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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, Martin Walls; 7, 14, Wouter Tansens; 8, Danie Pratt; 9, Ivo De Pauw; 10, Caetano Lacerda; 11, Ken Munyard;

12, Bieke Masselis; 13, Cornelia Roessing; p.25, p.95, Wouter Tansens; p.46, Wouter Verweirder; p.50, Leo Storme;

p.175, Bieke Masselis; p.314, Yu-Sung Chang.

D/2015/45/483 – ISBN 978 94 014 3204 7 – NUR 918

Layout: Jurgen Leemans, Peter Flynn and Bavo Langerock

Cover design: Stef Lantsoght, Keppie en 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 publiser.

Publisher LannooCampus

Erasme Ruelensvest 179 bus 101

B - 3001 Leuven

Belgium

www.lannoocampus.com.

Content

Acknow ledgment s 15

Chapte r 1 · Ar i thmet i c Re f r e she r 171.1 Algebra 18

Real Numbers 18Real Polynomials 23

1.2 Equations in one variable 25Linear Equations 25Quadratic Equations 26

1.3 Exercises 32

Chapte r 2 · L inear sy s t ems 352.1 Definitions 362.2 Methods for solving linear systems 38

Solving by substitution 38Solving by elimination 39

2.3 Exercises 43

Chapte r 3 · Tr igonomet ry 453.1 Angles 463.2 Triangles 483.3 Right Triangle 523.4 Unit Circle 533.5 Special Angles 55

Trigonometric ratios for an angle of 45°= π4 rad 56

Trigonometric ratios for an angle of 30°= π6 rad 56

Trigonometric ratios for an angle of 60°= π3 rad 57

Overview 573.6 Pairs of Angles 583.7 Sum Identities 583.8 Inverse trigonometric functions 613.9 Exercises 63

6 ANIMATION MATHS

Chapte r 4 · Funct i ons 654.1 Basic concepts on real functions 664.2 Polynomial functions 67

Linear functions 67Quadratic functions 69

4.3 Intersecting functions 714.4 Trigonometric functions 73

Elementary sine function 73General sine function 73Transversal oscillations 77

4.5 Inverse trigonometric functions 774.6 Exercises 80

Chapte r 5 · The Go lden Sec t i on 835.1 The Golden Number 845.2 The Golden Section 86

The Golden Triangle 86The Golden Rectangle 87The Golden Spiral 88The Golden Pentagon 90The Golden Ellipse 90

5.3 Golden arithmetics 91Golden Identities 91The Fibonacci Numbers 92

5.4 The Golden Section worldwide 945.5 Exercises 97

Chapte r 6 · Co ord ina te sy s t ems 996.1 Cartesian coordinates 1006.2 Parametric curves 1006.3 Polar coordinates 1036.4 Polar curves 106

A polar superformula 1076.5 Exercises 109

CONTENT 7

Chapte r 7 · Vector s 1117.1 The concept of a vector 112

Vectors as arrows 112Vectors as arrays 113Free Vectors 116Base Vectors 116

7.2 Addition of vectors 117Vectors as arrows 117Vectors as arrays 117Vector addition summarized 118

7.3 Scalar multiplication of vectors 119Vectors as arrows 119Vectors as arrays 119Scalar multiplication summarized 120Properties 120

7.4 Vector subtraction 121Creating free vectors 121Euler’s method for trajectories 122

7.5 Decomposition of vectors 123Decomposition of a plane vector 123Base vectors defined 124

7.6 Dot product 124Definition 124Geometric interpretation 126Orthogonality 128

7.7 Cross product 129Definition 129Geometric interpretation 132Parallelism 133

7.8 Normal vectors 1357.9 Exercises 137

Chapte r 8 · Paramete r s 1398.1 Parametric equations 1408.2 Vector equation of a line 1418.3 Intersecting straight lines 1458.4 Vector equation of a plane 1478.5 Exercises 151

8 ANIMATION MATHS

Chapte r 9 · Kinemat i c s 1539.1 Measures 154

Precision 154Units 154

9.2 Deltatime 1559.3 Translational motion 155

Rectilinear motion with constant velocity (RMCV) 158Rectilinear motion with constant acceleration (RMCA) 158Free Fall 161Summary 164

9.4 Circular motion 166Uniform circular motion (UCM) 166Nonuniform circular motion (NCM) 173Summary 176

9.5 Planar Curvilinear Motion 177Normal-tangential components 178Radial-angular components 181

9.6 Independence of Motion 184Combined rectilinear motions with constant velocity 184Projectile motion (PM) 185

9.7 Exercises 190

Chapte r 10 · Co l l i s i on de tec t i on 19310.1 Collision detection using circles and spheres 194

