Geometric Intuition

1. Geometric Intuition Randy Gaul

2. Talk Outline • Vectors, Points and Basis Matrices • Rotation Matrices • Dot product and how it’s useful • Cross product and how it’s useful • Extras

3. Prerequisites • Matrix multiplication • How to apply dot/cross product • Vector normalization • Basic understanding of sin/cos • Basic idea of what a plane is

4. Vectors • Formal definition • Input or output from a function of vector algebra • Informal definition • Scalar components representing a direction and magnitude (length) • Vectors have no “location”

5. Points and Vectors • A point P and Q is related to the vector V by: • P – Q = V • This implies that points can be translated by adding vectors

6. Points and Vectors (2) • Lets define an implied point O • O represents the origin • Any point P can be expressed with a vector V: • P = O + V • Vectors point to points, when relative to the origin!

7. Euclidean Basis • Standard Euclidean Basis (called E3) • Geometrically the x, y and z axes

8. Euclidean Basis (2) • Given 3 scalars i, j and k, any point in E3 can be represented

9. Linear Combination • Suppose we have a vector V • Consists of 3 scalar values • Below V written in “shorthand” notation

10. Linear Combination (2) • V can be written as a linear combination of E3 • This is the “longhand” notation of a vector

11. Basis Matrices • E3 is a basis matrix • Many matrices can represent a basis • Any vector can be represented in any basis Note: Different i, j and kvalues are used on leftand right http://en.wikipedia.org/wiki/Change_of_basis

12. Rotation Matrices • A rotation matrix can rotate vectors • Constructed from 3 orthogonal unit vectors • Can be called an “orthonormal basis” • E3 is a rotation matrix!

13. Rotation Matrices (2) • Rotation matrices consist of an x, y and z axis • Each axis is a vector y z X

14. Rotation Matrices (3) • Multiplying a rotation and a vector rotates the vector • This is a linear combination

15. Dot Product • Comes from law of cosines • Full derivation here • For two vectors uand v:

16. Shortest Angle Between 2 Vectors • Assume u and v are of unit length • Result in range of 0, 1 • No trig functions required

17. How Far in a Given Direction? • Given point P and vector V • How far along the V direction is P? x P V y

18. How Far in a Given Direction? (2) • Normalize V • Dot V and P (treat P as vector) • Explanation: • Cosine scaled by the length of P is equivalentto the distance P travels in the direction V

19. Planes • Here’s the 3D plane equation

20. Planes • Here’s the 3D plane equation • WAIT A SECOND

21. Planes • Here’s the 3D plane equation • WAIT A SECOND • THAT’S THE DOT PRODUCT

22. Planes (2) • a, b and c form a vector, called the normal • d is magnitude of the vector • Represents distance of plane from origin

23. Planes (3) • d is interesting • d is scaled by length n

24. Planes and the Dot Product • Dot P with the plane normal: • is distance along normal P travels • is scaled by length of normal

25. Planes and the Dot Product (2) • 2D visualization, assume normal () is unit plane x P y

26. Signed Distance P to Plane • Take subtract d • Explanation: • is distance P travels along normal • Subtract d (distance of plane from origin) • This gives distance of P to the plane

27. Project P onto Plane • 2D visualization, assume normal () is unit plane x = P y

28. Rotation Matrices and Dot Product • Given matrices A and B • A * B is to dot the rows of A with columns of B • Lets assume A and B are rotation matrices http://en.wikipedia.org/wiki/Matrix_multiplication

29. Rotation Matrices and Dot Product (2) • That’s a lot of dot products! • Try: A * B * vector V • V is rotated by A, and then rotated by B • What does A * B mean? • Each element is cos between a column of A and row of B

30. Rotation Matrices and Dot Product (3) • A = , B = • Lets start multiplying these guys

31. Rotation Matrices and Dot Product (4) • We start by performing: • is the x axis of A • is x components of the x, y and z axis of B

32. Rotation Matrices and Dot Product (5) • What does do? • Computes the contribution of B’s x components along the x axis of A! • Every single element of A * B can be thought of in this way

33. Point in Convex Hull • Test if point is inside of a hull • Compute plane equation of hull faces • Compute distance from plane with plane equation • If all distances are negative, point in hull • If any distance is positive, point outside hull • Works in any dimension • Can by used for basic frustum culling

34. Point in OBB • An OBB is a convex hull! • Hold your horses here… • Lets rotate point P into the frame of the OBB • OBB is defined with a rotation matrix, so invert it and multiply P by it • P is now in the basis of the OBB • The problem is now point in AABB

35. Point in Cylinder • Rotate cylinder axis to the z axis • Ignoring the z axis, cylinder is a circle on the xy plane • Test point in circle in 2D • If miss, exit no intersection • Get points A and B. A is at top of cylinder, B at bottom • See if point’s z component is less than A’s and greater than B’s • Return intersection • No intersection

36. Bumper Car Damage? • Two bumper cars hit each other • One car takes damage • How much damage is dealt, and to whom?

37. Bumper Car Damage Answer • Take vector from one car to another, T • Damage dealt: • 1.0 - Abs( Dot( velocity, T ) * collisionDamage )

38. Distance Point P to Line http://www.randygaul.net/2014/07/23/distance-point-to-line-segment/

39. Closest Point to Segment from P • Take and compute a plane C • C’s normal is / ||, and offset d is: • d = dot( / ||, A ) • Compute distance P from C • If C is negative, closest point is A • Repeat process for B

40. Visualization in 2D P C / ||

41. Cross Product • Operation between vectors • Produces a vector orthogonal to both input vectors

42. Cross Product Handedness http://en.wikipedia.org/wiki/Cross_product

43. Cross Product Details http://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Cross_product.gif/220px-Cross_product.gif

44. Plane Equation from 3 Points • Given three points A, B and C • Calculate normal with: Cross( C – A, B – A ) • Normalize normal • Compute offset d: dot( normal, A (or B or C) )

45. Intersection Segment and Plane • Compute distance A to Plane: da • Compute distance B to Plane: db • If da * db < 0 • Intersection = A + (da / (da – db)) * (B – A) • Else • No intersection

46. Segment Intersects Triangle UVW • If intersects plane triangle’s plane • Compute intersection point P with plane • For each edge along UVW, cross Va and P • Dot result with plane normal • If dot result < 0 • No intersection • Hit back side of UVW if dot( , normal ) > 0 • Else • No intersection

47. Segment Intersects Triangle UVW (2) normal P

48. Affine Transformations • Given matrix A, point x and vector b • An affine transformation is of the form: • Ax + b

49. Affine Transformations • Given 3x3 matrix A, point x and vector b • An affine transformation is of the form: • Ax + b • With a 4x4 matrix we can represent Ax + b in block formation:

50. Translation • Construct an affine transformation such that when multiplied with a point, translates the point by the vector b • I is the identity matrix, and means no rotation (or scaling) occurs