3D Computer Graphics. Mathem. Intro with OpenGL by Buss

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That is, a vector is the difference of two points. Rather than adopting a confusing and nonstandard notation that clearly distinguishes between points and vectors, we will instead follow the more common, but ambiguous, convention of using the same notation for points as for vectors. In view of the distinction between points and vectors, it can be useful to form the sums and differences of two vectors, or of a point and a vector, or the difference of two points, but it is not generally useful to form the sum of two points.

Example: Let M = 11 02 . 3. To find the matrix representation of its inverse M −1 , it is enough to determine M −1 i and M −1 j. It is not hard to see that M −1 1 1 = 0 −1/2 Hint: Both facts follow from M Therefore, M −1 is equal to M −1 and 0 = 1/2 1 0 −1/2 1/2 0 1 0 0 = . 1 1/2 and M . Team LRN 1 0 = 1 1 . 3. An F shape transformed by a linear transformation. The example shows a rather intuitive way to find the inverse of a matrix, but it depends on being able to find preimages of i and j. One can also compute the inverse of a 2 × 2 matrix by the well-known formula −1 a b c d 1 d −b , det(M) −c a = where det(M ) = ad − bc is the determinant of M.

0); pglTranslatef(0, r+1); drawThreePoints(); // // // // // // // // // // // Select model view matrix M = Identity M = M · Rθ M = M · T ,0 Save M on a stack M = M · T 0,r +1 Draw the three points Restore M from the stack M = M · R180◦ M = M · T 0,r +1 Draw the three points The new function calls glPushMatrix and glPopMatrix to save and restore the current matrix M with a stack. Calls to these routines can be nested to save multiple copies of the ModelView matrix in a stack. This example shows how the OpenGL matrix manipulation routines can be used to handle hierarchical models.

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