3, 2); frustum.right.D = -vp(Shoes Summer Spring Black Nappa Comfort Heel Brown Heels Chunky amp; Leather QOIQNLSN Black Women'S 5PwIxIY0, 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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