Lecture03-BackgroundMathematics

Lecture03-BackgroundMathematics - CS 455 Computer Graphics...

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CS 455 – Computer Graphics Background Mathematics
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Set Notation object a is an element of set S Cartesian product of sets A and B , i.e., the set that contains all ordered pairs ( a, b ) such that a A and b B . Sets of interest: : the set of real numbers + : the set of non-negative real numbers 2 : the set of ordered pairs in the real plane, i.e., the 2D plane n : the set of points in the n-dimensional Cartesian space Z : the integers S 2 : the set of 3D points (points in 3 ) on the unit sphere S a B A ×
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Mappings Notation for mappings: a function f which takes a real number as input and maps it to an integer This is equivalent to the programming notation Z f : ) real ( integer f
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Inverse Maps If we have a function the inverse of f is defined as where in order for a function to have an inverse, it needs to be a bijection one-to-one mapping between A and B B A f : a b f = - ) ( 1 ) ( a f b =
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Quadratic Formula Given a quadratic equation of the form The solutions are given by The discriminant D is Number of real roots: D > 0 : two real roots D = 0 : one real root D < 0 : no real roots 0 2 = + + c bx ax a ac b b x 2 4 2 - ± - = ac b D 4 2 - =
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Trigonometry Remember: ??? so: 360 radians 2 = π radians 180 degrees degrees 180 radians π π = =
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Trigonometry Given a right triangle with sides of length a, o, and h (adjacent, opposite, and hypotenuse): a h o θ h a = θ cos h o = θ sin a o = θ tan Theorem rean Pythago 2 2 2 h o a = + S ome O ld H ag C ame A round H ere T aking O ur A pples
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Trig identities sin(- α ) = -sin( α ) cos(- α ) = cos( α ) tan(- α ) = -tan( α ) sin 2 α + cos 2 α = 1 sin( α + β ) = sin α cos β + sin β cos α cos( α + β ) = cos α cos β - sin α sin β sin(2 α ) = 2sin α cos α cos(2 α ) = cos 2 α - sin 2 α c 2 = a 2 + b 2 – 2ab cos C law of cosines
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Vectors A vector has a length and direction Can be represented by an arrow Has NO position these are the same vector Can also be represented by an ordered pair in R 2 or an ordered triple in R 3 .
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