# 1- Axioms of Geometry

Euclidean space, denoted by , is a space that contains the elements of Euclidean geometry and satisfies the axioms (postulates) of Euclidean geometry. A geometry or a geometric system is a axiomatic system being a collection of the following entities.

- Undefined/abstract terms or primitives: These are abstract elements that can be interpreted based on context.
- Defined terms (if necessary to have): Terms that are defined using the primitives.
- Axioms: The statements that are accepted without proof. They set relations within and between the primitives.
- A system of logic.
- Theorems: The statements that can be proved using the axioms and the system of logic.

A mathematical model/representation can fit to a geometry/ a geometric system. A model contains elements that are explicit interpretations of the undefined/abstract terms of the geometry and compatible with the geometric axioms. One model for the Euclidean geometry is to define a point as (the imagination of) an exact location which has no dimension/size in the space. A line is defined as a straight line which is an infinitely long object in any (two) directions and has no width or thickness, and it uniquely exists (or can be defined/constructed) between any two points. In the materialistic space/world we can approximately visualize/consider points and lines according to what we see. In other words, we model physical objects or fit mathematical models to them. For example we can consider a point as a *relatively* small physical dot, a computer pixel or a light spot (modeling a physical dot by the notion of a point defined as a location in the space). The infinite trajectory of any object (light, trace of a pen, a long edge/ridge, etc) whose length, i.e. its longest dimension, follows the definition of the straight line can be considered/imagined as a straight line.

There are two main axiomatic systems for Euclidean geometry.

**1- Hilbert’s Postulates.**

The primitives are sets of points, lines, and planes. These are not generally and necessarily subsets of each other. For example a line may not considered as a set of points. Any physical object or non-physical notion that satisfies the axioms of Hilbert’s system can be recognized or interpreted as the primitives. However, notions like “a point lying on a line” is defined.

In Hilbert’s system, the notions of a ray, a line segment, *vector* as directed line segment, an angle, and polygons are also defined based on the primitives and axioms. Hilbert’s system is purely geometrical, in that nothing is postulated concerning numbers and arithmetic. Although this system has notions for comparing line segments, and comparing angles with each other, it does not have a metric for distance between two points and a measure of angle. If the axioms of real numbers are considered along with the continuity axiom of Hilbert’s system [1], then it can be proved that there is a bijective map between points *on *a line and real numbers. Thereby, a distance function between two points and a measure of length can be defined for line segments. A measure for angles can also be constructed. Theses measures inject numbers and arithmetic into the Euclidean geometry founded on Hilbert’s axioms.

**2- Birkhoff ’s postulates** **(axioms of a metric geometry)**

Birkhoff’s axioms [2] of Euclidean geometry directly has an axiom on the existence of a map between real numbers and points on a line. This brings real numbers and arithmetic into the system and quantifies notions and facilitates proofs. This is because axioms of real number can be used in relation with geometric terms. The Birkhoff’s system is as follows.

**Primitive/undefined objects: **the abstract geometry of Birkhoff, consists of a set whose elements are called points together with a collection of non-empty subsets of , called lines. So, .

**Primitive terms:** point, line, coordinate function of a line, half-line, bundle of half lines (BHLs), and coordinate functions of BHLs.

Note that *line* should not be considered as a straight line. It can be interpreted as a straight line but not limited to.

**1-** **Axiom on lines:** If A and B are two distinct points, then there exists one and only one line containing A and B. I.e and then .

**Definition:** A set of points is said to be collinear if this set is a subset of a line. Two sets are collinear if the union of these two sets is collinear.

**2-** **Axiom on coordinate function of a line:** There exists associated with each line , a nonempty class of one-to-one mappings of onto the field of real numbers. If and if is any one-to-one mapping, then if and only if for all ,

For a line , the elements of are called coordinate functions or ruler of .

The above axiom is called the ruler placement axiom and indicate that different maps are like rulers along a line. It does not matter where to put a ruler; all rulers gives the same value of for two fixed points and . This guarantees that the members of are well-defined.

As the result of the axiom, a line (and later a ray and a segment) has an infinitely uncountable number of points.

**precedence relation**: Because the coordinate function is one-to-one, any sequence of points is mapped to a monotonic sequence of real numbers. Therefore, we can define the precedence relation between to different points of a line as .

**Lemma:** For any two points on a line with a fixed ruler, either or .

**Lemma:** Assuming a precedence relation for two points of a line determines a class of rulers for the line such that the precedence relation (for all points) are the same for any of the rulers.

