04 – Geometric Transformations

1. Overview Geometric Primitives Points, Lines, Planes 2D Geometric Transformations Translation, Rotation, Scaling, Affine, Projective 3D Geometric Transformations Rotation About Arbitrary Axes 04 – Geometric Transformations

2. Before we can understand how digital images are formed we must review some geometry Geometric primitives 2D and 3D geometric transformations Then we will discuss how cameras create digital images of the world 3D to 2D projection Image sampling Overview

3. 2D points x=(x,y) in Euclidean coordinates x=(x,y,w) in homogeneous coordinates where x=x/w and y=y/w 3D points x=(x,y,z) in Euclidean coordinates x=(x,y,z,w) in homogeneous coordinates where x=x/w , y=y/w and z=z/w Geometric Primitives

4. 2D lines ax+by+c=0 is basic 2D line equation (a,b,c).(x,y,1)=0 using dot product notation (nx,ny,d) .(x,y,1)=0 where n=(nx,ny) is unit normal to line and d is distance from line to origin Geometric Primitives

5. 3D lines r = (1-l)p + lq is parametric line equation In this case p, q and r are 3D points When l = [0..1] we have 3D line segment Geometric Primitives

6. 3D planes ax+by+cz+d=0 is basic 3D plane equation (a,b,c,d).(x,y,z,1)=0 using dot product notation (nx,ny,nz,d) .(x,y,z,1)=0 where n=(nx,ny,nz) is unit normal and d is distance from plane to origin Geometric Primitives

7. The simplest geometric transformations that occur in the 2D plane are: translation, Euclidean, similarity, affine and projective 2D Geometric Transformations

8. 2D Translation: 2D Geometric Transformations

9. 2D Scaling: 2D Geometric Transformations

10. 2D Rotation: 2D Geometric Transformations

11. 2D Rotation around arbitrary point: Translate to origin, rotate, translate back 2D Geometric Transformations

12. 2D Scaling around arbitrary point: Translate to origin, scale, translate back 2D Geometric Transformations

13. 2D Euclidean (rotation + translation): Preserves distances between points 2D Geometric Transformations

14. 2D Similarity (scaling + rotation + translation): Preserves angles between lines 2D Geometric Transformations

15. 2D Affine: Parallel lines remain parallel 2D Geometric Transformations

16. 2D Projective: Straight lines remain straight 2D Geometric Transformations

17. 2D Transformation hierarchy 2D Geometric Transformations

18. 3D translation is very similar to 2D translation There are now 3 coordinates to translate (tx,ty,tz) We use 4x4 matrix to perform operation 3D scaling is very similar to 2D scaling We still only have one scale factor S We use 4x4 matrix to perform operation 3D Geometric Transformations

19. There are three ways to perform 3D rotations 1) Rotate about the X,Y,Z axes 2) Rotate about an arbitrary axis 3) Use unit quaternions to perform rotation Option 1 is easiest to understand Options 2 and 3 provide smoother motions 3D Geometric Transformations

20. 3D Rotation about Z axis: 3D Geometric Transformations

21. 3D Rotation about Y axis: 3D Geometric Transformations

22. 3D Rotation about X axis: 3D Geometric Transformations

23. 3D Rotation about arbitrary axis in X-Y plane Find angle a between rotation axis and X axis Rotate around Z axis by -a degrees Rotate around X axis by desired angle q Rotate around Z axis by a degrees Same approach works for rotations about an arbitrary axis in Y-Z plane or Z-X plane 3D Geometric Transformations

24. 3D Rotation about an arbitrary axis Project arbitrary axis onto Y-Z plane Find angle b between projected axis and Y axis Rotate around X axis by -b degrees Now rotate about axis in X-Y plane Rotate around X axis by b degrees Same approach can be used to project the arbitrary axis onto Y-Z plane or Z-X plane 3D Geometric Transformations

25. 3D Transformation hierarchy 3D Geometric Transformations