Computer Graphics University of Palestine Dr. Sana’a Wafa Al-Sayegh ITGD3107 Prepared By Niddal abu swereh Mahmoud elqedra Supervised By Dr. Sana’a Wafa Al-Sayegh

ITGD3107 Computer Graphics Chapter 11 Three-Dimensional Geometric and Modeling Transformations

Three-Dimensional Geometric and Modeling Transformations Some Basics 3D Translations. 3D Scaling. 3D Rotation. 3D Reflections. Transformations.

Some Basics Basic geometric types. Scalars s Vectors v Points p Transformations Types of transformation: rotation, translation, scale, Reflections, shears. Matrix representation Order P=T(P)

3D Point We will consider points as column vectors. Thus, a typical point with coordinates (x, y, z) is represented as:

3D Translations. P is translated to P' by T: T = Called the matrix T =

3D Translations.

3D Translations. An object is translated in 3D dimensional by transforming each of the defining points of the objects.

3D Translations.

3D Scaling P is scaled to P' by S: Called the Scaling matrix S =

3D Scaling Scaling with respect to the coordinate origin

3D Scaling Scaling with respect to a selected fixed position (xf, yf, zf) Translate the fixed point to origin Scale the object relative to the coordinate origin Translate the fixed point back to its original position

3D Scaling

About an axis: equivalent to 180˚rotation about that axis 3D Reflections About an axis: equivalent to 180˚rotation about that axis

3D Reflections

3D Shearing Modify object shapes Useful for perspective projections: E.g. draw a cube (3D) on a screen (2D) Alter the values for x and y by an amount proportional to the distance from zref

3D Shearing

Shears

Rotation Positive rotation angles produce counterclockwise rotations about a coordinate axis mshe1990@hotmail.com &&

Rotation mshe1990@hotmail.com &&

Coordinate-Axes Rotations mshe1990@hotmail.com &&

Coordinate-Axes Rotations mshe1990@hotmail.com &&

Coordinate-Axes Rotations mshe1990@hotmail.com &&

Coordinate-Axes Rotations mshe1990@hotmail.com &&

General Three-Dimensional Rotations An object is to be rotated about an axis that is parallel to one of the coordinate axes Translate the object so that the rotation axis coincides with the parallel coordinate axis Perform the specified rotation about that axis Translate the object so that the rotation axis is moved back to its original position mshe1990@hotmail.com &&

General Three-Dimensional Rotations An object is to be rotated about an axis that is not parallel to one of the coordinate axes Translate the object so that the rotation axis passes through the coordinate origin. Rotate the object so that the axis of rotation coincide with one of the coordinate axes. Perform the specified rotation about that coordinate axis. Apply inverse rotations to bring the rotation axis back to its original orientation. Apply the inverse Translation to bring the rotation axis back to its original position. mshe1990@hotmail.com &&

Quiz Draw any shape, then moving translation matrix. Good Luck