2IV60 Computer Graphics Basic Math for CG Jack van Wijk TU/e

Overview Coordinates, points, vectors Matrices H&B A-1

nD Space nD Space: n (typically) n : number of dimensions Examples: 1D space: time, along a line or curve 2D space: plane, sphere 3D space: the world we live in 4D space: 3D + time H&B A-2

Coordinates 2D Cartesian coordinates: Standard Screen (output, input) x y (x,y) (x,y) y x Standard Screen (output, input) H&B A-1

Polar coordinates y (x,y) r  x H&B A-1

3D coordinates 1 3D Cartesian coordinates: Right-handed Left-handed (x,y,z) z (x,y,z) z x y x y Right-handed Left-handed H&B A-1

3D coordinates 2 3D Cartesian coordinates: Right-handed Left-handed (x,y,z) z (middle) z (middle) (x,y,z) y (index) x (thumb) x (thumb) y (index) Right-handed Left-handed H&B A-1

3D coordinates 3 Cylinder coordinates: (x,y,z) z y x  H&B A-1

3D coordinates 4 Spherical coordinates: (x,y,z) z f r y x  H&B A-1

Points Point: position in nD space Notation: P (H&B), also P, p, p en p (x,y,z) (H&B), also (x1, x2, x3), (Px, Py, Pz), (r, , z), ( r, , ), … H&B A-2

Vectors 1 Vector: Notation: V (H&B), also V, v, v en v “arrow” multiple interpretations (displacement, velocity, force, …) has a magnitude and direction has no position Notation: V (H&B), also V, v, v en v (Vx, Vy, Vz) (H&B), also (x, y, z), (x1, x2, x3) H&B A-2

Vectors 2 y P2 y2 P1 y1 V: directed line segment, or difference between two points x1 x2 x H&B A-2

Vectors 3 Length of a vector: H&B A-2

Vectors 4 Direction of a vector: Direction angles. Unit vector V : Magnitude info is removed, direction is kept. z g b y x a H&B A-2

Vector addition Add components, put vector head to tail y y W V+W W V x x H&B A-2

Scalar multiplication of vector Multiplication components with scalar s y y 2V V x x H&B A-2

Combining vector operations Infinite line through P with direction V: L(t) = P + Vt, t  V y P t x t : parameter along line t [a, b]: line segment H&B A-2

Vector multiplication 1 Scalar product or dot product: Product of parallel components, gives 1 real value W V q |W| cos q |V| H&B A-2

Vector multiplication 2 Scalar product: H&B A-2

Vector multiplication 3 Vector product or cross product: gives a vector (in 3D) n perpendicular to V and W VW W q n V H&B A-2

Vector multiplication 5 Scalar product: H&B A-2

Vector multiplication 6 Scalar product: scalar Test if vectors are perpendicular cos project,… Vector product: vector Get a vector perpendicular to two given vectors sin surface area,… H&B A-2

Exercise 1 Given a point P. Requested: Reflect a point Q with respect to P. Q W = P – Q Q’ = Q + 2W = 2P – Q or: = P + (P – Q ) W P x y Q’ We don’t need coordinates!

Exercise 1 Given a point P. Requested: Reflect a point Q with respect to P. Q Alternative P is halfway Q and Q’: P = (Q + Q’)/2 2P = Q + Q’ Q’ = 2P – Q P Q’

Exercise 2 Given a line L: L(t) = P + Vt . Requested: Reflect a point Q with respect to L. Q Q’ We know: Q’ = Q + 2 W W = L(t) – Q W. V = 0 W V L(t) y P t x

Exercise 2 We know: L(t) = P + Vt Q’ = Q + 2 W W = L(t) – Q W. V = 0 Substitute to get t: (L(t) – Q).V = 0 (P + Vt – Q).V = 0 V .V t + (P – Q).V = 0 t = ((Q – P).V) / (V .V) Then: Q’ = Q + 2 (P + Vt – Q) = 2P – Q + V((Q – P).V / V .V)

Steps to be made Write down what you know Eliminate, substitute, etc. to get he result Check the result Does it make sense? Is there a simpler derivation?

