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The View Matrix

The View Matrix. Lecture 21 Fri, Oct 17, 2003. The View Matrix. The function gluLookAt() creates a matrix representing the transformation from world coordinates to eye coordinates. This is called the view matrix.

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The View Matrix

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  1. The View Matrix Lecture 21 Fri, Oct 17, 2003

  2. The View Matrix • The function gluLookAt() creates a matrix representing the transformation from world coordinates to eye coordinates. • This is called the view matrix. • The model matrix is the one that places the objects in position in world coordinates.

  3. The View Matrix • The current transformation is post-multiplied by the matrix created by gluLookAt(). • For this reason, gluLookAt() should be called before the model transformations, such as rotating or translating individual objects. • Thus, it will affect the entire scene.

  4. The Eye Coordinate System • The matrix created by the gluLookAt() function transforms world coordinates system into eye, or camera, coordinates. • Let the vectors u, v, n be the unit vectors of the eye coordinate system (corresponding to i, j, k in the world coordinate system).

  5. The Eye Coordinate System • Let E be the eye position, L the look point, and up the up vector. up E L

  6. The Eye Coordinate System • We will base our calculations on the facts that • i j = k • j  k = i • k  i = j • |i| = |j| = |k| = 1. • Therefore, we should end up with • u v = n • v  n = u • n  u = v • |u| = |v| = |n| = 1.

  7. The Eye Coordinate System • Form the normalized vector n = (E – L)/|E – L|. up E n L

  8. The Eye Coordinate System • The vector u must be perpendicular (and to the right) of n. • Define u to be the unit vector u = (upn)/|upn|. up E n u L

  9. The Eye Coordinate System • We cannot assume that up is perpendicular to n. • Therefore, let v be the unit vector v = (nu)/|nu| up v E n u L

  10. The View Matrix • The coordinate system of the camera is determined by u, v, n. • The view matrix V must transform u, v, n into i, j, k. • Vu = i • Vv = j • Vn = k

  11. The View Matrix • We know from an earlier discussion that this means that the view matrix will be of the form V = • For some values of a, b, and c.

  12. The View Matrix • To determine a, b, and c, use that fact that V also transforms E to the origin: VE = O. • Thus, • a = –(uxex + uyey + uzez) = –ue • b = –(vxex + vyey + vzez) = –ve • c = –(nxex + nyey + nzez) = –ne where e = E – O.

  13. The View Matrix • Therefore, the matrix created by gluLookAt() is V =

  14. The View Matrix • Verify that V transforms the points • E  (0, 0, 0) • E + u  (1, 0, 0) • E + v  (0, 1, 0) • E + n  (0, 0, 1)

  15. Example • LookMover.cpp • mesh.cpp • Remove the call to gluLookAt(). • Translate the cone 5 units in the negative z-direction. • Reinstate gluLookAt(). • Change the up vector.

  16. Example: Modelview Matrix • Let eye = (10, 5, 5), look = (0, 5, 0), up = (1, 1, 0). • Then • eye – look = (10, 0, 5). • n = (2, 0, 1)/5. • up  n = (1, -1, -2)/5. • u = (1, -1, -2)/6. • v = n  u = (1, 5, -2)/30.

  17. Example: Modelview Matrix • Also • e = eye – O = (10, 5, 5). • –eu = -5/6. • –ev = 25/30. • –en = 25/5.

  18. Example: Modelview Matrix • Therefore, the view matrix is

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