Vector algebra Christian Miller CS Fall 2011

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1 Vector algebra Christian Miller CS Fall 2011

2 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority of computer graphics We ll deal mostly with 3D, but also a bit of 4D

3 Vectors A vector is a direction and a magnitude Does NOT include a point of reference Usually thought of as an arrow in space Vectors can be added together and multiplied by scalars Zero vector has no length or direction

4 Vectors

5 Points A point is a location in space Cannot be added or multiplied together Subtract two points to get the vector between them

6 Coordinates It s easier to deal with objects if you turn them into numbers Do this by using a coordinate system Define some reference Decompose objects into numbers by projecting onto that reference

7 Cartesian coordinates The most common and simple coordinate system A coordinate frame is: A point (the origin) A set of ordered vectors spanning space (axes) Usually orthographic (vectors at right angles) and normalized (unit length vectors) Called orthonormal in that case

8 Cartesian coordinates We get coordinates for vectors by projecting them onto the axes Coordinates for points come from subtracting the origin, then getting coordinates for the resulting vector

9 Cartesian coordinates

10 Names for axes The axes are sometimes called the basis vectors Sometimes named Often called in 3D

11 Writing coordinates Vector and point coordinates usually written as 3x1 matrices (column vectors): Sometimes shorthanded as:

12 Basic math Addition: Scalar multiplication: Length / magnitude: Normalizing a vector:

13 Linear interpolation Can smoothly interpolate between two points α [0, 1]

14 Dot product Formula: Alternately: Where φ is the angle between the vectors

15 Dot product intuition Several important properties: = 0 means vectors are at right angles > 0 means vectors are acute < 0 means vectors are obtuse If inputs are normalized, Linear, commutative

16 Projection / rejection Projection: W is the part of U that lies on V Rejection: Just U - W

17 Cross product Formula: Creates a vector that is: Perpendicular to the inputs Length Right-hand oriented

18 Cross product intuition If U & V point along the same line, W = 0 Useful for constructing local coordinate frames Length of cross product is area of parallelogram spanned by U and V (divide by 2 for area of the triangle)

19 More cross product Don t assume the cross product behaves like regular multiplication! NOT commutative: NOT associative:

20 Scalar triple product Formula: Computes the volume of a parallelpiped Divide by 6 to get volume of a tetrahedron

21 Planes Defined implicitly: N is the normal vector (perpendicular to plane) d is the distance to the origin at the closest point Given any point p on the plane:

22 Triangle normals Every triangle lies in a plane 2 choices of normal, pick one by convention CCW winding is usually used Formula:

23 Rays A directed, semi-infinite line segment Often used to approximate light paths Ray equation: o, the origin of the ray d, the ray direction (usually normalized) t > 0, the distance along the ray

24 Matrices A matrix is a 2D array of numbers Used to represent linear operators with coordinates Size: M x N M rows, N columns (example is 3 x 4)

25 Matrix operations Addition: component-wise Scalar multiplication: also component-wise Trace: sum up diagonal elements Transpose: flip over diagonal

26 Matrix multiplication Two matrices: A is P x M and B is M x N Inner dimension (M) must be the same Every element of the result matrix is the dot product of a row of the first matrix with a column of the second Non-commutative in general: Associative:

27 Matrix multiplication

28 Identity matrix Square matrix that, when multiplied by any compatible matrix, does nothing 1 on diagonal, 0 elsewhere Size is usually implicit

29 Determinant Scalar value associated with a matrix Can be recursively computed by minors Some properties:

30 Cross products and determinants Useful mnemonic trick for cross products:

31 Matrix inverse A matrix multiplied by its inverse is the identity Exists iff matrix is square and nonsingular A singular matrix has a determinant of 0

32 Computing inverses Closed form inverses exist for all sizes Quickly gets impractical to compute Also Gauss-Jordan elimination, factorizations, etc.

33 One more useful property Transposes and inverses can be reversed:

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