CS32310

1. CS32310 Affine Transformations

2. Motivation • For a large class of transformations, • straight lines map to straight lines (and do so in proportion), • and planes map to planes. • We investigate this property. • The implication is that the construction of lines and planes can be carried out by transforming only a few control points, • complete the construction in the transformed space using the transformed control points.

3. Linear transformations • The above properties are consequence of the transformations having linear properties. • Linearity: superposition, no cross-talk • Matrices are part of linear algebra, • for a 3x3 matrix A and column matrices r1, r2

4. Linear transformations • Unfortunately, linearity does not hold when a displacement is involved:

5. (Pseudo) Linear transformations return • then • and

6. Linear transformations • In order to retain the benefits of linearity, even when displacements are present, we restrict the discussion to certain combinations of points to be transformed.

7. Definition • Given npoints P0 , P1 , … Pn-1 via their coordinates, point Q is said to be affinely connected to P0 , P1 , … Pn-1if there exist weights wisuch that

8. Affine transformations • A transformation T is affine if

9. Affine transformations • It turns out that linear transformations with displacements are affine transformations:

10. Affine transformations • It turns out that linear transformations with displacements are affine transformations:

11. Affine transformations • It turns out that linear transformations with displacements affine transformations:

12. Affine point properties • These concepts will be useful in establishing transformation properties for lines and planes.

13. Case n =1 • In this case, point Q is affinely connected to P0 , P1when

14. Case n =1 • Geometric interpretation • The unit sum implies w0=1-w1 , so

15. Case n =1 • Rewrite this as • Interpret this as position vector plus a multiple of vector , leading to points on the join of and .

16. Case n =1

17. Theorem • Let be affinely connected to • i.e.

18. Theorem • Then all points on the join of are also affinely connected to • i.e. for any we can find weights wisuch that

19. Theorem • The proof is straight forward (omitted here). • Example • The weights are non-negative, their sum is 1, as can also be seen from

20. Definition • The convex hull of a set of points is their set of affinely connected points.

21. Affine transformations • An affine transformation preserves affine connectedness • Let Qbe affinely connected to P0 , P1 : then an affine transformation T of Q will give

22. Straight lines -> straight lines • i.e. lies on the straight line join of and . We conclude that under the mapping T, straight lines transform to straight lines. Moreover, since the weights w0 ,w1 are preserved under T, any movement of Q from P0 to P1 is associated with the same proportionate movement of from to .

23. Parallel lines -> parallel lines • Let us displace points P0 , P1 by the same vectors, • the join points between the displaced points are given by This represents the sameoffsetsfor all join points Q between P0 , P1, i.e. we obtain a line segment parallel to the join of P0toP1 .

24. Parallel lines -> parallel lines • Under affine transformation T, (see slide 5) • The points have a constant offset from the points that lie on a straight line. Hence they lie on a line parallel to the line formed by

25. Planes -> planes • The points Q in the triangular domain (convex hull) formed by are given by • Under transformation T, these points become

26. Planes -> planes • Hence the points T(Q) lie in the convex hull of control points i.e points T(Q) also lie in a plane. • We conclude that planes transform into planes