CAMPUS HANDBOOK

CAMPUS HANDBOEK

Praktische theologie.indd 3

ANNEMIE DILLEN EN STEFAN GÃ„RTNER

Praktische theologie VERKENNINGEN AAN DE GRENS

27/04/15 20:12

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

Ac k n ow l e d g m e n t s

15

Chapter 1 · Arithmetic Refresher 1.1

1.2

1.3

17

Algebra Real Numbers Real Polynomials Equations in one variable Linear Equations Quadratic Equations Exercises

18 18 23 25 25 26 32

C h a p t e r 2 · L i n e ar s y s t e m s 2.1 2.2

2.3

35

Definitions Methods for solving linear systems Solving by substitution Solving by elimination Exercises

36 38 38 39 43

C h a p t e r 3 · Tr i g o n o m e t r y 3.1 3.2 3.3 3.4 3.5

3.6 3.7 3.8 3.9

Angles Triangles Right Triangle Unit Circle Special Angles Trigonometric ratios for an angle of 45°= Trigonometric ratios for an angle of 30°= Trigonometric ratios for an angle of 60°= Overview Pairs of Angles Sum Identities Inverse trigonometric functions Exercises

45

π 4 π 6 π 3

rad rad rad

46 48 52 53 55 56 56 57 57 58 58 61 63

6

A N I M AT I O N M AT H S

C h a p t e r 4 · Fu n c t i o n s 4.1 4.2

4.3 4.4

4.5 4.6

Basic concepts on real functions Polynomial functions Linear functions Quadratic functions Intersecting functions Trigonometric functions Elementary sine function General sine function Transversal oscillations Inverse trigonometric functions Exercises

Chapter 5 · The Golden Section 5.1 5.2

5.3

5.4 5.5

The Golden Number The Golden Section The Golden Triangle The Golden Rectangle The Golden Spiral The Golden Pentagon The Golden Ellipse Golden arithmetics Golden Identities The Fibonacci Numbers The Golden Section worldwide Exercises

C h a p t e r 6 · C o or d i n a t e s y s t e m s 6.1 6.2 6.3 6.4

6.5

Cartesian coordinates Parametric curves Polar coordinates Polar curves A polar superformula Exercises

65 66 67 67 69 71 73 73 73 77 77 80

83 84 86 86 87 88 90 90 91 91 92 94 97

99 100 100 103 106 107 109

CONTENT

C h a p t e r 7 · Ve c t or s 7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8 7.9

The concept of a vector Vectors as arrows Vectors as arrays Free Vectors Base Vectors Addition of vectors Vectors as arrows Vectors as arrays Vector addition summarized Scalar multiplication of vectors Vectors as arrows Vectors as arrays Scalar multiplication summarized Properties Vector subtraction Creating free vectors Euler’s method for trajectories Decomposition of vectors Decomposition of a plane vector Base vectors defined Dot product Definition Geometric interpretation Orthogonality Cross product Definition Geometric interpretation Parallelism Normal vectors Exercises

C h a p t e r 8 · Par a m e t e r s 8.1 8.2 8.3 8.4 8.5

Parametric equations Vector equation of a line Intersecting straight lines Vector equation of a plane Exercises

7

111 112 112 113 116 116 117 117 117 118 119 119 119 120 120 121 121 122 123 123 124 124 124 126 128 129 129 132 133 135 137

139 140 141 145 147 151

8

A N I M AT I O N M AT H S

Chapter 9 Âˇ Kinematics 9.1

9.2 9.3

9.4

9.5

9.6

9.7

Measures Precision Units Deltatime Translational motion Rectilinear motion with constant velocity (RMCV) Rectilinear motion with constant acceleration (RMCA) Free Fall Summary Circular motion Uniform circular motion (UCM) Nonuniform circular motion (NCM) Summary Planar Curvilinear Motion Normal-tangential components Radial-angular components Independence of Motion Combined rectilinear motions with constant velocity Projectile motion (PM) Exercises

C h a p t e r 10 Âˇ C o l l i s i o n d e t e c t i o n

10.1 Collision detection using circles and spheres Circles and spheres Intersecting line and circle Intersecting circles and spheres 10.2 Collision detection using vectors Location of a point with respect to other points Altitude to a straight line Altitude to a plane Frame rate issues Location of a point with respect to a polygon 10.3 Exercises

153 154 154 154 155 155 158 158 161 164 166 166 173 176 177 178 181 184 184 185 190

