#2: Geometry & Homogeneous Coordinates

1. #2: Geometry & Homogeneous Coordinates CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006

2. Outline for Today More math… • Finish linear algebra: Matrix composition • Points, Vectors, and Coordinate Frames • Homogeneous Coordinates

3. Matrix Multiplication • Each entry is dot product of row of M with column of N

4. Matrix Multiplication

5. Multiple Transformations • If we have a vector v, and an x-axis rotation matrix Rx, we can generate a rotated vector v′: • If we wanted to then rotate that vector around the y-axis, we could multiply by another matrix:

6. Multiple Transformations • We can extend this to the concept of applying any sequence of transformations: • Because matrix algebra obeys the associative law, we can regroup this as: • This allows us to compose them into a single matrix:

7. Order matters! • Matrix multiplication does NOT commute: • (unless one or the other is a uniform scale) • Try this:rotate 90 degrees about x then 90 degrees about z, versusrotate 90 degrees about z then 90 degrees about x. • Matrix composition works right-to-left. • Compose:Then apply it to a vector:It first applies C to v, then applies B to the result, then applies A to the result of that.

8. Quick Matrix algebra summary

9. What good is this? • Composition of transformations, by matrix multiplication, is a basic technique • Used all the time • You’ll probably use it for Project 1 • All linear operations on vectors can be expressed as composition of rotation and scale (even “shear”) • But there’s a limit to what we can do only having linear operations on vectors….

10. Outline for Today • Finish linear algebra: Matrix composition • Points, Vectors and Coordinate Frames • Homogeneous Coordinates

11. Geometric objects • Interesting Objects • Points • Vectors • Transformations • Coordinate Frames • Also: Lines, Rays, Planes, Normals, …

12. Points and vectors • You know linear algebra, vector spaces • Why am I talking about this? • Emphasize differences: • between a point and a vector • between a point or vector and the representation of a point or vector

13. Points and vectors • in R3, can represent point as 3 numbers • in R3, can represent vector as 3 numbers • Easy to think they’re the same thing… • …but they’re not! • different operations, different behaviors • many program libraries make this mistake • easy to have bugs in programs

14. In 1D, consider time • point in time: a class meets at 2PM • duration of time: a class lasts 2 hours • operations: • class at 2PM + class at 3PM≠ class at 5PM !! • 2 hour class + 3 hour class = 5 hours of classes • class ends at 5PM – starts at 2PM = 3 hour class • class starts at 2PM + lasts 3 hours = ends at 5PM • 2 classes at 3PM≠ one class at 6PM !! • 2 classes last 3 hours = 6 hours of classes • Class from 2PM to 10PM, half done at

15. “Coordinate Systems” for time Knowing just the hour number doesn’t tell you everything… • AM vs. PM (or use 8h00 vs 20h00) • Time zones: same point, many representations • 10 (Paris) == 9 (London) • to remove ambiguity, often use GMT • If always staying in local time zone, not important • if scheduling globally, must be careful. • convert from one time zone to another. • (Also, hours only good within one day • need to specify date & time • UNIX time: seconds since 01/01/1970, 00h00 GMT) • Notice: time durations are unaffected by all this!

16. Geometry, analogously • Point describes a location in space • can subtract points to get a vector • can take weighted average of points to get a point • Vector describes a displacement in space • has a magnitude and direction • can add, subtract, scale vectors • can add vector to a point to get a point • To represent them as three numbers, you must specify which coordinate system

17. Vector and point arithmetic • C++ classes can support these operations

18. y O x z Coordinate Frames • Origin point, and 3 orthonormal vectors for x,y,z axes (right-handed) • In CG, often work with many different frames simultaneously c b a h P g f Q