**Definition:** A precedence relation on a line can define a direction for a line. Let . Then we can say the direction or the sense of direction of the line is from the point to the point . Similarly, the direction of the line is from to if . Therefore, two different directions can be defined for a line. The direction of a line can be represented by an arrow, or a *directed line segment* which latter is called a geometric vector.

The distance between two point is denoted by or and is defined to be the unique non-negative number where is an arbitrary member of . As a result, the distance between two arbitrary point is calculated through constructing a line between them and using the coordinate function of the line. Obviously, for any pair of points and their line in the geometry we should use the same ruler if we want to be consistent.

**Betweennees relation**: For , i.e. on the same line, the point is between the points and if either or . It can be proved that this relation is independent of the coordinate function. As a result, .

if and are two distinct points of a line, we call the set of points on that line such that is not between and , a half-line or a ray with end-point . In speaking of a ray , the first point always represents the end point. Two rays are collinear if their points belong to the same line.

For , the sets of points containing and and all between them is called a segment of . A segment without its end point is called an interval. A segment is denoted as and its length is .

**Definition:** Two line segments and of a line or two different lines are called congruent if . The relation being a binary relation is denoted as .

**3-** **Axioms on bundles and coordinates of the rays of bundles**

**Definitions:** *Certain* subclasses (sub sets) of the class (set) of all rays with the same end point are called bundles. The common end point of the rays of a bundle is called the vertex of the bundle. A bundle with a vertex is denoted as . An angle is an unordered couple of rays with the same end point which is called the vertex of the angle, and the rays are called the sides of the angle. An angle is straight if the sides are distinct and collinear (on the same line, i.e. being the subset of the same line).

The word *certain* in the above definition is important. It says only specific subsets of the set of ALL RAYS with THE SAME end point/vertex is categorized as bundles. Firstly, it means different bundles can have/share the same vertex. Secondly, by the following axiom on the coordinate functions of bundles and later the definition of a plane, it will be clear that the rays of a bundle must all belong to the same plane.

**Axiom on bundles:** If and are two non-collinear rays with the same endpoint , then there exists one and only one bundle containing these rays. If they are collinear then they (their points) can belong to other bundles with different vertices.

**Axiom on the coordinate functions of bundles (protractor axiom):** There exists, associated with each bundle , a nonempty class of one-to-one mappings of onto the equivalence classes of real numbers modulo (i.e. . If is a member of and if is any one-to-one mapping of onto the equivalence classes of real numbers modulo , then , is a member of if and only if for all modulo . The elements of are called the coordinate functions of . If then is defined to denote the equivalence class modulo containing , and is the real number of class such that .

Measure of an angle: if , and , the the measure of the angle is denoted by and defined as the minimum of and where is the coordinate function associated with the bundle . The number is independent of the coordinate function and It can be proved that the measure of an angle is independent of the bundle it belongs to.

**Continuity axiom:** If is a bundle of vertex , and if are distinct nonvertex points of noncollinear rays of the bundle, then to every point on the segment , there exists a ray of containing such that . Conversely if a ray of the bundle is such that then there exist a point belonging simultaneously to the ray and to the segment .

**Theorem:** The measure of an angle is if and only if this angle is straight.

**Theorem:** If and are two noncollinear rays of a bundle , then there exists one and only one coordinate function such that and . This theorem says that the measurement of angles can be intuitively obtained in the usual way as the application of a protractor (plain or half-disk) in such a way that one side of the angle coincides with the zero of the protractor and the other side corresponds to a number less than (the measure of the angle).

**Corollary: **If , and are three distinct rays of a bundle and if and are collinear, then .

**Corollary: **If is a ray of a bundle and if then there exist two and only two distinct rays and of the bundle such that .

**Lemma:** Let be an element of a bundle and let , If such that , then there exists a ray with the following properties: (a) , (b) for all points and for all points such that the segment has a point in common with the unique line containing the ray .

Two distinct lines having a point in common determine six angles. Two have for the measure and we the four remaining ones form two sets, each set consisting of two distinct angles with the same measure.

**Definition:** Two distinct lines having a point in common are said to be perpendicular if the four angles with measures not equal to 0 or have the same measure i.e., .

**Lemma:** If is a line and is a point not on , then there exists one and only one line containing and perpendicular to .

**Definition:** A plane is defined to be the set of all points belonging to the rays of a bundle ; this set will be denoted by .

