Ministry of Higher Education and Scientific Research
University of Technology
Computer Engineering Department
Geometric Transformation
3D Transformations
Computer Graphics
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2019 - 2020
Summary
The geometric transformations play a vital role in generating images of three Dimensional
objects with the help of these transformations.
The location of objects relative to others can be easily expressed.
Sometimes viewpoint changes rapidly, or sometimes objects move in relation to
each other.
For this number of transformation can be carried out repeatedly.
Introduction
Translation it is the movement of an object from one position to another position.
Translation is done using translation vectors.
There are three vectors in 3D instead of two.
These vectors are in x, y, and z directions. Translation in the x-direction is represented
using Tx.
The translation is y-direction is represented using Ty.
The translation in the z- direction is represented using Tz.
If P is a point having co-ordinates in three directions (x, y, z) is translated, then after
translation its coordinates will be (x1 y1 z1) after translation. Tx Ty Tz are translation
vectors in x, y, and z directions respectively.
x1=x+ Tx , y1=y+Ty , z1=z+ Tz
Three-dimensional transformations are performed by transforming each vertex of the
object.
If an object has five corners, then the translation will be accomplished by translating
all five points to new locations.
Following figure 1 shows the translation of point figure 2 shows the translation of the cube.
figure 1 figure 2
Scaling
Scaling is used to change the size of an object. The size can be increased or decreased.
The scaling three factors are required Sx Sy and Sz.
Sx=Scaling factor in x- direction , Sy=Scaling factor in y-direction , Sz=Scaling factor in z-direction
Following are steps performed when scaling of objects with fixed point (a, b, c). It can be
represented as below:
1- Translate fixed point to the origin 2- Scale the object relative to the origin
3- Translate object back to its original position.
Rotation
It is moving of an object about an angle, movement can be anticlockwise or clockwise.
3D rotation is complex as compared to the 2D rotation.
For 2D we describe the angle of rotation, but for a 3D angle of rotation and axis of rotation
are required. The axis can be either x or y or z.
rotation of the object about the Y axis rotation of the object about the Z axis
Rotation about Arbitrary Axis :
When the object is rotated about an axis that is not parallel to any one of co-ordinate
axis, i.e., x, y, z. Then additional transformations are required. First of all, alignment
is needed, and then the object is being back to the original position.
Following steps are required:
1-Translate the object to the origin.
2-Rotate object so that axis of object coincide with any of coordinate axis.
3-Perform rotation about co-ordinate axis with whom coinciding is done.
4-Apply inverse rotation to bring rotation back to the original position.
5- Apply inverse translation to bring rotation axis to the original position.
Step3: Rotate P" to z axis so that it
Step1: Initial position of P' and P" Step2: Translate object P' to origin aligns along the z-axis
Step4: Rotate about around z- axis Step5: Rotate axis to the original position Step6: Translate axis to the original position.
Reflection
It is also called a mirror image of an object. For this reflection axis and reflection of
plane is selected.
Three-dimensional reflections are similar to two dimensions. Reflection is 180° about
the given axis.
For reflection, plane is selected (xy,xz or yz).
Following matrices show reflection respect to all these three planes.
Reflection relative to XY plane :
Shearing
It is change in the shape of the object. It is also called as deformation.
Change can be in the x -direction or y -direction or both directions in case of 2D.
If shear occurs in both directions, the object will be distorted. But in 3D shear can occur
in three directions.
Conclusion
The reasoning used in the plane can be transferred to space. Again we use the fact that
we can make compound transformation matrices that represent compound geometric
transformations.
In the plane the situation is well arranged and it is easy to follow the operations graphically.
In space this quickly gets complicated, especially when we rotate.
A quite demanding exercise, which often is repeated in graphical literature, is rotating
around an arbitrary axis.
We will not go through the reasoning for this here. It is seldom we need to construct so
complicated transformations in our head when we write code.
Usually we get away with simpler solutions if we make some rational choices in the
description of the objects we want to represent.
References
Geometric Transformations for 3D Modeling 2nd Edition
by Michael Mortenson
Euclidean and Affine Transformations: Geometric Transformations Paperback
– January 1, 1965 by P. S. Modenov , Henry Booker , D. Allan Bromley ,
Nicholas DeClaris , A. S. Parkhomenko
Geometric Transformations II (NEW MATHEMATICAL LIBRARY)
by I. M. Yaglom, Allen Shields
Episodes in Nineteenth and Twentieth Century Euclidean Geometry
(Anneli Lax New Mathematical Library)
by Ross Honsberger
Geometric Inequalities (New Mathematical Library) 1st Edition
by Nicholas D. Kazarinoff
Geometric Transformations I (Number 8) First Edition
by I. M. Yaglom
Geometric Transformations, Vol. 2: Projective Transformations Paperback
– January 1, 1965
by P.S. Modenov , A.S. Parkhomenko
Rotation Transforms for Computer Graphics Paperback – January 11, 2011
by John Vince
Geometry for Computer Graphics: Formulae, Examples and Proofs Hardcover
– January 5, 2005 by John Vince
