Multimedia Maths

Bieke Masselis and ivo de Pauw

VERKENNINGEN AAN DE GRENS

Praktische theologie

ANNEMIE DILLEN EN STEFAN GÄRTNER

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Chapter 1, David Ritter; 2, John Evans; 3, Wouter Verweirder; 4, Daryl Beggs, Juan Pablo Arancibia Medina; 5, Stephanie

Berghaeuser; 6, 10, Wouter Tansens; 7, Danie Pratt; 8, Ken Munyard; 9, Bieke Masselis; p.21, p.93, Wouter Tansens;

p.44, Wouter Verweirder; p.48, Leo Storme; p.214, Yu-Sung Chang.

D/2016/45/400 – ISBN 978 94 014 3821 6 – NUR 918

Layout: Jurgen Leemans, Peter Flynn and Bavo Langerock

Cover design: Studio Lannoo and Keppie & Keppie

© Bieke Masselis, Ivo De Pauw and Publisher Lannoo n.v., Tielt, 2016.

LannooCampus is part of the Lannoo Publishing Group

All rights reserved

No part of this book may be reproduced,

in any form or by any means,

without permission in writing from the publisher.

Publisher LannooCampus

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Belgium

www.lannoocampus.com.

Content

Acknow ledgment s 11

Chapte r 1 · Ar i thmet i c Re f r e she r 131.1 Algebra 14

Real Numbers 14Real Polynomials 19

1.2 Equations in one variable 21Linear Equations 21Quadratic Equations 22

1.3 Exercises 28

Chapte r 2 · L inear sy s t ems 312.1 Definitions 322.2 Methods for solving linear systems 34

Solving by substitution 34Solving by elimination 35

2.3 Exercises 40

Chapte r 3 · Tr igonomet ry 433.1 Angles 443.2 Triangles 463.3 Right Triangle 503.4 Unit Circle 513.5 Special Angles 53

Trigonometric ratios for an angle of 45°= π4 rad 54

Trigonometric ratios for an angle of 30°= π6 rad 54

Trigonometric ratios for an angle of 60°= π3 rad 55

Overview 553.6 Pairs of Angles 563.7 Sum Identities 563.8 Inverse trigonometric functions 593.9 Exercises 61

6 MULTIMEDIA MATHS

Chapte r 4 · Funct i ons 634.1 Basic concepts on real functions 644.2 Polynomial functions 65

Linear functions 65Quadratic functions 67

4.3 Intersecting functions 694.4 Trigonometric functions 71

Elementary sine function 71General sine function 71Transversal oscillations 75

4.5 Inverse trigonometric functions 754.6 Exercises 79

Chapte r 5 · The Go lden Sec t i on 815.1 The Golden Number 825.2 The Golden Section 84

The Golden Triangle 84The Golden Rectangle 85The Golden Spiral 86The Golden Pentagon 88The Golden Ellipse 88

5.3 Golden arithmetics 89Golden Identities 89The Fibonacci Numbers 90

5.4 The Golden Section worldwide 935.5 Exercises 95

Chapte r 6 · Vector s 976.1 The concept of a vector 98

Vectors as arrows 98Vectors as arrays 99Free Vectors 102Base Vectors 102

6.2 Addition of vectors 103Vectors as arrows 103Vectors as arrays 103Vector addition summarized 104

6.3 Scalar multiplication of vectors 105Vectors as arrows 105Vectors as arrays 105

CONTENT 7

Scalar multiplication summarized 106Properties 106

6.4 Vector subtraction 107Creating free vectors 107Euler’s method for trajectories 108

6.5 Decomposition of vectors 109Decomposition of a plane vector 109Base vectors defined 110

6.6 Dot product 110Definition 110Geometric interpretation 112Orthogonality 114

6.7 Cross product 115Definition 116Geometric interpretation 118Parallelism 120

6.8 Normal vectors 1216.9 Exercises 123

Chapte r 7 · Paramete r s 1257.1 Parametric equations 1267.2 Vector equation of a line 1277.3 Intersecting straight lines 1317.4 Vector equation of a plane 1337.5 Exercises 137