**Theorem:** If two distinct points of a line are in a plane, then the whole line is in the plane.

**Theorem:** A plane is uniquely defined by 3 non-collinear points. In other words, two planes coincide if and only if they have three non-collinear points in common.

**Theorem:** If two planes have two collinear points in common, they have a line in common.

**Definition: **If two lines of a plane have no point in common, then they are parallel.

**Theorem:** In a plane, from a given point not on a given line there exists one and only one perpendicular to that line, and from a given point not on a given line,there exists one and only one parallel to that line.

**4- Axiom and theorems on triangles**

A triangle is an unordered set of three distinct points. The points are the vertices of the triangle. The three segments defined by the vertices of a triangle are the sides of the triangle. The three angles defined by the sides of a triangle are the angles of the triangle. A triangle is assumed to be a proper one meaning that the vertices are non-collinear. triangle. In the context of triangles, for instance a triangle , the measure of an angle, say angle with vertex , will be denoted . Two triangles are similar if the vertices can be labelled and in such a way that, AND , , . The notation / denotes the ratio of the lengths.

**Axiom of similarity:** If two triangles and are such that and , then they are similar.

Birkhoff wrote the above statement as an axiom. But, I think it can be proved; see Ref 1. Based on the above axiom, several theorems on the similarity of triangles can be obtained. They are skipped here; see Ref 2.

Two important theorems on triangles are as follows.

**Theorem (Euclidean geometry) :** The sum of the measures of the angles of a triangles is equal to .

**Theorem:** If a triangle is such that , then (proof based on similarity of triangles).

**5-** **Axiom of 3-dimensional euclidean space:** There exists a point not on a given plane.

**Geometric forms:** A geometric form in a geometric space is a subset of points satisfying a certain condition. This subset with its condition is also called a locus of points. For example lines, triangles, and planes are geometric forms. Other geometric forms are circles, polygons, or any imaginable form. As an example, an Euclidean circle is a set of points in a Euclidean plane such that they all have the same distance from a fixed point. A sphere is a set of points in a Euclidean 3D space such that they all have the same distance from a fixed point.

# 2- Some definitions and theorems

## 2-1 general stuff

**Definition:** For a straight line intersecting a plane at a point . The line is said to be perpendicular to the plane if it is perpendicular to all straight lines lying on the plane and passing through the point .**Theorem:** A line intersecting a plane at a point is perpendicular to this plane if and only if it is perpendicular to some two distinct straight lines lying on the plane and passing through the point (for a proof see [1] page 84).

## 2-2 Congruent transformation (translation)*

* Congruent transformation is referred to as congruent translation in Ref 1.

**Definition:** Let be two (different) lines. Then, a mapping is called a congruent transformation of the straight line to the straight line if for any two points and on the line the condition of congruence is fulfilled. Congruent transformation can also be composed of general rotation and pure translation as explained in page 134 Ref 1.

Assume two lines with particular directions and points an shown. We can define two mappings and of congruent transformation of the first line (black) to the second line (blue) as

**Theorem:** For any point on a straight line with a distinguished direction and for any point on another straight line with a distinguished direction there are exactly two mappings performing congruent transformation of the line to the line . The first of them preserves the precedence of points, i. e. implies . The second mapping inverts the precedence of points, i. e. implies .

### 2-2-1 Translation and inversion

Let be a line and two points forming a vector . Fixing a direction for the line using a vector , we can define a mapping as . This mapping is called congruent translation by the vector . The vector determines the order of points on the line. The mapping preserves the precedence of the points and is uniquely defined by its action being mapping the initial point of the vector to its terminal point . Therefore, it maps any point on the line to another point on the line such that and construct a vector and .

The proof is as follows. Given that , and an arbitrary point such that , we can write . Without loss of generality we can assume therefore . Now assume that . Therefore, . Since , we should have . Otherwise, is between and and hence contradicting with . Now, we can write . Hence,

which completes the proof.

This mapping is called congruent translation by a vector because it models the translation of a particle that moves from a point to a point along a line.

**Lemma:** Because preserve the precedence over the line, it will preserve the initial and terminal points marked on a vector. Note that the direction of a line is set independently; the initial and terminal points of a vector on a line does not show their precedence.

**Lemma:** if , then is the identity map.

**Remark:** Later that we define the affine space, any map that take a point and add it to a fixed free vector and returns another point is a congruent translation along the line passing through the point and defined (including its direction) by the free vector.

If we let , the this map does not preserve the precedence and is called inversion.

