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**Linear Algebra (Mathematics for CG) Reading: HB Appendix A**

Computer Graphics Last Updated: 13-Jan-12 Linear Algebra (Mathematics for CG) Reading: HB Appendix A

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**Notation: Scalars, Vectors, Matrices**

(lower case, italic) Vector (lower case, bold) Matrix (upper case, bold)

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Vectors Arrow: Length and Direction Oriented segment in 2D or 3D space

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**Column vs. Row Vectors Row vectors Column vectors**

Switch back and forth with transpose

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**Vector-Vector Addition**

Add: vector + vector = vector Parallelogram rule tail to head, complete the triangle geometric algebraic examples:

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**Vector-Vector Subtraction**

Subtract: vector - vector = vector

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**Vector-Vector Subtraction**

Subtract: vector - vector = vector argument reversal

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**Scalar-Vector Multiplication**

Multiply: scalar * vector = vector vector is scaled

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**Vector-Vector Multiplication**

Multiply: vector * vector = scalar Dot product or Inner product

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**Vector-Vector Multiplication**

Multiply: vector * vector = scalar dot product, aka inner product

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**Vector-Vector Multiplication**

Multiply: vector * vector = scalar dot product or inner product geometric interpretation lengths, angles can find angle between two vectors

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**Dot Product Geometry Can find length of projection of u onto v**

as lines become perpendicular,

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Dot Product Example

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**Vector-Vector Multiplication**

Multiply: vector * vector = vector cross product

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**Vector-Vector Multiplication**

multiply: vector * vector = vector cross product algebraic geometric parallelogram area perpendicular to parallelogram

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**RHS vs. LHS Coordinate Systems**

Right-handed coordinate system Left-handed coordinate system right hand rule: index finger x, second finger y; right thumb points up left hand rule: index finger x, second finger y; left thumb points down

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Basis Vectors Take any two vectors that are linearly independent (nonzero and nonparallel) can use linear combination of these to define any other vector:

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**Orthonormal Basis Vectors**

If basis vectors are orthonormal (orthogonal (mutually perpendicular) and unit length) we have Cartesian coordinate system familiar Pythagorean definition of distance Orthonormal algebraic properties

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**Basis Vectors and Origins**

Coordinate system: just basis vectors can only specify offset: vectors Coordinate frame (or Frame of Reference): basis vectors and origin can specify location as well as offset: points It helps in analyzing the vectors and solving the kinematics.

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Working with Frames F1 F1 Coefficient of i: Start from O, move along i and count until the line passing through P parallel to j. Coefficient of j: Start from O, move along j and count until the line passing through P parallel to i. Coefficient of i or j will be –ve if the movement is opposite to the direction of i or j, resp.

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Working with Frames F1 F1 p = (3,-1)

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Working with Frames F1 F1 p = (3,-1) F2 F2

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Working with Frames F1 F1 p = (3,-1) F2 F2 p = (-1.5,2)

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Working with Frames F1 F1 p = (3,-1) F2 F3 F2 p = (-1.5,2) F3

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**Working with Frames F1 p = (3,-1) F2 p = (-1.5,2) F3 p = (1,2) F1 F2**

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**Named Coordinate Frames**

Origin and basis vectors Pick canonical frame of reference then don’t have to store origin, basis vectors just convention: Cartesian orthonormal one on previous slide Handy to specify others as needed airplane nose, looking over your shoulder, ... really common ones given names in CG object, world, camera, screen, ...

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**Lines Slope-intercept form Implicit form y = mx + b y – mx – b = 0**

Ax + By + C = 0 f(x,y) = 0

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**Implicit Functions Find where function is 0**

An implicit function f(x,y) = 0 can be thought of as a height field where f is the height (top). A path where the height is zero is the implicit curve (bottom). Find where function is 0 plug in (x,y), check if = 0 on line < 0 inside > 0 outside

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**Implicit Circles circle is points (x,y) where f(x,y) = 0**

points p on circle have property that vector from c to p dotted with itself has value r2 points p on the circle have property that squared distance from c to p is r2 points p on circle are those a distance r from center point c

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**Parametric Curves Parameter: index that changes continuously**

(x,y): point on curve t: parameter Vector form

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2D Parametric Lines start at point p0, go towards p1, according to parameter t p(0) = p0, p(1) = p1

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**Linear Interpolation Parametric line is example of general concept**

p goes through a at t = 0 p goes through b at t = 1 Linear weights t, (1-t) are linear polynomials in t

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**Matrix-Matrix Addition**

Add: matrix + matrix = matrix Example

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**Scalar-Matrix Multiplication**

Multiply: scalar * matrix = matrix Example

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**Matrix-Matrix Multiplication**

Can only multiply (n, k) by (k, m): number of left cols = number of right rows legal undefined

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**Matrix-Matrix Multiplication**

row by column

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**Matrix-Matrix Multiplication**

row by column

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**Matrix-Matrix Multiplication**

row by column

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**Matrix-Matrix Multiplication**

row by column

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**Matrix-Matrix Multiplication**

row by column Non commutative: AB != BA

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**Matrix-Vector Multiplication**

points as column vectors: post-multiply

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**Matrices Transpose Identity Inverse**

Note: Not all matrices are invertible

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**Matrices and Linear Systems**

Linear system of n equations, n unknowns Matrix form Ax=b

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Exercise 1 Q 1: If the vectors, u and v in figure are coplanar with the sheet of paper, does the cross product v x u extend towards the reader or away assuming right-handed coordinate system? For u = [1, 0, 0] & v = [2, 2, 1]. Find Q 2: The length of v Q 3: length of projection of u onto v Q 4: v x u Q 5: Cos t, if t is the angle between u & v

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Exercise 2 F1 F3 F2 F1 p = (?, ?) F2 p = (?, ?) F3 p = (?, ?)

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Exercise 3 Given P1(3,6,9) and P2(5,5,5). Find a point on the line with end points P1 & P2 at t = 2/3 using parametric equation of line, where t is the parameter.

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Things to do Reading for OpenGL practice: HB 2.9 Ready for the Quiz

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References

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