Chapte r 8 · Matr i ce s 1398.1 The concept of a matrix 1408.2 Determinant of a square matrix 1418.3 Addition of matrices 1438.4 Scalar multiplication of a matrix 1458.5 Transpose of a matrix 1468.6 Dot product of matrices 146

Introduction 146Condition 148Definition 148Properties 149

8.7 Inverse of a matrix 151Introduction 151Definition 151

8 MULTIMEDIA MATHS

Conditions 152Row reduction 152Matrix inversion 153Inverse of a product 156Solving systems of linear equations 157

8.8 The Fibonacci operator 1598.9 Exercises 161

Chapte r 9 · L inear t rans format ions 1639.1 Translation 1649.2 Scaling 1699.3 Rotation 172

Rotation in 2D 172Rotation in 3D 174

9.4 Reflection 1769.5 Shearing 1789.6 Composing standard transformations 180

2D rotation around an arbitrary center 1833D scaling about an arbitrary center 1852D reflection over an axis through the origin 1862D reflection over an arbitrary axis 1883D combined rotation 191

9.7 Row-representation 1929.8 Exercises 193

Chapte r 10 · Bez i e r cu rve s 19510.1 Vector equation of segments 196

Linear Bezier segment 196Quadratic Bezier segment 197Cubic Bezier segment 198Bezier segments of higher degree 200

10.2 De Casteljau algorithm 20110.3 Bezier curves 202

Concatenation 202Linear transformations 204Illustrations 204

10.4 Matrix representation 206Linear Bezier segment 206Quadratic Bezier segment 207Cubic Bezier segment 208

CONTENT 9

10.5 B-splines 210Cubic B-splines 210Matrix representation 211De Boor’s algorithm 213

10.6 Exercises 215

Annex A · Rea l numb er s in compute r s 217A.1 Scientific notation 217A.2 The decimal computer 217A.3 Special values 218

Annex B · Notat i ons and Convent ions 219B.1 Alphabets 219

Latin alphabet 219Greek alphabet 219

B.2 Mathematical symbols 220Sets 220Mathematical symbols 221Mathematical keywords 221Remarkable numbers 222

Annex C · Compan ion webs i t e 223C.1 Interactivities 223C.2 Answers 223

Bib l i og raphy 224

I ndex 227

Acknowledgments

We hereby insist to thank a lot of people who made this book possible: Prof. Dr. Leo Storme,Wim Serras, Wouter Tansens, Wouter Verweirder, Koen Samyn, Hilde De Maesschalck,Ellen Deketele, Conny Meuris, Hans Ameel, Dr. Rolf Mertig, Dick Verkerck, ir. GoseFischer, Prof. Dr. Fred Simons, Sofie Eeckeman, Dr. Luc Gheysens, Dr. Bavo Langerock,Wauter Leenknecht, Marijn Verspecht, Sarah Rommens, Prof. Dr. Marcus Greferath,Dr. Cornelia Roessing, Tim De Langhe, Niels Janssens, Peter Flynn, Jurgen Leemans,Jan Middendorp, Hilde Vanmechelen, Jef De Langhe, Ann Deraedt, Rita Vanmeirhaeghe,Prof. Dr. Jan Van Geel, Dr. Ann Dumoulin, Bart Uyttenhove, Rik Leenknegt, PeterVerswyvelen, Roel Vandommele, ir. Lode De Geyter, Bart Leenknegt, Olivier Rysman,ir. Johan Gielis, Frederik Jacques, Kristel Balcaen, ir. Wouter Gevaert, Bart Gardin, DieterRoobrouck, Dr. Yu-Sung Chang (WolframDemonstrations), Steven Verborgh, IngridViaene, Thomas Vanhoutte, Fries Carton and anyone whom we might have forgotten!