# 3- Geometric vectors and Euclidean geometric vector space

A geometric vector is a *directed* line segment (abstract) in a sense that one of the segment’s end point is distinguished/marked with respect to the other. A line segment is then can define two distinct geometric vectors. For a line segment , its two vectors are denoted as and . The first and last points of a vector are respectively called the initial and the terminal points.

**Definition:** The/a zero vector is a vector whose initial and terminal points coincides.

**Note:** Saying that a vector belongs to or lies on a line or a plane means its end points belong to/lies on the line/plane.

**Definition [EV1]:** Two vectors and lying on one line are called equal if they are co-directed and if the segment is congruent (pure coincident if translated) to the segment .

There can be 3 types of vectors:

1- Position vectors or (pure) geometric vectors: A position vector is a vector whose position is fixed in the space.

2- Sliding vectors: A sliding vector is a vector that is free to slide along its line. In other words, a sliding vector is a class of mutually equivalent vectors in the sense of the definition EV1. A slipping vector has infinite representatives lying on a given line. They are called geometric realizations of this slipping vector.

3- Free vectors: The classes of mutually *equal* vectors in Euclidean geometry space are called free vectors. Geometric vectors composing a class (of free vectors) are called geometric realizations of a free vector. The concept of equal vectors in space is as follows.

**Definition:** Two vectors and NOT lying on one line are called codirected if they lie on parallel lines, and the segment connecting their end points does not intersect the segment connecting their initial points.**Definition:** Two vectors and in the space are called called equal if they are codirected and if the segment is congruent to the segment . The congruency of straight line segments can be inferred by their lengths. The equality of vectors in the space is reflexive, symmetric, and transitive.

Noe that the concept of parallel lines are in a plane, i.e. two different lines are parallel if they are in a plane and have no intersection.

ِFor any free vector and for any point there is a geometric realization of with the initial point .

**Free vectors, lines and planes:** By realization of a free vector with a particular initial point, a directed line segment (with the initial and the terminal points) is born and hence the corresponding line containing the point and the vector. By realization of two non-collinear vectors with the same initial point, a plane containing the point and the two vectors is defined.

**Euclidean geometric vector space: **Generally a vector is not limited to geometric vectors. A vector is in fact a member of a vector space. Geometric vectors defined as directed line segments also belong to a vector space. A vector space is a set on which addition operation and multiplication by a scalar (or any other field) are defined and the set is closed under these operations (some other axiom also exist in the definition of a vector space). Geometric vectors also belong to a space called the Euclidean geometric vector space denoted as .

**Remark:** since is closed, its members are free vectors, i.e. the classes of mutually *equal* vectors. This means the space is not restricted to a set of realizations of geometric vectors.

**Remark:** When working with , i.e the space of free geometric vectors, their arbitrary realizations in the geometric space are recruited. In this regard, they are treated as directed line segments. Also, we have all the properties of the geometric space (points, lines, planes, axioms, theorems, etc). So, we can develope theorems and show that they are independent of a particular realization.

In the geometric vector space the multiplication by a scalar scales the length of the vector (line segment) in the same direction if the scalar is positive. The direction of the vector is reversed and then scaled if the scalar is negative. The addition of vectors and is by treating them as free vectors and translating such that the end point get coincided with the end point , then the new vector is the result of .

Like any other vector spaces, the Euclidean geometric vector space have bases and hence vectors can be written in terms of the basis vectors. This generates coordinates for the vectors. The dimension of the Euclidean vector space is at most 3, i.e. three dimensional physical space denoted as .

Other than properties and operations regarding vectors, the definition of angle also exists in the Euclidean geometric vector space. The angle between two vectors is the angle between they lines when both vectors are treated as free vectors and their geometric realizations share the same starting point.

3- Conditionally geometric vectors:

If we considers the locations in the physical space as the notion of points and construct the Euclidean geometry, then a vector from a spatial point to another is called a displacement vector. This vector binds some two points of the geometric space. The lengths of this vector is a geometric length defined based on the ruler placement axiom. The unit of the length is defined by a length scale like meter.

A displacement vector is a geometric vector as it’s in the geometric space. A geometric vector has direction and a relative orientation/angle (with other geometric objects). Some physical phenomena like velocity, acceleration, force, etc have also the senses of geometric directions and geometric orientations in the physical space. Moreover, they are bound to some points in the space, i.e. the physical effect is bound or recognized at a point. For example we say “force at a point/location” or “velocity of an object at a point/location”. Therefore, in the physical space recognized as a model of the Euclidean geometric space, the bounding point, orientation, and direction of this type of vectors have geometric representations, i.e. a point, angle, sense of direction. However, these vectors do not have geometric length, in other words, their bounding point can be recognized as either the initial or the terminal point of a vector. Therefore, there is not a direct geometric representation of such a vector unless a geometric length through choosing some scale factor is assumed. Any non-geometric vector that can have a geometric representation upon setting a geometric orientation, direction, and length is called a Conditionally geometric vector.

