The Modelview Matrix Lecture 8 Mon, Sep 10, 2007.

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Presentation transcript:

The Modelview Matrix Lecture 8 Mon, Sep 10, 2007

Geometric Transformations and Matrices Each geometric transformation can be represented by a matrix. If a point P is represented by a vector p and a transformation is represented by a matrix M, then the product Mp represents the transformed point. That is, the transformed point P is represented by the vector p = Mp.

Sequences of Transformations Suppose that we perform two transformations in sequence. If M1 represents the first transformation and M2 represents the second transformation, then M2M1 represents the sequence of transformations. The same principle applies to any number of transformations in sequence.

The Modelview Matrix OpenGL maintains a matrix called the Modelview matrix. It records the current transformation, as determined by calls to glTranslatef(), glRotatef(), glScalef(), etc.. It should be initialized to the identity matrix.

The Modelview Matrix OpenGL code to initialize the Modelview matrix. glMatrixMode(GL_MODELVIEW); glLoadIdentity();

The Modelview Matrix If M is the current matrix or current transformation and we wish to apply transformation M1, then OpenGL computes MM1 and makes that the new current matrix. The old M is gone. Note that M was multiplied on the right by M1.

The Modelview Matrix Suppose the current transformation is M and we wish to apply a sequence of transformations M1, M2, and M3 to an object. We need to create MM3M2M1 on the stack. In other words, we call the various transformation functions in reverse order.

Example Suppose we begin with the unit sphere centered at the origin and we wish to Scale it by a factor of 2 in the y-direction, Rotate it 45 about the positive x-axis, Translate it +2 units in the z-direction.

Example OpenGL code for this sequence of transformations. glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glTranslatef(0.0, 0.0, 2.0); glRotatef(45.0, 1.0, 0.0, 0.0); glScalef(1.0, 2.0, 1.0); glutSolidSphere(1.0, 40, 40);

Example OpenGL code for this sequence of transformations. glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glTranslatef(0.0, 0.0, 2.0); glRotatef(45.0, 1.0, 0.0, 0.0); glScalef(1.0, 2.0, 1.0); glutSolidSphere(1.0, 40, 40); Apply transformations to the object in this order

Example Read Run

Combined Transformations Suppose we want to rotate an object about the point (x, y, z) (not the origin). We perform A translation by (-x, -y, -z). The rotation (about the origin). The inverse translation by (x, y, z). The first translation may not be necessary if we draw the object in object coordinates.

Rotation about an Arbitrary Axis Read Run