Circles and spheres 194Intersecting line and circle 196Intersecting circles and spheres 198

10.2 Collision detection using vectors 201Location of a point with respect to other points 201Altitude to a straight line 202Altitude to a plane 204Frame rate issues 206Location of a point with respect to a polygon 207

10.3 Exercises 210

CONTENT 9

Chapte r 11 · Matr i ce s 21311.1 The concept of a matrix 21411.2 Determinant of a square matrix 21511.3 Addition of matrices 21711.4 Scalar multiplication of a matrix 21911.5 Transpose of a matrix 22011.6 Dot product of matrices 220

Introduction 220Condition 222Definition 222Properties 223

11.7 Inverse of a matrix 225Introduction 225Definition 225Conditions 226Row reduction 226Matrix inversion 227Inverse of a product 230Solving systems of linear equations 231

11.8 The Fibonacci operator 23311.9 Exercises 235

Chapte r 12 · L inear t rans format ions 23712.1 Translation 23812.2 Scaling 24312.3 Rotation 246

Rotation in 2D 246Rotation in 3D 248

12.4 Reflection 25012.5 Shearing 25112.6 Composing basic transformations 254

2D rotation around an arbitrary center 2563D scaling about an arbitrary center 2592D reflection over an axis through the origin 2602D reflection over an arbitrary axis 2613D combined rotation 264

12.7 Conventions 26512.8 Exercises 266

10 ANIMATION MATHS

Chapte r 13 · Hyp e rcomp lex numb er s 26913.1 Complex numbers 27013.2 Complex number arithmetics 273

Complex conjugate 273Addition and subtraction 274Multiplication 275Division 277

13.3 Complex numbers and transformations 27913.4 Complex continuation of the Fibonacci numbers 281

Integer Fibonacci numbers 281Complex Fibonacci numbers 282

13.5 Quaternions 28313.6 Quaternion arithmetics 284

Addition and subtraction 285Multiplication 285Quaternion conjugate 287Inverse quaternion 288

13.7 Quaternions and rotation 28813.8 Exercises 293

Chapte r 14 · Bez i e r cu rve s 29514.1 Vector equation of segments 296

Linear Bezier segment 296Quadratic Bezier segment 297Cubic Bezier segment 298Bezier segments of higher degree 300

14.2 De Casteljau algorithm 30114.3 Bezier curves 302

Concatenation 302Linear transformations 304Illustrations 304

14.4 Matrix representation 306Linear Bezier segment 306Quadratic Bezier segment 307Cubic Bezier segment 308

14.5 B-splines 310Cubic B-splines 310Matrix representation 311De Boor’s algorithm 313

14.6 Exercises 315

CONTENT 11

Annex A · Rea l numb er s in compute r s 317A.1 Scientific notation 317A.2 The decimal computer 317A.3 Special values 318

Annex B · Notat i ons and Convent ions 319B.1 Alphabets 319

Latin alphabet 319Greek alphabet 319

B.2 Mathematical symbols 320Sets 320Mathematical symbols 321Mathematical keywords 321Numbers 322

Annex C · The In te rna t i ona l Sys tem of Un i t s (S I ) 323C.1 SI Prefixes 323C.2 SI Base measures 324C.3 SI Supplementary measure 324C.4 SI Derived measures 325

Annex D · Compan ion webs i t e 327D.1 Interactivities 327D.2 Solutions 327

Bib l i og raphy 328

I ndex 331

This book is dedicated to Malaika.

“Sometimes I’m black, sometimes I’m whiteit all depends on who is on the other sidethere are things they can not seeand there are things I can not hide”

Bruno Deneckere (Someday, June 2006)

Acknowledgments

We hereby insist to thank a lot of people who made this book possible: Prof. Dr. LeoStorme, Wim Serras, Wouter Tansens, Wouter Verweirder, Koen Samyn, Hilde DeMaesschalck, Ellen Deketele, Conny Meuris, Hans Ameel, Dr. Rolf Mertig, DickVerkerck, ir. Gose Fischer, 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, 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,Dieter Roobrouck, Dr. Yu-Sung Chang (WolframDemonstrations), Prof. Dr. Sy Blinder(WolframDemonstrations), Prof. Dr. Mark McClure (WolframDemonstrations), Dr. FelipeDimer de Oliveira (WolframDemonstrations), Steven Verborgh, Ingrid Viaene, KaylaChauveau, Angelika Kirkorova, Thomas Vanhoutte, Fries Carton and anyone whom wemight have forgotten!

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

18 ANIMATION 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 19

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.

20 ANIMATION 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‘Stairway of Distributivity’ by going 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 21

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.

22 ANIMATION 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