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The rotation angle is θ (measured in degrees), and the direction of the rotation is determined by the right-hand rule. The vector u must not equal 0. 7 ,  (1 − c)u 1 u 3 − su 2 (1 − c)u 2 u 3 + su 1 (1 − c)u 23 + c 0 0 0 0 1 Team LRN 38 Transformations and Viewing where c = cos θ and s = sin θ . OpenGL does not require that u be passed in as a unit vector: OpenGL will automatically compute the normalization of u in order to compute the rotation matrix. 3. glScalef(float α1 , float α2 , float α3 ).

The code fragment draws wireframe spheres with commands glutWireSphere( radius, slices, stacks ); The value of radius is the radius of the sphere. The integer values slices and stacks control the number of “wedges” and horizontal “stacks” used for the polygonal model of the sphere. The sphere is modeled with the “up” direction along the z-axis, and thus “horizontal” means parallel to the x y-plane. Team LRN 40 Transformations and Viewing The glColor3f(red, green, blue) commands are used to set the current drawing color.

3 Appendix A contains a review of elementary facts from linear algebra, including a discussion of dot products and cross products. 7. A generalized rotation Rθu . The center of rotation is u = 0, 3 . The angle is θ = 45◦ . 3 Every rigid, orientation-preserving, affine transformation can be (uniquely) expressed as the composition of a translation and a rotation. Definition A generalized rotation is a transformation that holds a center point u fixed and rotates all other points around u through a fixed angle θ .

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