## 3-1 Coordinates of a vector in the spacial Euclidean geometric vector space

Let be an ordered set (sequence) of different independent geometric (free) vectors in ; i.e. they are not co-planar all together. Then, they form a basis for the space and any vector is a linear combination of them, i.e. . The scalars are called the coordinates of . For any vector and a basis , there a unique coordinate function defined as .

**Example:**

## 3-2 Dot/scalar product

Noting that there are a definition/notion of angle and definition of triangular functions in , we can define a function, called the dot or scalar product, as acting as , where returns the length of the vector, i.e. the line segment of the vector. As a result perpendicularity of vector implies .

### 3-2-1 Orthogonal projection onto a line

**Theorem:** For any nonzero (geometric) vector and for any vector , there are two unique vectors and such that is collinear to , and is perpendicular to , and they both satisfy .

Proof is by geometric realization of the vectors, and using the theorem that says there is a plane containing and , and the theorem that states there is perpendicular line to another line from any point not on .

**Definition: **For any vectors and , the mapping is called the orthogonal projection of onto the direction of or onto the line containing (collinear with) .

**Theorem [OP1]:** For each vector and for a vector :

Proof: to prove that two geometric are equal, we should prove that they lengths are equal and also they are co-directed. The first part can be proved by the definition of dot product and | as in the previous figure. The second part is proved by showing that is in the same direction as the RHS for , , and .

**Lemma [OP2]:** For any the sum of two vectors collinear to is a vector collinear to and the sum of two vectors (each) perpendicular to is a vector perpendicular to .

Proof: The first part is straight forward. For the second part we assume and are not either collinear or parallel as the proof becomes straight forward. So, we assume . Now, build a realization for as at an arbitrary point and a the realization of as ; Since the realizations are not parallel, they construct a plane. The resultant vector is in the plane and it is the realization of . Now consider another realization of at . Each realization corresponds to a line; since the line and (since ), then . Therefore, is perpendicular to two lines in a plane, and hence it is perpendicular to all lines including in the plane. This implies that is perpendicular to .

**Theorem:** is a linear mapping, i.e. and .

Proof: Let . Using the orthogonal projection, we can write, , , and with respect to . Therefore,

By the lemma (OP2), and . Therefore, and . Using theorem OP1,we can deduce,

The second property can be proved by theorem OP1.

### 3-2-2 Properties of the dot product (theorem)

1-

2a-

2b-

3a-

3b-

4- and

Proof: Only the 3rd one will be proved as the rest can be readily proved through using the definition.

If , then it is a trivial case. If not, we can use theorem OP1 and write

By the linearity of the orthogonal projection, the LHS of the above equation is zero, hence the proof is implies by setting the RHS equal to zero. The property 3b is then proved by the using the first property and 3a which is already proved.

In terms of type, the dot product is a bilinear map.

### 3-2-3 Calculation of the scalar product through the coordinates of vectors in a skew-angular basis

**Definition:** a skew-angular basis (SAB) for is an orered set of non-coplanar vectors. The angle between each pair is not necessarily a right angle.

Let be a SAB. Also consider two free vectors and . Each vector can then be written in terms of the basis vectors as,

where and are the coordinates of and respectively.

Using the *Einstein notation* and the properties of the dot product, the dot product is written as

The terms do not depend on the vectors and ; but on the lengths of s and the angles between them. If the values are collected in a symmetric matrix, the matrix is called the Gram matrix of the basis . The above equation can be written using matrix multiplications as

where , and . This equation expresses the dot product through the coordinates of vectors in a skew-angular basis.

**Proposition: **If the basis of the space is an orthonormal basis, i.e. for any basis vector and for any pair of the basis vectors , then becomes the identity matrix and

## 3-3 Concept of orientation and the cross/vector product

**ِDefinition:** An ordered triple of non-coplanar vectors is called a right triple if, when they are geometrically realized with the same initial point and observed from the end/terminal of the third vector, the shortest rotation from the first vector toward the second vector is seen as a counterclockwise rotation. If the rotation is seen clockwise, then the triple is called a left triple.

