1. COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS 2003 1 3D Geometric Transformations Graphics Systems / Computer Graphics and Interfaces

2. COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS 2003 2 3D Geometric Transformations Extension of 2D methods now including the Z coordinate Transformations: Translation Scaling Rotation 3D coordinate system: Right-Hand Rule Axis rotation Positive direction of rotation x y z for for x y z Z x to y

3. COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS 2003 3 Translation Translation of a Point y x z x, y, z xTYTZTxTYTZT T = T x, T y, T z The translation of an object is carried out by applying the operation to each of its vertices. y x z

4. COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS 2003 4 Scaling Regarding the origin: y x z

5. COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS 2003 5 Scaling T(X F Y F Z F ).S(S x, S y, S z ).T(X- F,-Y F,-Z F ) = In relation to an arbitrary point: y x z x y x z y x z xFYFZFxFYFZF

6. COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS 2003 6 Rotation In 2D the axis of rotation is perpendicular to the XY plane 3D rotation axis may be –x, y or z –An axis arbitrarily placed in the space Around z-axis  constant z y x z +

7. COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS 2003 7 Rotation y x z + y x z + Around x-axis  x constant Around y-axis  constant y

8. COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS 2003 8 z Rotation Rotation about an axis placed arbitrarily in 3D space: 1.Applying the translation to place the axis of rotation passing through the origin of the coordinate system. 2.Rotating the object so that the axis of rotation coincides with one of the coordinate axes. 3.Apply the desired rotation about this axis. 4.Apply the inverse rotation of point 2. 5.Apply the inverse translation of 1. y x z y x z P2 P1 y x z P2 ' P1 ' y x z P2'' P1'' y x P2'' P1'' y x z P2 P1

9. COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS 2003 9 Exercises