Vectors, Points, Lines and Planes

1. Vectors, Points, Lines and Planes Jim Van Verth (jim@essentialmath.com) Lars M. Bishop (lars@essentialmath.com)

2. Geometric Representation • Problem • Represent geometric data in computer? • Solution • Vectors and points • Can represent most abstract objects as combinations of these • E.g. lines, planes, polygons Essential Math for Games

3. What Is a Vector? • Geometric object with two properties • direction • length (if length is 1, is unit vector) • Graphically represented by Essential Math for Games

4. Scale • Graphically • change length of vector v by  Essential Math for Games

5. b a a + b Addition • Graphically • put tail of second vector at head of first • draw new vector from tail of first to head of second Essential Math for Games

6. Algebraic Vectors • Algebraically, vectors are more than this • Any entity that meets certain rules (lies in vector space) can be called ‘vector’ • Ex: Matrices, quaternions, fixed length polynomials • Mostly mean geometric vectors, however Essential Math for Games

7. Vector Space • Set of vectors related by +,· • Meet rules • v + w = w + v (commutative +) • (v + w) + u = v + (w + u) (associative +) • v + 0 = v (identity +) • v + (-v) = 0 (inverse +) • () v =  (v) (associative ·) • (+)v = v + v (distributive ·) • (v + w) = v + w(distributive ·) Essential Math for Games

8. Number Spaces • Cardinal – Positive numbers, no fractions • Integer – Pos., neg., zero, no fractions • Rational – Fractions (ratios of integers) • Irrational – Non-repeating decimals (,e) • Real – Rationals+irrationals • Complex – Real + multiple of -1 a+bi Essential Math for Games

9. Real Vector Spaces • Usually only work in these • Rn is an n-dimensional system of real numbers • Represented as ordered list of real numbers (a1,…,an) • R3is the 3D world, R2is the 2D world Essential Math for Games

10. Linear Combination • Combine set of n vectors using addition and scalar multiplication • v = 1v1 + 2v2 + … + nvn • Collection of all possible linear combinations for given v1… vn is called a span • Linear combination of 2 perpendicular vectors span a plane Essential Math for Games

11. Linear Dependence • A system of vectors v1, … ,vnis called linearly dependant if for at least one vi • vi = 1v1 +…+ i-1vi-1 + i+1vi+1 +…+ nvn • Otherwise, linearly independent • Two linearly dependant vectors are said to be collinear • I.e. w = .v • I.e. they point the “same” direction Essential Math for Games

12. Vector Prerequisites Linear Dependence • Example • Center vector can be constructed from outer vectors Essential Math for Games

13. Vector Basis • Ordered set of n lin. ind. vectors •  = {v1,v2, …, vn} • Span n-dimensional space • Represent any vector as linear combo • v = 1v1 + 2v2 + … + nvn • Or just components • v = (1,2, …, n) Essential Math for Games

14. v3 v2 v1 Vector Representation • 3D vector vrepresented by (x, y,z) • Use standard basis { i, j, k } • Unit length, perpendicular (orthonormal) • v = xi + yj + zk • Number of units in each axis direction Essential Math for Games

15. Vector Operations • Addition: +,- • Scale: · • Length: ||v|| • Normalize: Essential Math for Games

16. Addition • Add a to b b a a + b Essential Math for Games

17. Scalar Multiplication • change length of vector v by  Essential Math for Games

18. Length • Length • ||v|| gives length (or Euclideannorm) of v • if ||v|| is 1, v is called unit vector • usually compare length squared • Normalize • v scaled by 1/||v|| gives unit vector Essential Math for Games

19. Vector Operations • Games tend to use most of the common vector operations • Addition, Subtraction • Scalar multiplication • Two others are extremely common: • Dot product • Cross product Essential Math for Games

20. Dot product • Also called inner product, scalar product a  b Essential Math for Games

21. Dot Product: Uses • a • a equals ||a||2 • can test for collinear vectors • if a and b collinear & unit length, |a • b| ~ 1 • Problems w/floating point, though • can test angle/visibility • a • b > 0 if angle < 90° • a • b = 0 if angle = 90° (orthogonal) • a • b < 0 if angle > 90° Essential Math for Games