Chapte r 1 · Ar i thmet i c Re f r e she r

14 MULTIMEDIA MATHS

As this chapter offers all necessary mathematical skills for a full mastering of all furthertopics explained in this book, we strongly recommend it. To serve its purpose, the succes-sive paragraphs below refresh some required aspects of mathematical language as usedon the applied level.

1.1 Algebra

Real Numbers

We typeset the set of:

� natural numbers (unsigned integers) as N including zero,

� integer numbers as Z including zero,

� rational numbers as Q including zero,

� real numbers (floats) as R including zero.

All the above make a chain of subsets: N⊂ Z⊂Q⊂ R.

To avoid possible confusion, we outline a brief glossary of mathematical terms. We recallthat using the correct mathematical terms reflects a correct mathematical thinking. Puttingdown ideas in the correct words is of major importance for a profound insight.

Sets� We recall writing all subsets in between braces, e.g. the empty set appears as {}.

� We define a singleton as any subset containing only one element, e.g. {5} ⊂ N, asa subset of natural numbers.

� We define a pair as any subset containing just two elements, e.g. {115,−4} ⊂ Z,as a subset of integers. In programming the boolean values true and false make upa pair {true, f alse} called the boolean set which we typeset as B.

� We define Z− = {. . . ,−3,−2,−1} whenever we need negative integers only. Weexpress symbolically that −1234 is an element of Z− by typesetting −1234 ∈ Z−.

� We typeset the setminus operator to delete elements from a set by using a back-slash, e.g. N\{0} reading all natural numbers except zero, Q\Z meaning all purerational numbers after all integer values left out and R \ {0,1} expressing all realnumbers apart from zero and one.

ARITHMETIC REFRESHER 15

Calculation basics

operation example a b c

to add a+b = c term term sum

to subtract a−b = c term term difference

to multiply a ·b = c factor factor product

to divide ab = c, b �= 0 divisor quotient

numerator or denominator or fraction

to exponentiate ab = c base exponent power

to take root b√

a = c radicand index radical

We write the opposite of a real number r as −r, defined by the sum r+(−r) = 0. Wetypeset the reciprocal of a nonzero real number r as 1

r or r−1, defined by the productr · r−1 = 1.

We define subtraction as equivalent to adding the opposite: a−b = a+(−b). We definedivision as equivalent to multiplying with the reciprocal: a : b = a ·b−1.

When we mix operations we need to apply priority rules for them. There is a fixed prioritylist ‘PEMDAS’ in performing mixed operations in R that can easily be memorized by‘Please Excuse My Dear Aunt Sally’.

� First process all that is delimited in between Parentheses,

� then Exponentiate,

� then Multiply and Divide from left to right,

� finally Add and Subtract from left to right.

16 MULTIMEDIA MATHS

Now we discuss the distributive law rulingwithin R, which we define as threading a ‘su-perior’ operation over an ‘inferior’ operation.Conclusively, distributing requires two differentoperations.

Hence we distribute exponentiating over multi-plication as in (a ·b)3 = a3 ·b3. Likewise rulesmultiplying over addition as in 3 · (a+b) =3 ·a+3 ·b.

However we should never stumble on this‘Staircase of Distributivity’ by stepping it too fast:

(a+b)3 �= a3 +b3,

√a+b �=

√a+

√b,

√x2 + y2 �= x+ y.

FractionsA fraction is what we call any rational number written as t

n given t,n ∈ Z and n �= 0,wherein t is called the numerator and n the denominator. We define the reciprocal of a

nonzero fraction tn as 1

tn= n

t or as the power(

tn

)−1. We define the opposite fraction as

− tn = −t

n = t−n . We summarize fractional arithmetics:

sum tn +

ab = t·b+n·a

n·b ,

difference tn −

ab = t·b−n·a

n·b ,

product tn ·

ab = t·a

n·b ,

divisiontnab= t

n ·ba ,

exponentiation( t

n

)m= tm

nm ,

singular fractions 10 =±∞ infinity,00 =? indeterminate.

