By Dave Shreiner, Ed Angel, Vicki Shreiner

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**Extra info for An Interactive Introduction to OpenGL Programming**

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In particular, affine transformations will not alter the type of object. A transformed line (polygon, quadric) is still a line (polygon, quadric). Any composition of affine transformations is still affine. For example, a rotation followed by a translation followed by a projection preserves lines and polygons. 0 37 A 3D vertex is represented by a 4-tuple vector (homogeneous coordinate system). Why is a 4-tuple vector used for a 3D (x, y, z) vertex? To ensure that all matrix operations are multiplications.

The command-manipulation menu allows you to change the order of the glTranslatef() and glRotatef() commands. Later, we will see the effect of changing the order of modeling commands. 48 An Interactive Introduction to OpenGL Programming Connection: Viewing and Modeling Moving camera is equivalent to moving every object in the world towards a stationary camera Viewing transformations are equivalent to several modeling transformations gluLookAt() has its own command can make your own polar view or pilot view 49 Instead of gluLookAt(, one can use the following combinations of glTranslate() and glRotate() to achieve a viewing transformation.

For example, a rotation followed by a translation followed by a projection preserves lines and polygons. 0 37 A 3D vertex is represented by a 4-tuple vector (homogeneous coordinate system). Why is a 4-tuple vector used for a 3D (x, y, z) vertex? To ensure that all matrix operations are multiplications. 0, we can recover x, y and z by division by w. Generally, only perspective transformations change w and require this perspective division in the pipeline. 37 An Interactive Introduction to OpenGL Programming 3D Transformations A vertex is transformed by 4 x 4 matrices • all affine operations are matrix multiplications • all matrices are stored column-major in OpenGL • matrices are always post-multiplied v • product of matrix and vector is M v M= m0 m 1 m2 m3 m4 m8 m12 m5 m6 m9 m10 m13 m14 m7 m11 m15 38 Perspective projection and translation require 4th row and column, or operations would require addition, as well as multiplication.