In mathematics, homogeneous coordinates or projective coordinates. This ppt is about the use of coordinates in graphics

1. HOMOGENEOUS
CO-ORDINATES
Presented By:
Ahtesham Ullah Khan
CS-3rd yr
1604610013

2. HOMOGENEOUS COORDINATES
◈ In mathematics, homogeneous coordinates or projective
coordinates, introduced by August Ferdinand Möbius in his 1827.
◈ They have the advantage that the coordinates of points, including
points at infinity, can be represented using finite coordinates.
◈ Homogeneous coordinates have a range of applications,
including computer graphics and 3D computer vision, where they
allow affine transformations and, in general, projective
transformations to be easily represented by a matrix.
◈ If the homogeneous coordinates of a point are multiplied by a non-
zero scalar then the resulting coordinates represent the same point.

3. Why we need Homogeneous
Coordinates?
◈ One of the many purposes of using homogeneous
coordinates is to capture the concept of infinity.
◈ If we don’t use homogeneous coordinates, it would be
difficult to design certain classes of very useful curves
and surfaces. These curves and surfaces are very
crucial in developing algorithms in computer vision,
graphics, CAD, etc.
◈ We have seen that basic transformations can be
expressed in matrix form. But many graphic application
involve sequences of geometric transformations. Hence
we need a general form of matrix to represent
such transformations.

4. OPERATIONS:-
◈ Translation
◈ Shearing
◈ Rotation
◈ Reflection

5. TRANSLATION

6. X
Y
Z
TRANSLATION

7. ROTATION

8. COORDINATE AXIS
ROTATION

9. X-AXIS ROTATION

10. X-AXIS ROTATION
Y
Z
X

11. Y-AXIS ROTATION

12. Y-AXIS ROTATION
Y
Z
X

13. Z-AXIS ROTATION

14. Z-AXIS ROTATION
Y
Z
X

15. SCALING

16. SCALING

17. SCALING
Y
Z
X

18. REFLECTION

19. REFLECTION

20. REFLECTION

21. SHEARING

22. SHEARING

23. SHEARING

24. THANK
YOU