PowersWe define a power as any real number written as gm , wherein g is called its base and mits exponent. The opposite of gm is simply −gm. The reciprocal of gm is 1

gm = g−m, giveng �= 0.

ARITHMETIC REFRESHER 17

According to the exponent type we distinguish between:

g3 = g ·g ·g 3 ∈ N,

g−3 = 1g3 = 1

g·g·g −3 ∈ Z,

g13 = 3

√g = w ⇔ w3 = g 1

3 ∈Q,

g0 = 1 g �= 0.

Whilst calculating powers we may have to:

multiply g3 ·g2 = g3+2 = g5,

divide g3

g2 = g3 ·g−2 = g3−2 = g1,

exponentiate(g3)2

= g3·2 = g6 them.

We insist on avoiding typesetting radicals like 7√

g3 and strongly recommend their con-temporary notation using radicand g and exponent 3

7 , consequently exponentiating g to

g37 . We recall the fact that all square roots are non-negative numbers,

√a = a

12 ∈ R+ for

a ∈ R+.

As well knowing the above exponent types as understanding the above rules to calculatethem are inevitable to use powers successfully. We advise memorizing the integer squaresrunning from 12 = 1, 22 = 4, . . ., up to 152 = 225, 162 = 256 and the integer cubes runningfrom 13 = 1, 23 = 8, . . ., up to 73 = 343, 83 = 512 in order to easily recognize them.

Recall that the only way out of any power is exponentiating with its reciprocal exponent.For this purpose we need to exponentiate both left hand side and right hand side of anygiven relation (see also paragraph 1.2).

Example: Find x when 7√x3 = 5 by exponentiating this power.

x37 = 5 ⇐⇒

(x

37

) 73= (5)

73 ⇐⇒ x ≈ 42.7494.

We emphasize the above strategy as the only successful one to free base x from its expo-nent, yielding its correct expression numerically approximated if we like to.

Example: Find x when x2 = 5 by exponentiating this power.

x2 = 5 ⇐⇒(x2) 1

2 = (5)12 or − (5)

12 ⇐⇒ x ≈ 2.23607 or−2.23607.

We recall the above double solution whenever we free base x from an even exponent,yielding their correct expression as accurate as we like to.

18 MULTIMEDIA MATHS

Mathematical expressionsComposed mathematical expressions can often seem intimidating or cause confusion.To gain transparancy in them, we firstly recall indexed variables which we define assubscripted to count them: x1,x2,x3,x4, . . . ,x99999,x100000, . . ., and α0,α1,α2,α3,α4, . . . .It is common practice in industrial research to use thousands of variables, so just pickingunindexed characters would be insufficient. Taking our own alphabet as an example, itwould only provide us with 26 characters.

We define finite expressions as composed of (math-ematical) operations on objects (numbers, variablesor structures). We can for instance analyze the ex-pression (3a+ x)4 by drawing its tree form. Thisexample reveals a Power having exponent 4 and asubexpression in its base. The base itself yields asum of the variable x Plus another subexpression.This final subexpression shows the product 3 Timesa.

Let us also evaluate this expression (3a+ x)4. Saya = 1, then we see our expression partly collapsto (3+ x)4. If we on top of this assign x = 2, ourexpression then finally turns to the numerical value(3+2)4 = 54 = 625.

When we expand this power to its pure sum expression 81a4 + 108a3x + 54a2x2 +

12ax3 + x4, we did nothing but reshape its pure product expression (3a+ x)4.

We warn that trying to solve this expression - which is not a relation - is completely invain. Recall that inequalities, equations and systems of equations or inequalities are theonly objects in the universe we can (try to) solve mathematically.

Relational operatorsWe also refresh the use of correct terms for inequalities and equations.

We define an inequality as any variable expression comparing a left hand side to a righthand side by applying the ‘is-(strictly)-less-than’ or by applying the ‘is-(strictly)-greater-than’ operator. For example, we can read (3a+ x)4 � (b+4)(x+3) containing variablesa, x, b. Consequently we may solve such inequality for any of the unknown quantities a,xor b.

