Maths for Computer Graphics
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1 Trigonometry Is concerned wit te analysis of triangles. Degrees and radians Te degree (or sexagesimal unit of measure derives from defining one complete rotation as 360. Eac degree divides into 60 minutes, and eac minute divides into 60 seconds. Te radian is te angle created y a circular arc wose lengt is equal to te circle s radius. Te perimeter of a circle equals πr, terefore π radians correspond to one complete rotation. 360 correspond to π radians, terefore radian corresponds to π 80, approximately Try to memorie te following relationsips etween radians and degrees π 3π 90, π 80, 70, π 360
2 Te trigonometric ratios Te Hindu word arda-jya meaning alf-cord was areviated to jya ( cord, wic was translated y te Aras into jia, and corrupted to j. Oter translators converted tis to jai, meaning cove, ulge or ay, wic in Latin is us. Today, te trigonometric ratios are known as,, tan, ec, sec and cot. Te trigonometric ratios are given y ( β ( β tan( β ec ( β opposite ypotenuse ( β sec ( β adjacent ypotenuse ( β cot ( β opposite adjacent tan ( β ypotenuse opposite β adjacent
3 0 Example Given a triangle were te ypotenuse and one angle are known. Te oter sides are calculated as follows: ( ( ( ( θ 6.79
4 Inverse trigonometric ratios Te, and tan functions convert angles into ratios, and te inverse functions -, - and tan - convert ratios into angles. For example, ( , terefore - ( Altoug e and coe functions are cyclic functions (i.e. tey repeat indefinitely te inverse functions return angles over a specific period tan - - ( x θ were (x θ were π θ π and (x θ were 0 θ π and ( θ x ( θ π θ π and tan( θ x x
5 Trigonometric relationsips Tere is an intimate relationsip etween te and definitions and are formally related y ( β ( β 90 Te Teorem of Pytagoras can e used to derive oter formulae suc as ( β tan( β ( β ( β ( β tan cot ( β sec ( β ( β ec ( β
6 ( β ( β tan ( β y β x (β y (β x ( β ( β y x y x tan( β ( β ( β tan ( β
7 ( β ( β y β y x x y x y x ( β ( β
8 tan ( β sec ( β y β x ( β ( β ( β ( β ( β ( β ( β tan ( β sec ( β
9 cot ( β ec ( β y β x ( β ( β ( β ( β ( β ( β ( β cot ( β ec ( β
10 Te Sine Rule c B a A C a ( A ( B c ( C Example A 50, B 30, a 0, find (30 0 (50 0(30 (
11 Te Coe Rule c B a A C a c c ( A c a ca ( B c a a( C a c c a ( C c ( B ( A a ( C ( B ( A
12 Compound angles Two sets of compound trigonometric relationsips sow ow to add and sutract two different angles and multiples of te same angle. Te following are some of te most common relationsips ( A± B ( A ( B ( A ( B A± B ( A ( B m ( A ( B ± ( tan ( A± B tan m tan ( A ± tan( B ( A tan( B ( β ( β ( β ( β ( β ( β ( β ( β ( β ( β 3 ( 3β 3( β ( β 3 β ( β 3( β ( β ( ( β (3 ( β ( ( β
13 Perimeter relationsips c B a A ( a c s C A ( ( c c B ( c( a ca C ( a( a
14 Perimeter relationsips c B a A ( a c s C A s( a c B s( ca C s( c a
15 Perimeter relationsips c B a A ( a c s C c ( A s( a( ( c ca ( B s( a( ( c a ( C s( a( ( c
16 By te way. K! 3!!! 3 e e.g ! 3!!! 3 e e e K K ut K K 8! 6!!! ( 9! 7! 5! 3! ( Euler's formula ( ( i e i
17 e and compound interest P te sum of money in a deposit account. R te % rate of interest for monts. If interest is awarded eac monts At te end of monts te alance is P R P P( R 5 00 e.g If interest is awarded eac 6 monts At te end of te st 6 monts te alance is P R P P( R At te end of te nd 6 monts te alance is P ( R R( R P(
18 If interest is awarded eac 3 monts At te end of te st 3 monts te alance is P R P P( R At te end of te nd 3 monts te alance is P ( R R( R P( At te end of te 3 rd 3 monts te alance is P ( R 3 R ( R P( At te end of te t 3 monts te alance is P ( R 3 R ( R P( In te limit, te alance is P ( R k k But e k ( k and x x k e ( k In te limit, te alance is R Pe
19 Let P 5,000 and R 0% alance P Ug e we get ( R k k k Balance 5, ,5.50 5, , , ,55.85 alance Pe R e alance
20 Compound interest Wat interest rate do I need to turn 5,000 into 5,700 in one year? alance R Pe R 5000 e 5700 R e ln( e R ln(. R 0.30 R 3.%
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