الفهرس 
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Abstract
Summary:
In this dissertation we will define Minkowski normed space, this space is an affine 3-
space endowed with a metric which is supported by a centrally symmetric convex unit
ball B . This means that the boundary of the unit ball ¶B contains no line segment
and each point of B has a unique supporting plane.
Also we study the concept of ruled surfaces in Minkowski space. Ruled surfaces
described by three values (curvature, torsion and striction) as functions of the arc
length parameter, these values called Kruppa invariants and we will find a definition
of striction curve in M-space.
Furthermore, we redefine some important theorems about angle measure, involute
curve and theorems related to Bertrand curves.
The main results of dissertation is devoted to:
- Define the Minkowski striction curve and the left-orthogonal moving frame of
a ruled surface.
- Define a semi-inner product in B M and the trigonometric functions
(Minkowski-Cosine and Minkowski-Sine).
- Translate Brauner’s angle measure in Minkowski space.
- Define and calculate Minkowski substitutes for the classical Euclidean
Kruppa-curvature and –torsion as coefficients of the system of Frenet-Serret
equations.
- Define a “deformation vector”.
- Give a comparison between the concept of involute in Euclidean space and
Minkowski space, and get the definition of left and right involute in
Minkowski space.
- Discuss some important theorem about Bertrand curve in Minkowski space.
The dissertation is structured as:
Chapter 1 is an introductory chapter with an explanation of the problem and giving
basic definitions and presenting the aim of the dissertation.
Chapter 2 presents some of the different orthogonality concepts in Minkowski spaces.
For example the orthogonalities which are given by Roberts, Birkhoff, Carlsson,
James, Pythagorean and Singer.
Birkhoff orthogonality is not symmetric, this means that if the vector x leftorthogonal
to the vector y ( x ⊣y ) it dese not implies that ( y ⊣x ), and we show
relations between Birkhoff orthogonality and other concepts like Diminnie (2-norm)
and Area orthogonality.
The definition of left orthogonality is symmetric in the case of Radon plane and we
will also discuss some theorems about it, furthermore we define the dual Minkowski
space ( )*
n M and its unit ball o B , this space is the set of all linear functionals onto the
one dimensional normed space.
Also we talk about the support theorem and some related definitions in Minkowski
space which play an importance role for the definition of the Minkowski surface area,
also we talk about support function ( ) K h f of a convex sets K and its different
properties and discuss the concepts of volume and mixed volume of Borel sets (closed
convex sets).
We give also a summary about isoperimetric problem in Minkowski space n M where
n ³ 2 . Isoperimetric problem is stated as the simply closed curve that contains a
given area such that its length is a minimum, for example in Euclidean plane this
curve is a circle.
We discuss the concept of surface area following different approaches in for a given
n-dimensional Minkowski space ( n ³ 3). There are mainly two definitions of the
surface area, one by Busemann and the other by Holmes-Thompson. Finally, we study
some definitions and special theorems about Minkowski length μ (¶K ) of a convex
body K and the relation between it and Haar measure l in Minkowski plane where
( ) ( ) B B μ K =s l K
such that
( ) * o
B
l B
s
p
= .
Chapter 3 In this chapter we present the concept of angle measure in projective space
(Brauner’s theorem of angle measure) and we try to find this value in Minkowski
space. Since the angle measure between two vectors a,b in Minkowski space is not
symmetric because of the definition of orthogonality, then we need to define so-called
Minkowski cosine function and study important properties of it.
We need also to define semi-inner product between two vectors 1 2 x ,x in M-space. The
angle measure here is depend on the cross ratio of some ideal points in a plane at
infinity.
Chapter 4 we talk about the definition of ruled surface f (u,v ) such that
f (u,v ) = {x(u,v ) x = P(u ) +v e(u )}
where P(u ) is called a director curve and e(u ) is a unit vector in the direction of the
generator.
A surface f is called ruled if through every point of f there is a straight line that lies
on f . Plane is the most famous example of ruled surface, but a developable ruled
surface is a surface that can be unrolled onto a flat plane without tearing or stretching
it, for examples cylinder and cone.
After that we discuss the orthonormal frame in Minkowski three space (o eeɺ Z) such
that e(u ) ⊣eɺ (u ), Z ⊣e(u ) and Z ⊣eɺ (u )
and using it to prove the equation of
striction curve in Minkowski space which lies on the central plane.
Finally, we define the deformation vector xɶ which describes the deviation of the
Minkowski space from Euclidean space and helps us to get Frenet-Serret formulae for
the moving frame of a ruled surface in Minkowski space.
Chapter 5 This chapter is very important because we talk about some special curves
like Bertrand curve and Involute curve and we try to translate it into Minkowski
space.
According to Bertrand curve, there is only one curve having the same principle
normal, the two curves are Bertrand mates. In Euclidean space, the angle between the
tangents of two Bertrand mates at corresponding points is constant, but in Minkowski
space, r and * r are two Bertrand mates if:
(( ) ( )) * 1 1 , .
ds
cm const
ds
−lt −lt t b =
where t and 1 t are the 3rd and 4th Minkowski torsions, we assume that the angle
between the tangents of two associated Bertrand curves is constant when the
Minkowski cosine is also constant.
About involute curve, a curve *
1 C is called involute of a curve C if it lies on the
tangent surface of a curve C and intersect the tangent lines orthogonally. Since
Birkhoff orthogonaliy in M-spaces is different from Euclidean one then the definition
of involute is become different to get two new definitions left-involute curve and
right-involute curve.