22. Dot Product: Example • Suppose have view vector v and vector t to object in scene (t = o - e) • If v • t < 0, object behind us, don’t draw e v t o Essential Math for Games

23. Dot Product: Uses • Projection of a onto b is a b Essential Math for Games

24. Dot Product: Uses • Example: break a into components collinear and perpendicular to b a b Essential Math for Games

25. Cross Product • Cross product: definition • returns vector perpendicular to a and b • right hand rule • length = area of parallelogram c b a Essential Math for Games

26. Cross Product: Uses • gives a vector perpendicular to the other two! • ||a  b|| = ||a|| ||b|| sin() • can test collinearity • ||a  b|| = 0 if a and b are collinear • Better than dot – don’t have to be normalized Essential Math for Games

27. Other Operations • Several other vector operations used in games may be new to you: • Scalar Triple Product • Vector Triple Product • These are often used directly or indirectly in game code, as we’ll see Essential Math for Games

28. Scalar Triple Product • Dot product/cross product combo • Volume of parallelpiped • Test rotation direction • Check sign u v w Essential Math for Games

29. Triple Scalar Product: Example • Current velocity v, desired direction d on xy plane • Take • If > 0, turn left, if < 0, turn right v v  d d d v  d v Essential Math for Games

30. Vector Triple Product • Two cross products • Useful for building orthonormal basis • Compute and normalize: Essential Math for Games

31. Points • Points are positions in space — anchored to origin of coordinate system • Vectors just direction and length — free-floating in space • Can’t do all vector operations on points • But generally use one class in library Essential Math for Games

32. Point-Vector Relations • Two points related by a vector • (Q - P) = v • P + v = Q Q v P Essential Math for Games

33. Affine Space • Vector, point related by origin • (P - O) = v • O + v = P • Vector space, origin, relation between them make an affine space P e3 v O e2 e1 Essential Math for Games

34. Cartesian Frame • Basis vectors {i, j, k}, origin (0,0,0) • 3D point Prepresented by (px, py, pz) • Number of units in each axis direction relative to origin pz o px py Essential Math for Games

35. Affine Combination • Like linear combination, but with points • P = a1P1 + a2P2 + … + anPn • a1,…,an barycentric coord., add to 1 • Same as point + linear combination • P = P1 + a2 (P2-P1) + … + an (Pn-P1) • If vectors (P2-P1), …, (Pn-P1) are linearly independent, {P1, …, Pn} called a simplex (think of as affine basis) Essential Math for Games

36. Convex Combination • Affine combination with a1,…,an between 0 and 1 • Spans smallest convex shape surrounding points – convex hull • Example: triangle Essential Math for Games

37. Points, Vectors in Games • Points used for models, position • vertices of a triangle • Vectors used for velocity, acceleration • indicate difference between points, vectors Essential Math for Games

38. Parameterized Lines • Can represent line with point and vector • P + tv • Can also represent an interpolation from P to Q • P + t(Q-P) • Also written as (1-t)P + tQ Q v P Essential Math for Games

39. Planes • 2 non-collinear vectors span a plane • Cross product is normaln to plane n Essential Math for Games

40. Planes • Defined by • normal n = (A, B, C) • point on plane P0 • Plane equation • Ax+By+Cz+D = 0 • D=-(A·P0x + B·P0y + C·P0z) Essential Math for Games

41. Planes • Can use plane equation to test locality of point • If n is normalized, gives distance to plane n Ax+By+Cz+D > 0 Ax+By+Cz+D = 0 Ax+By+Cz+D < 0 Essential Math for Games

42. References • Anton, Howard and Chris Rorres, Elementary Linear Algebra, 7th Ed, Wiley & Sons, 1994. • Axler, Sheldon, Linear Algebra Done Right, Springer Verlag, 1997. • Blinn, Jim, Notation, Notation, Notation, Morgan Kaufmann, 2002. Essential Math for Games