1 By Dr. HANY ELSALAMONY

 We have seen how to create models in the 3D world. We discussed transforms in lecture 3, and we have used some transformations to view our 3D models effectively.  Let us look into 3D transformations in more detail. 2 By Dr. HANY ELSALAMONY

 Just as 2D transformations can be represented by 3 x 3 matrices using homogenous coordinates.  3D transformations can be represented by 4 x 4 matrices.  3D homogenous coordinates are represented as a quadruplet (x, y, z, W).  We discuss the representation of the three most commonly used transformations: translation, scaling, and rotation. 3 By Dr. HANY ELSALAMONY

 The translation matrix T(Tx, Ty, Tz) is defined as:  so any vertex point: 4 By Dr. HANY ELSALAMONY

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 Scaling is similarly extended to be defined as: By Dr. HANY ELSALAMONY 6

 Recall from Lecture 3 that scaling occurs about a point. This point is called the pivot point. The scaling matrix defined above scales about the origin.  You can define models to have different pivot points by setting the transformation stack appropriately. By Dr. HANY ELSALAMONY 7

 Rotation matrices are defined uniquely based on the axis of rotation and the pivot point.  The axis of rotation is a normalized vector defining the axis in 3D space along which the rotation will occur. By Dr. HANY ELSALAMONY 8

 (A 3D vector will simply be the triplet (x, y, z) defining the direction of the axis). The 2D rotation we saw earlier is a 3D rotation about the z-axis (the vector (0,0,1)), and the world origin (0,0,0) and is defined as: By Dr. HANY ELSALAMONY 9

 Similarly the rotation matrix about the x- axis is defined to be:  and about the y-axis as: By Dr. HANY ELSALAMONY 10

 Just as with 2D transforms, 3D transforms can be composed together to provide desired effects.  The order in which the transforms are applied is important.  Recall from our previous section that the OpenGL commands for matrix operations By Dr. HANY ELSALAMONY 11

 sets the current matrix to be the 4 x 4 identity matrix.  sets the matrix passed in to be the current transformation matrix.  multiplies the current matrix by the matrix passed in as the argument. By Dr. HANY ELSALAMONY 12

 Additionally, OpenGL provides three utility functions for the most commonly used transformations.  multiplies the current transformation matrix by the matrix to move an object by (Tx, Ty, Tz) units By Dr. HANY ELSALAMONY 13

 multiplies the current transformation matrix by a matrix that rotates an object in a counterclockwise direction by theta degrees about the axis defined by the vector (x, y, z).  multiplies the current transformation matrix by a matrix that scales an object by the factors, (Sx, Sy, Sz). By Dr. HANY ELSALAMONY 14

 All three commands are equivalent to producing an appropriate translation, rotation, or scaling matrix and then calling glMultMatrixf0 with that matrix as the argument.  For example, Let us apply some transformations to the famous pyramid that we created earlier. By Dr. HANY ELSALAMONY 15

 We rotate the pyramid about the y-axis and use double buffering to display the rotation as an animation. By Dr. HANY ELSALAMONY 16

 In lecture 3, we saw how to map a two- dimensional world onto the two dimensional screen.  The process was a simple mapping of 2D world coordinates onto the 2D viewport of the window.  Model positions are first clipped against the 2D world coordinates and then mapped into the viewport of the window for display. By Dr. HANY ELSALAMONY 17