193 194 194 196 198 201 201 202 204 206 207 210

CONTENT

C h a p t e r 11 Âˇ M a t r i c e s 11.1 11.2 11.3 11.4 11.5 11.6

The concept of a matrix Determinant of a square matrix Addition of matrices Scalar multiplication of a matrix Transpose of a matrix Dot product of matrices Introduction Condition Definition Properties 11.7 Inverse of a matrix Introduction Definition Conditions Row reduction Matrix inversion Inverse of a product Solving systems of linear equations 11.8 The Fibonacci operator 11.9 Exercises

C h a p t e r 12 Âˇ L i n e ar t r a n s f or m a t i o n s 12.1 Translation 12.2 Scaling 12.3 Rotation Rotation in 2D Rotation in 3D 12.4 Reflection 12.5 Shearing 12.6 Composing basic transformations 2D rotation around an arbitrary center 3D scaling about an arbitrary center 2D reflection over an axis through the origin 2D reflection over an arbitrary axis 3D combined rotation 12.7 Conventions 12.8 Exercises

9

213 214 215 217 219 220 220 220 222 222 223 225 225 225 226 226 227 230 231 233 235

237 238 243 246 246 248 250 251 254 256 259 260 261 264 265 266

10

A N I M AT I O N M AT H S

C h a p t e r 13 · H y p e r c o m p l e x n u m b e r s

13.1 Complex numbers 13.2 Complex number arithmetics Complex conjugate Addition and subtraction Multiplication Division 13.3 Complex numbers and transformations 13.4 Complex continuation of the Fibonacci numbers Integer Fibonacci numbers Complex Fibonacci numbers 13.5 Quaternions 13.6 Quaternion arithmetics Addition and subtraction Multiplication Quaternion conjugate Inverse quaternion 13.7 Quaternions and rotation 13.8 Exercises

C h a p t e r 14 · B e z i e r c u r v e s

14.1 Vector equation of segments Linear Bezier segment Quadratic Bezier segment Cubic Bezier segment Bezier segments of higher degree 14.2 De Casteljau algorithm 14.3 Bezier curves Concatenation Linear transformations Illustrations 14.4 Matrix representation Linear Bezier segment Quadratic Bezier segment Cubic Bezier segment 14.5 B-splines Cubic B-splines Matrix representation De Boor’s algorithm 14.6 Exercises

269 270 273 273 274 275 277 279 281 281 282 283 284 285 285 287 288 288 293

295 296 296 297 298 300 301 302 302 304 304 306 306 307 308 310 310 311 313 315

CONTENT

11

Annex A 路 Real numbers in computers A.1 Scientific notation A.2 The decimal computer A.3 Special values

Annex B 路 Notations and Conventions B.1 Alphabets Latin alphabet Greek alphabet B.2 Mathematical symbols Sets Mathematical symbols Mathematical keywords Numbers

Annex C 路 The International System of Units (SI) C.1 C.2 C.3 C.4

SI SI SI SI

Prefixes Base measures Supplementary measure Derived measures

A n n e x D 路 C o m p a n i o n we b s i t e D.1 Interactivities D.2 Solutions

317 317 317 318

319 319 319 319 320 320 321 321 322

323 323 324 324 325

327 327 327

Bibliography

328

Index

331

This book is dedicated to Malaika.

“Sometimes I’m black, sometimes I’m white it all depends on who is on the other side there are things they can not see and 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. 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. 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, Peter Verswyvelen, 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. Felipe Dimer de Oliveira (WolframDemonstrations), Steven Verborgh, Ingrid Viaene, Kayla Chauveau, Angelika Kirkorova, Thomas Vanhoutte, Fries Carton and anyone whom we might have forgotten!

Chapter 1 Â· Arithmetic Refresher

18

A N I M AT I O N M AT H S

As this chapter offers all necessary mathematical skills for a full mastering of all further topics explained in this book, we strongly recommend it. To serve its purpose, the successive paragraphs below refresh some required aspects of mathematical language as used on 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 recall that using the correct mathematical terms reflects a correct mathematical thinking. Putting down 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, as a 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 up a 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. We express 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 backslash, e.g. N \ {0} reading all natural numbers except zero, Q \ Z meaning all pure rational numbers after all integer values left out and R \ {0, 1} expressing all real numbers 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

factor

factor

product

numerator

divisor or denominator

quotient or fraction

=c

base

exponent

power

a=c

radicand

index

radical

to multiply to divide to exponentiate to take root

a·b = c a b

= c, b = 0

ab √ b

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

We define subtraction as equivalent to adding the opposite: a − b = a + (−b). We define division 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 priority list ‘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

A N I M AT I O N M AT H S

Now we discuss the distributive law ruling within R, which we define as threading a â€˜superiorâ€™ operation over an â€˜inferiorâ€™ operation. Conclusively, distributing requires two different operations. Hence we distribute exponentiating over multiplication as in (a Âˇ b)3 = a3 Âˇ b3 . Likewise rules multiplying over addition as in 3 Âˇ (a + b) = 3 Âˇ a + 3 Âˇ b. However we should never stumble on this â€˜Stairway of Distributivityâ€™ by going too fast: (a + b)3 = a3 + b3 , âˆš âˆš âˆš a + b = a + b, x2 + y2 = x + y.