We define an equation as any variable expression comparing a left hand side to a righthand side by applying the ‘is-equal-to’ operator. For example (3a+ x)4 = (b+4)(x+3)is an equation containing variables a, x, b. Consequently we also may solve equations for

ARITHMETIC REFRESHER 19

any of the unknown quantities a,x or b.

We define an equality as a constant relational expression being true, e.g. 7= 7. We definea contradiction as a constant relational expression being false, e.g. −10 > 5.

Real Polynomials

We elaborate upon the mathematical environment of polynomials over the real numbersin their variable or indeterminate x, a set we denote with R[x].

� Monomials

We define a monomial in x as any product axn, given a ∈ R and n ∈ N. We canextend this concept to several indeterminates x,y,z, . . . like the monomials 3(xy)6

and 3(x2y3z6) are.

We define the degree of a monomial axn as its natural exponent n ∈ N to the inde-terminate part x. We say constant numbers are monomials of degree 0 and linearterms are monomials of degree 1. We say squares to have degree 2 and cubes tohave degree 3, followed by monomials of higher degree.

For instance the real monomial −√

12x6 is of degree 6. Extending this concept, themonomial 3(xy)6 is of degree 6 in xy and the monomial 3(x2y3z6)9 is of degree 9 inx2y3z6.

We define monomials of the same kind as those having an identical indeterminatepart. For instance both 5

7 x6 and −√

12x6 are of the same kind. Extending theconcept, likewise 5

7 x3y5z2 and −√

12x3y5z2 are of the same kind.

All basic operations on monomials emerge simply from applying the calculationrules of fractions and powers.

� Polynomials

We define a polynomial V (x) as any sum of monomials. We define the degree ofV (x) as the maximal exponent m ∈ N to the indeterminate variable x. For instancethe real polynomial

V (x) = 17x2 +14

x3 +6x−7x2 −√

12x6 −13x−1,

is of degree 6.

Whenever monomials of the same kind appear in it, we can simplify the polynomial.For instance our polynomial simplifies to V (x) = 10x2 + 1

4 x3 −7x−√

12x6 −1.

Moreover, we can sort any given polynomial either in an ascending or descendingway according to its powers in x. Sorting our polynomial V (x) in an ascending way

20 MULTIMEDIA MATHS

yields V (x) = −1− 7x+ 10x2 + 14 x3 −

√12x6. Sorting V (x) in a descending way

yields V (x) =−√

12x6 + 14 x3 +10x2 −7x−1.

Eventually we are able to evaluate any polynomial, getting a numerical value fromit. For instance evaluating V (x) in x=−1, yields V (−1)=−

√12(−1)6+ 1

4 (−1)3+

10(−1)2 −7(−1)−1 =−√

12− 14 +16 = 63

4 −2√

3 ∈ R.

� Basic operations

Adding two monomials of the same kind: we add their coefficients and keep theirindeterminate part

5a2 −3a2 = (5−3)a2 = 2a2.

Multiplying two monomials of any kind: we multiply both their coefficients andtheir indeterminate parts

−5ab · 74

a2b3 =−5 · 74·a1+2b1+3 =

−354

a3b4.

Dividing two monomials: we divide both their coefficients and their indeterminateparts

−8a6b4

−4a4 =−8−4

a6−4b4−0 = 2a2b4.

Exponentiating a monomial: we exponentiate each and every factor in the mono-mial (

−2a2b4)3= (−2)3(a2)3(b4)3 =−8a6b12.

Adding or subtracting polynomials: we add or subtract all monomials of the samekind

(x2 −4x+8)− (2x2 −3x−1) = x2 −4x+8−2x2 +3x+1 =−x2 − x+9.