Exercise 3 Given a triangle with vertices P, Q en R, where the angle PQR is perpendicular. Requested: Rotate the triangle around the line PQ over an angle a . What is the new position R’ of R? P  R’ Q R

Circle in space |A|=1, |B|=1, A.B = 0 y C() C()  P A B r P A B  x |A|=1, |B|=1, A.B = 0 C() = (r cos , r sin , 0) = (0,0,0) + r cos  (1,0,0) + r sin  (0,1,0) = P + r cos  A + r sin  B

Exercise 3 A = (R – Q) / |R – Q| B = (R – Q)(P – Q) / | (R – Q)(P – Q) | R’ = Q + |R – Q| cos  A + |R – Q| sin  B P B  R’ Q A R

Use: Scalar product, cross product coordinate independent definitions vector algebra Don’t use: arccos, arcsin y = f(x)

Very short intro to Linear Algebra System of linear equations: Such systems occur in many, many applications. They are studied in Linear Algebra. H&B A-5

Very short intro to Linear Algebra System of linear equations: Typical questions: Given u, v, w, what are x, y, z? Can we find a unique solution? H&B A-5

Very short intro to Linear Algebra System of linear equations: Crucial in computer graphics: Transforming geometric objects Change of coordinates H&B A-5

Example transformation y P: (x,y) x y’ P’ V U y x x’ H&B A-5

What is a matrix? Matrix: Mathematical objects with operations Matrix in computer graphics: Defines a coordinate frame Defines a transformation Handy tool for manipulating transformations H&B A-5

Matrix Matrix: rectangular array of elements Element: quantity (value, expression, function, …) Examples: H&B A-5

Matrix r  c matrix: r rows, c columns mij : element at row i and column j. H&B A-5

Matrix r  c matrix: r rows, c columns mij : element at row i and column j. r  c elements H&B A-5

Matrix r  c matrix: r rows, c columns mij : element at row i and column j. c columns H&B A-5

Matrix r  c matrix: r rows, c columns mij : element at row i and column j. r rows H&B A-5

Matrix Column vector: matrix with c =1 Used for vectors (and points) Row vector: matrix with r =1 Scalar: matrix with r=1and c=1 H&B A-5

Matrix Matrix as collection of column vectors: W Matrix contains an axis-frame: H&B A-5

Operations on matrices Multiplication with scalar simple Addition Matrix-matrix multiplication More difficult, but the most important H&B A-5

Scalar Matrix multiplication Matrix M multiplied with scalar s: H&B A-5

Scalar Matrix addition H&B A-5

Scalar Matrix addition Just add elements pairwise A and B must have the same number of rows and columns Generalization of vector and scalar addition H&B A-5

Matrix Matrix multiplication i = 2 ; k j = 1 ; k cij: dot product of row vector ai and column vector bj #columns A must be the same as #rows B: n = p C: m  q matrix H&B A-5

Matrix Matrix multiplication Example: i = 3 j = 2 i=3, j=2 H&B A-5

Matrix Matrix multiplication Example: H&B A-5

Matrix Matrix multiplication Example: H&B A-5

Matrix Matrix multiplication Example: AB  BA H&B A-5

Matrix Matrix multiplication AB  BA Matrix matrix multiplication is not commutative! Order matters! A(B+C) = AB+AC Matrix matrix multiplication is distributive H&B A-5

Example transformation sequence (coordinate version) Unclear! Error-prone! H&B A-5

Example transformation sequence (matrix vector version) H&B A-5

Example transformation sequence (compact matrix vector version) Matrix vector notation allows for compactness and genericity! H&B A-5

Matrix Transpose Transpose matrix: Interchange rows and columns. r  c matrix M  c  r matrix MT H&B A-5

Matrix Inverse Simple algebra puzzle. Let ax = b. What is x? Linear algebra puzzle. Let MU = V. What is U? H&B A-5

Matrix Inverse H&B A-5

Matrix Inverse examples H&B A-5

Matrix Inverse examples Does not exist, cannot be inverted. H&B A-5

Matrix Inverse Does not always exist In general: if determinant = | M | = 0, matrix cannot be inverted Inverse for n = 1 and n = 2: easy Inverse for higher n: use library function Important special case: orthonormal matrix. H&B A-5

Overview Coordinates, points, vectors Matrices definitions, operations, examples Matrices