UNIT - 5 3D transformation and viewing

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

3 Representation of 3D Transformations Z axis represents depth Right Handed System When looking “down” at the origin, positive rotation is CCW Left Handed System When looking “down”, positive rotation is in CW More natural interpretation for displays, big z means “far” (into screen)

Translation Objects are usually defined relative to their own coordinate system. We can translate points in space to new positions by adding offsets to their coordinates, as shown in the following vector equation. P’ = T. P

x’ = x + tx y’ = y + ty z’ = z + tz Translating a point with translation with vector T = (tx,ty,tz).

6 3D Translations.  An object is translated in 3D dimensional by transforming each of the defining points of the objects.

Rotation Rotations in three-dimensions are considerably more complicated than two-dimensional rotations. In general, rotations are specified by a rotation axis and an angle. In two-dimensions there is only one choice of a rotation axis that leaves points in the plane.

Rotation about x axis

Rotation about z axis Rotation is in the following form :

12 3D Transformations: Rotation One rotation for each world coordinate axis

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

14 3D Scaling Scaling with respect to the coordinate origin

3D Scaling Scaling with respect to a selected fixed position (x f, y f, z f ) 1. Translate the fixed point to origin 2. Scale the object relative to the coordinate origin 3. Translate the fixed point back to its original position

3D Scaling

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

18 3D Reflections

19 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 z ref

20 Shears