Rovenskii - Modeling of Curves and Surfaces with MATLAB

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Part 1
Functions and Transformations

Vladimir Rovenski Springer Undergraduate Texts in Mathematics and Technology Modeling of Curves and Surfaces with MATLAB 10.1007/978-0-387-71278-9_1 Springer Science+Business Media, LLC 2010

1. Functions and Graphs

Vladimir Rovenski 1

(1)

Department of Mathematics and Computer Science, University of Haifa, Mount Carmel, Haifa, 31905, Israel

Vladimir Rovenski

Email:

Abstract

Section 1.1 discusses the geometry of (integer, real, complex, and quaternion) numbers. and the necessary notations from algebra. In Section 1.2 we plot graphs of some elementary, special, and piecewise functions, and investigate functions using derivatives. In Section 1.3 we study piecewise functions of one variable that are defined by several formulae for different values (intervals) of that variable. Section 1.4 is an excursion to into remarkable curves (graphs) in polar coordinates. In Sections 1.5, and 1.6 we use several MATLAB functions that implement various interpolation and approximation algorithms. Chapter 1 can be considered as the introduction to MATLAB symbolic/numeric calculations, programming, and basic graphing as needed in later chapters.

1.1 Numbers

When MATLAB starts, its desktop opens with the Menu window, the Command window (where the special > > prompt appears), the Workspace window, the Command History window, and the Current Directory window.

Throughout this book we usually type/edit the code in the Editor window, then place it (copypaste) in the Command window, and execute. To save space, we present code and MATLAB answers in compact form (not identical to the view on the display).

MATLAB can be used in two distinct modes (see Example, p. ):

• it offers immediate execution of statements in the Command, or
• it also offers programming by means of script and function M-files.

A script M-file collects a sequence of commands that constitute a program. It is executed when one enters the name of the script M-file.

We type % to designate a group of words (in a line) as a comment. The M-files created by ourselves we type with a bold font. The reader should place them in the MATLAB Current Directory,say D:/work ; one may choose it manually or type in the Command window cd d : / work

Integers, rationals, and reals

Let be integers , and be natural numbers . The following table contains some scalar and array arithmetic in MATLAB :

Example 1.1.

Type 2+3 after the > > prompt, followed by Enter (i.e., press the Enter key). Next try the following: 2*3, 42, 1/2, 2 1

N	Symbol	Operation	MATLAB
	+ and	addition/subtraction	a + b - c
	and.	multiplication	a * b
	and.	right division	a / b
	and.	left division	b / a
	and.	exponentiation	a b

Try the commands x = [1, 2, 3]

x.2 % a dot in front of *, /, is used when we work with arrays.

ans = 1 4 9 % next we present MATLAB answers in a compact form.

Compound numbers of the form n 2 are called squares for obvious reasons. The triangular numbers are , etc.

The MATLAB command for allows a statement or a group of statements to be repeated. One may type two lines in the Command window: for n = 1 : 10; x(n) = n2; end; % squares x % Answer: x = 1 4 9 16 25 36 49 64 81 100

Alternatively, we write a program my_ squares.m with one line above in the file edit window. For practice, create an M-file my_ triangles.m : for n = 1 : 10; t(n) = n*(n + 1)/2; end; % triangle numbers

To run this script (after saving my_ triangles.m in the Current Directory), go to the MATLAB Command window and enter two commands: my_ triangles ;

t % Answer: t = 1 3 6 10 15 21 28 36 45 55

( MATLAB executes the instructions in the order in which the commands are stored in the my_ triangles.m file).

If , j 0, and i = kj for some integer k , then we say that jdividesi and that j is a divisor of i . We write j i . From the Euclidean algorithm for integers there follows: if i , j are integers that have a greatest common divisord , then there exist integers s, t such that s i + t j = d .

Two integers i , j are said to be congruent modulo m if m divides i j . In that case we shall write i j (mod m). For example, mod(8, 3) % 8 2 (mod 3). Answer: 2

Modular arithmetic. A set of all remainders of any integer divided by n is denoted by , where . The operations of addition and multiplication modulo n , n, n: , are defined naturally: a n b = a + b (mod n), a n b = a b (mod n ).

For example, 5 7 6 = 4, 0 n ( 1) = n 1, and 3 6 7 = 4, 3 6 4 = 0. mod(5 + 6, 7), mod(3 * 7, 6), mod(3 * 4, 6) % Answer: 4, 3 and 0

Note that is a cyclic group C n (see the definition in Section 2.4).

Definition 1.1.

A skew field is a triple ( K , +, ), where

(1) ( K , +) is an abelian group (with identity denoted as 0 ),

(2) ( K {0}, ) is a group,

(3) multiplication is left and right distributive over + .

A skew field in which multiplication is commutative is called a field .

Rational numbers form a field. If p is a prime, then is a field.

The fundamental theorem of arithmetic reads that

Every integer n 2 can be factorized into a product of primes in only one way apart from the order of the factors.

For example, 2 0 0 9 = 7 2 4 1, 2 0 1 0 = 2 3 5 6 7 ,

factor(2009), factor(2010) % Answer: 7 7 4 1 and 2 3 5 6 7

If a , b are members of a set A then (instead of a = b ) we write a b and say that this is an equivalence relation

if it satisfies the following three properties: (1) reflexivity : always a a ; (2) symmetry : if a b then b a ; (3) transitivity : if a b and b c then a c .

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