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.
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Slide 1
UNIT - 5 3D transformation and viewing
Slide 2
3D Point We will consider points as column vectors. Thus, a
typical point with coordinates (x, y, z) is represented as:
Slide 3
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)
Slide 4
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
Slide 5
x = x + tx y = y + ty z = z + tz Translating a point with
translation with vector T = (tx,ty,tz).
Slide 6
6 3D Translations. An object is translated in 3D dimensional by
transforming each of the defining points of the objects.
Slide 7
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.
Slide 8
Rotation about x axis
Slide 9
Rotation about z axis Rotation is in the following form :
Slide 10
Slide 11
Slide 12
12 3D Transformations: Rotation One rotation for each world
coordinate axis
Slide 13
3D Scaling P is scaled to P' by S: Called the Scaling matrix S
=
Slide 14
14 3D Scaling Scaling with respect to the coordinate
origin
Slide 15
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
Slide 16
3D Scaling
Slide 17
3D Reflections About an axis: equivalent to 180rotation about
that axis
Slide 18
18 3D Reflections
Slide 19
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