Algebraic Geodesy and Geoinformatics - 2009 PART I - METHODS

1 Basics of Numbers

1-1 Integers

When undertaking Geodetic and Geoinformatics operations such as measurements of angles, distances, gravity, photo

coordinates, digitizing of points etc., numbers are often used.

To circumvent the failure of the natural numbers to be closed under subtraction, negative numbers were introduced where

they were added in front of a natural number n i.e. - n. This expanded set is the set of integers. The letter Z is adopted from

the first letter of the German word for integers �Zahl�,

Clear@"Global‘*"D

n1 = 89; n2 = 123;

Operations,

n3 = n1 + n2; n4 = n1 - n2; n5 = n1 n2;

One can check the type of a variable, x with function Head[x]

Map@Head@ðD &, 8n3, n4, n5<D

8Integer, Integer, Integer<

1- 2 Rationals

Incorporating all products of integers and multiplicative inverses m × 1

n =

m

n which are ratios of integers resulting into a set

of rational numbers Q. Rational numbers always consist of a ratio of two integers. The set Q is suitable as it permits addition,

subtraction, multiplication and division.

nr = 67 � 49

67

49

Head@nrD

Rational

1- 3 Reals

The set of quotient Q is enlarged to the set of real numbers R where the positive real numbers are the ones required to

measure distances as it shall be seen later. Negative real numbers are included to provide additive inverses. Approximate real

numbers are distinguished by the presence of an explicit decimal point, can have any number of digits.

nar = N@nr, 25D

1.367346938775510204081633

Head@narD

Real

The set R is also closed under multiplicative inverses and enables the solution of equations of the form

y2 - 3 = 0 ® y = – 3 which is neither integer nor rational. Solving equation

eq = y2 - 3 � 0;

we get

Solve@eq, yD

::y ® - 3 >, :y ® 3 >>

From computational point of view 3 is not a number but a symbol, however operations can be carried out

3

3

3

but using decimal point

HeadB 3. F

Real

Real number can be approximated by rationals within a given tolerance

RationalizeB 3 , 10-6F

1351

780

1- 4 Complexes

The set R is however not large enough to provide a solution to equations of the form y2 + 1 = 0 and gives way to the set C of

complex number where ä2 = -1. The solutions of this equation are complexes

eq = y2 + 1 � 0;

Solve@eq, yD

88y ® -ä<, 8y ® ä<<

Head@äD

Complex

A complex number Α is said to be algebraic if there exists an algebraic polynomial P, such that P(Α) = 0. Algebraic numbers form a subfield of C.

2 Basics_of_Numbers_01.nb

A complex number Α is said to be algebraic if there exists an algebraic polynomial P, such that P(Α) = 0. Algebraic numbers form a subfield of C.

1- 5 Quaternions

The set C can be developed further into a set H of quaternions, which were discovered by Hamilton. In Geodesy, we can

apply the quaternions to solve the three-dimensional resection problem. Quaternions can also be employed to solve the

similarity transformation problem in Geoinformatics. Quaternions are extension of complex numbers and can be defined as

matrices,

h = K a + d ä b + c ä

-b + c ä a - d äO;

The real representation of h matrix is h = a L + b i + c J + d K, where

L = K 1 0

0 1O ; i = K 0 1

-1 0O ; J = K 0 ä

ä 0O; K = K ä 0

0 -äO;

indeed

h - Ha L + b i + c J + d KL �� MatrixForm

K 0 0

0 0O

The followings are true

i.i � J.J � K.K � -L.L � i.J.K � -IdentityMatrix@2D

True

The norm of the quaternion h is the determinant of the matrix h

Det@hD

a2 + b2 + c2 + d2

and the norm of L, i, J and K

Map@Det@ðD &, 8L, i, J, K<D

81, 1, 1, 1<

Multiplication of the quaternions is possible via matrix multiplication

MatrixForm@h1 = 81, 2, 3, 4<.8L, i, J, K<D

K 1 + 4 ä 2 + 3 ä

-2 + 3 ä 1 - 4 äO

MatrixForm@h2 = 85, 6, 7, 8<.8L, i, J, K<D

K 5 + 8 ä 6 + 7 ä

-6 + 7 ä 5 - 8 äO

MatrixForm@h12 = h1. h2D

K -60 + 24 ä 12 + 30 ä

-12 + 30 ä -60 - 24 äO

Therefore it is not surprising that this operation is not commutative,

Basics_of_Numbers_01.nb 3

MatrixForm@h21 = h2.h1D

K -60 + 32 ä 20 + 14 ä

-20 + 14 ä -60 - 32 äO

4 Basics_of_Numbers_01.nb