19. Coordinates • If you have coordinate triples such as: • Then, with frame such as you can construct a point or vector: • Same coordinates, different frame  different point or vector • Coordinates have no real meaning without a frame • CG programs often have lots of frames--you have to keep track! • (It’s possible to write C++ classes that keep track of frames. But it’s hard for them to be time- and memory-efficient, so it’s rarely done in practice.) • Typically have “World Coordinates” as implicit base frame • Notice: vectors don’t depend on the origin of the frame, only the axes

20. Coordinates of a Frame • Suppose you have a frame • In world coordinates, might have • But in itself always have coordinates:

21. O Coordinate equivalences • Given a frame: • Can have a point with some coords: • Can have a vector with same coords: • Formally, we have: • Informally, p and v look and act about the same • People often sloppy, don’t distinguish point from vector • Can only get away with it if you stay in same frame! • And even then need to be careful…

22. Outline for Today • Finish linear algebra: Matrix composition • Points, Vectors and Coordinate Frames • Homogeneous Coordinates • But first: transforming points

23. Linear Transformations

24. Rotating an object • Object defined as collection of points • Apply rotation matrix to every point: • Rotates object about origin of frame • Also rotates all vectors in object:

25. Scaling an object • Apply scale matrix to every point: • Scale object about origin of frame • Also scales all vectors in object

26. d Moving an object • Add displacement vector to each point: • Translates the object: • Vectors don’t change:

27. General Object Transformation • Some matrix M and displacement d: • Math note: the transformation for p isn’t linear • It’s an affine transformation • Points and their transforms form an affine space • Not a vector space

28. But it’s very inconvenient • Different rule for points vs. vectors • Hard to compose transformations: • Hard to invert, etc.…so introduce Homogeneous Coordinates

29. Homogeneous coordinates • Basic: a trick to unify/simplify computations. • Deeper: projective geometry • Interesting mathematical properties • Good to know, but less immediately practical • We will use some aspect of this when we do perspective projection (in a few weeks)

30. Homogeneous coordinates • Add an extra component. 1 for a point, 0 for a vector: • combine M and d into single 4x4 matrix: • And see what happens when we multiply…

31. Homogeneous point transform • Transform a point: • Top three rows are the affine transform! • Bottom row stays 1

32. Homogeneous vector transform • Transform a vector: • Top three rows are the linear transform • Displacement d is properly ignored • Bottom row stays 0

33. Homogeneous arithmetic • Legal operations always end in 0 or 1!

34. Homogeneous Transforms • Rotation, Scale, and Translation of points and vectors unified in a single matrix transformation: • Matrix has the form: • Last row always 0,0,0,1 • Transforms compose by matrix multiplication! • Same caveat: order of operations is important • Same note: Transforms operate right-to-left

35. Primitive Transforms

36. Programming in practice • Everyone uses homogeneous matrices. • built into low-level software, hardware • In practice, almost never explicitly use 4th component of vector or point. • Waste of memory & computation • Instead, keep track of points vs. vectors explicitly • E.g. by C++ classes • Separate matrix methods “transform point” and “transform vector” that implicitly use the 1 or 0.

37. More Coordinate Equivalences • Translate an object by d Translate the coordinate frame by -d • Either way, get the same coordinates out • Rotate object about the frame’s origin Rotate the frame oppositely out its own origin • Either way, get the same coordinates out • Duality: • Matrix transforms objects vs. changes coordinates • Either can be handy (we’ll talk about next class) • Can be confusing, make sure you know which you want…

38. Transformation as coordinate frame • Build matrix from vectors a, b, c, point d • Notice effect on coordinate frame: • Any transform M describes change of frame: • If a,b,c are right-handed orthonormal, transformation is rigid • Pure rotation, pure translation, or mix of rotation and translation • No scale

39. Useful tidbit: Rotate about a “pivot” • Q: How to rotate about arbitrary pivot? • Rotation matrix always rotates about origin! • A: Sequence of operations: • This is a handy “primitive” to implement • Food for thought: what’s the structure of the resulting matrix?

40. Next class: • Hierarchical Transformations • Geometric Calculations