Fractions

A fraction is what we call any rational number written as nt given t, n âˆˆ Z and n = 0, wherein t is called the numerator and n the denominator. We define the reciprocal of a âˆ’1 . We define the opposite fraction as nonzero fraction nt as 1t = nt or as the power nt âˆ’ nt =

âˆ’t n

=

t âˆ’n .

n

We summarize fractional arithmetics: sum

t n

+ ab =

tÂˇb+nÂˇa nÂˇb ,

difference

t n

tÂˇbâˆ’nÂˇa nÂˇb ,

product

t n

âˆ’ ab =

division

t n a b

exponentiation singular fractions

Âˇ ab =

tÂˇa nÂˇb ,

= nt Âˇ ba , t m t m = nm , n 1 0 0 0

= Âąâˆž infinity,

=? indeterminate.

Powers We define a power as any real number written as gm , wherein g is called its base and m its exponent. The opposite of gm is simply âˆ’gm . The reciprocal of gm is g1m = gâˆ’m , given g = 0.

ARITHMETIC REFRESHER

21

According to the exponent type we distinguish between: g3 = g Âˇ g Âˇ g

3 âˆˆ N,

1 gâˆ’3 = g13 = gÂˇgÂˇg 1 âˆš g 3 = 3 g = w â‡” w3 = g

âˆ’3 âˆˆ Z, 1 3

g0 = 1

Whilst calculating powers we may have to: multiply

âˆˆ Q,

g = 0.

g3 Âˇ g2 = g3+2 = g5 , g3 g2

= g3 Âˇ gâˆ’2 = g3âˆ’2 = g1 , 2 exponentiate g3 = g3Âˇ2 = g6 them. divide

We insist on avoiding typesetting radicals like 7 g3 and strongly recommend their contemporary notation using radicand g and exponent 37 , consequently exponentiating g to âˆš 3 1 g 7 . We recall the fact that all square roots are non-negative numbers, a = a 2 âˆˆ R+ for a âˆˆ R+ . As well knowing the above exponent types as understanding the above rules to calculate them are inevitable to use powers successfully. We advise memorizing the integer squares running from 12 = 1, 22 = 4, . . ., up to 152 = 225, 162 = 256 and the integer cubes running from 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 any given relation (see also paragraph 1.2). âˆš 7 Example: Find x when x3 = 5 by exponentiating this power. 3 7 7 3 3 x 7 = 5 â‡?â‡’ x 7 = (5) 3 â‡?â‡’ x â‰ˆ 42.7494. We emphasize the above strategy as the only successful one to free base x from its exponent, yielding its correct expression numerically approximated if we like to. Example: Find x when x2 = 5 by exponentiating this power. 1 1 1 x2 = 5 â‡?â‡’ x2 2 = (5) 2 or âˆ’ (5) 2 â‡?â‡’ x â‰ˆ 2.23607 or âˆ’ 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

A N I M AT I O N M AT H S

Mathematical expressions Composed mathematical expressions can often seem intimidating or cause confusion. To gain transparancy in them, we firstly recall indexed variables which we define as subscripted 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 picking unindexed characters would be insufficient. Taking our own alphabet as an example, it would only provide us with 26 characters. We define finite expressions as composed of (mathematical) operations on objects (numbers, variables or structures). We can for instance analyze the expression (3a + x)4 by drawing its tree form. This example reveals a Power having exponent 4 and a subexpression in its base. The base itself yields a sum of the variable x Plus another subexpression. This final subexpression shows the product 3 Times a. Let us also evaluate this expression (3a + x)4 . Say a = 1, then we see our expression partly collaps to (3 + x)4 . If we on top of this assign x = 2, our expression then finally turns to the numerical value (3 + 2)4 = 54 = 625. When we expand this power to its pure sum expression 81a4 + 108a3 x + 54a2 x2 + 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 in vain. Recall that inequalities, equations and systems of equations or inequalities are the only objects in the universe we can (try to) solve mathematically. Relational operators We 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 right hand side by applying the ‘is-(strictly)-less-than’ or by applying the ‘is-(strictly)-greaterthan’ operator. For example, we can read (3a + x)4 (b + 4)(x + 3) containing variables a, x, b. Consequently we may solve such inequality for any of the unknown quantities a, x or b. We define an equation as any variable expression comparing a left hand side to a right hand 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