Multiplying two polynomials: we multiply each monomial of the first polynomialwith each monomial of the second polynomial and simplify all those products tothe resulting product polynomial

(2x2 +3y) · (4x2 − y) = 2x2(4x2 − y)+3y(4x2 − y)

= 2x2 ·4x2 +2x2 · (−y)+3y ·4x2

+3y · (−y)

= 8x4 − 2x2y + 12x2y −3y2

= 8x4 +10x2y−3y2.

ARITHMETIC REFRESHER 21

1.2 Equations in one variable

Anticipating this paragraph we refresh some vocabulary for it. A solution is any valueassigned to the variable that turns the given equation into an equality (being true). Thescope of an equation is any number set in which the equation resides, realizing it will bemost likely R. We define the solution set as the set containing all legal solutions to anequation. This solution set always is a subset of the scope of the equation.

Linear Equat ions

A linear equation is an algebraic equation of degree one, referring to the maximumnatural exponent of the unknown quantity. By simplifying we can always standardize anylinear equation to

ax+b = 0, (1.1)

given a∈R\{0} and b∈R. We cite 3x+7= 22,5x−9d = c and 5(x−4)+x=−2(x+2)as examples of linear equations, and 3x2 +7 = 22 and 5ab−9b = c as counterexamples.The adjective ‘linear’ originates from the Latin word ‘linea’ meaning (straight) line asreferring to the graph of a linear function (see chapter 4).

We solve a linear equation for its unknown part by rewriting the entire equation until itsshape exposes the solution explicitly.

We recall easily the required rules for rewriting alinear equation by the metaphor denoting a linearequation as a ‘pair of scales’. This way we shouldneverforget tokeeptheequation’sbalance: whateveroperation we apply, it has to act on both sides of theequals-sign. If we add (or subtract) to the left hand‘scale’ than we are obliged to add (or subtract) thesame term to the right hand ‘scale’. If we multiply(or divide) the left hand side, than we are likewiseobliged to multiply (or divide) the right hand side with the same factor. If not, our equationwould loose its balance just like a pair of scales would. We realize that our metaphor coversall usual ‘rules’ to handle linear equations.

The reason we perform certain rewrite steps depends on which variable we are aiming for.This is called strategy. Solving the equation for a different variable implies a differentsequence of rewrite steps.

22 MULTIMEDIA MATHS

Example: We solve the equation 5(x− 4)+ x = −2(x+ 2) for x. Firstly, we apply thedistributive law: 5x− 20+ x = −2x− 4. Secondly, we put all terms dependent of x tothe left hand side and the constant numbers to the right hand side 5x+ x+2x =−4+20.Thirdly, we simplify both sides 8x = 16. Finally, we find x = 2 leading to the solutionsingleton {2}.

Quadrat ic Equat ions

Handling quadratic expressions and solving quadratic equations are useful basics in orderto study topics in multimedia, digital art and technology.

� Expanding products

We refresh expanding a product as (repeatedly) applying the distributive law untilthe initial expression ends up as a pure sum of terms. Note that our given polyno-mial V (x) itself does not change: we just shift its appearance to a pure sum. Weillustrate this concept through V (x) = (2x−3)(4− x).

(2x−3)(4− x) = (2x−3) ·4+(2x−3) · (−x)

= (8x−12)+(−2x2 +3x)

=−2x2 +11x?12.

Other examples are

5a(2a2 −3b) = 5a ·2a2 −5a ·3b = 10a3 −15ab

and

4(

x− 12

)(x+

132

)= (4x−2)

(x+

132

)

= (4x−2) · x+(4x−2) · 132

= 4x2 −2x+26x−13 = 4x2 +24x−13.

� Factoring polynomials

We define factoring a polynomial as decomposing it into a pure product of (asmany as possible) factors. Note that our given polynomial V (x) itself does notchange: we just shift its appearance to a pure product. Our trinomial V (x) =−2x2+11x−12 just shifts its appearance to the pure product V (x)= (2x−3)(4−x)when factored. It merely shows that the product (2x− 3)(4− x) is a factorizationof the trinomial −2x2 +11x−12.