Affine and Projective Geometry

1. Affine and Projective Geometry Chand T. John

2. R2 • Imagine a bucket filled with vectors. • A vector is an ordered pair of real numbers. • These real numbers are called coordinates of the vector. • A vector is usually written by listing its coordinates in parentheses. • For example, a vector consisting of 3 and then 5 is written as (3, 5).

3. R2 • Pretend our bucket contains all possible vectors: that is, it contains all possible ordered pairs of real numbers. • This bucket is called R2, because each vector in R2 has 2 real numbers. • This bucket has an infinite number of vectors, since there are an infinite number of real numbers.

4. R2 • A vector with coordinates (x, y) can be drawn as an arrow that goes horizontally by x units and vertically by y units. R2 y x

5. Operations in R2 • There are some operations we can do with vectors in R2—addition, subtraction, and scalar multiplication—to produce another vector in R2. • Pretend we have two vectors u = (ux, uy) and v = (vx,vy). We can add these vectors by adding their coordinates. Adding these vectors results in another vector:r = u + v = (ux+ uy, vx + vy)

6. Operations in R2 • We can also subtract these vectors by sutracting their coordinates. This produces another vector:s = u – v = (ux– uy, vx – vy) • Finally, we can multiply a vector by any real number c (called a scalar) to get another vector:t = cu = (cux, cvx)

7. Pictures of Operations in R2 Subtraction: s = u – v Addition: r = u + v v u r –v s u

8. Pictures of Operations in R2 Scalar multiplication: t = cu t u

9. E2 • Imagine we also have an infinitely large square, which we will call E2. • Each exact location on E2 is called a point. Points E2

10. Moving around on E2 • Suppose we are at point p on E2. • Suppose we pick a vector v out of the bucket R2. • We can addv to p to move to another point q on E2. To do this, simply place the arrow v on E2 so that the tail end of v is at p. Then the head of v is at q: R2 q v p E2

11. Subtraction on E2 • We can also subtract point p from point q to get the vector v as a result: v = q – p R2 q v p E2

12. Center of Mass • Seesaw…

13. Center of Mass on E2

14. Calculating Center of Mass

15. Barycentric Combination • Change units so sum of masses = 1 • Barycenter = center of mass • Mobius and der barycentrische calcul • Possible on E2 even though “addition” of points is not allowed

16. Linear and Affine Space • Explain one-to-one correspondence between from R2 to E2 and back: pick a point called origin and two nonparallel vectors called basis vectors—this origin + basis vectors is called a coordinate system • Key difference between E2 and R2 is that E2 has a notion of position, whereas R2 is just a bucket full of vectors with no other structure to it • R2 is called a vector space or linear space, and E2 is called an affine space

17. Linear versus Affine • Difference between degrees Celsius (oC) vs. Celsius degrees (Co) • Affine relationship: oF = 1.8oC + 32 • Linear relationship: Fo = 1.8Co • Again: linear = difference between affine points

18. Linear versus Affine • mechanical design: align coord systems => vectors have same coordinates in all of those coordinate systems, so vectors instead of points is convenient representation for forces/velocities/etc for computation

19. Standard E2 Coordinate System • Show standard coordinate system on E2 and some example points. Related to R2-E2 one-to-one correspondence. Can show how points are actually represented as origin + basis vectors with scalar multiplication.

20. Linear Transformations • Emphasize that linear transformations move us from one vector to another. • But remember, vectors in R2 are just floating around all the time, they have no fixed position.

21. Affine Transformations • Affine transformations move us from one point to another, or another way of looking at it is one coordinate system to another. • Points in E2 have a fixed position, so affine maps move us from one specific point to another, unlike linear maps which move us from one vector to another, with no fixed position at any time. • Affine transformation is linear transformation + translation; so just like linear transformation but with notion of position (hence translation).

22. The Big Picture So Far • Linear spaces have no notion of position. • Linear maps transform one vector into another. • Affine spaces have a notion of position. • Coordinate systems can be defined on affine spaces to give special numbers representing each point. • Any coordinate system is transformed into another using an affine map.

23. E3 • Just to introduce real projective plane: show standard coordinate system with basis vectors pulled out of R3 and z = 1 plane, which we call RP2

24. RP2 • Show point coordinates and vector coordinates on the real projective plane (in homogeneous coordinates)

25. Projective… • Projective maps, line-point duality, homogeneous coordinates, line at infinity • Weak perspective projection, perspective projection, orthogonal projection, parallel projection, cameras • Affine transformations in E2 as 3 x 3 matrices in homogeneous coordinates