**Definition:** The property of ordered triples of non-coplanar vectors to be right or left is called their orientation.

**Definition:** The cross/vector product is defined as acting as where and is a vector perpendicular to both and and pointing in the direction defined by the right hand rule, i.e. the vectors form a right triple.

**Definition: **The the orthogonal complement a free vector is the set of all free vectors perpendicular to ; i.e. .

The orthogonal complement of a vector can be visualized as a plane or vectors in a plane if one of the geometric realizations of the vector is considered as . Indeed, all vectors starting from the initial point and perpendicular to has ending points in (belongs to) the plane .

In 1-1-3, it was shown that if a vector is given, then any vector can be written as

where is the orthogonal projection of onto the direction of and is the component of perpendicular to and hence . The vector is called the orthogonal projection onto a plane perpendicular to , or the orthogonal projection onto the orthogonal complement of .

Denoting as , we can write,

\vec b = \pi_{\vec a}(\vec b) + \pi_{\perp \vec a}(\vec b) \tag{OP1}

\]

**Theorem [OPP]:** The function is a linear mapping, i.e. and .

Proof: Use Eq. OP1 and write and use the linearity of and then arrange the terms and use Eq. OP1 again. The same procedure can be done for the second condition.

**Rotation about an axis:**

Let be two non-zero vectors. Lay the vector at some arbitrary point and get the realization . Then lay the vector . This non-zero vector defines a line . We take this line as a rotation axis. For this setup, we define a function and for now we call it the function of rotation of about the axis . The rotation function works as follows.

. Otherwise,

such that:

1- and belong to a plane being perpendicular to .

2-

3- The angle between the vectors and where is the intersection point of and the plane . The angle is a signed angle with respect to the counterclockwise direction.

The vector is a realization of any vector being equivalent to by parallel translation. This is also true for being the realization of . Moreover, the parallel translation of indicates that the rotation function does not depend on the realization of the rotation axis vector. Therefore, we can re-define the rotation function over the vector space as:

**Remark:** If , then .

**Theorem [RF]:** The rotation function is a linear mapping.

**The relation of the cross product with projections and rotations:**

For two non-colinear vectors and we can write and sketch the following realization.

The resultant vector lies in the plane perpendicular to (because therefore there is a unique plane that contains AND perpendicular to ). The orthogonal projection of onto the plane is whose length is

### 3-3-1 Properties of the cross product (theorem)

# 2- Affine geometric space

A geometric vector is defined as a directed line segment which has two points. Here, the final produced object, vector, belongs to a vector space like , but its fundamental components which are the endpoints belong to the geometric space, like . The line segment can be implied by the two end points because of the axiom of the Euclidean geometry stating that there is a unique line for every two distinct points. So, there should be a connection between the geometric space and the vector space. This is defined by the concept of an affine space as follows.

**Definition:** Let be a vector space over the field , and let be a set. The addition written as is defined for any and with the following conditions:

1- p + \vec 0 = p

2- (p + \vec v) + \vec u = p + (\vec v + \vec u)

3- For any there exists a unique vector such that .

Then, , is called an affine space.

With the above abstract definition, the affine Euclidean geometric space is defined by letting the and where the geometric space is inherently a set of points (an axioms). The notion is clear that a point plus a vector takes us to another points; these points *become* the end points of the realization of the geometric vector.

# 3- Models of Euclidean geometry

One model for Euclidean geometry is to define a point as (the imagination of) an exact location which has no dimension/size in the space. A line is defined as a straight line which is an infinitely long object in any (two) directions and has no width or thickness, and it uniquely exists (or can be defined/constructed) between any two points. A flat surface that is infinitely wide and has zero thickness is a model of a plane. A flat surface is an object whose any of its two points defines a line segment that is completely contained in the surface. This model of geometry can be referred to as the physical space geometry.

Another model for Euclidean geometry is as follows. Let be the set of points, i.e. a point is now defined as an ordered pair in . Define lines as: 1- a vertical line , 2- a non-vertical line as . Then, it can be proved that the structure satisfies the axioms of lines in Euclidean geometry. The model is called the Euclidean plane. A coordinate function (axiom of ruler) of a line in this space is constructed as follows.

For a line , assume that there is a distance function on . As previously defined, a function is called a ruler or a coordinate system for if (1) f is a bijection, and (2) for each pair of points and on , . This equation is called the ruler equation and the value and is called the coordinate of with respect to .

For this space, if the Euclidean distance defined as is considered as the distance function, the following rulers can be obtained for lines in the space.

Case 1: If and , i.e. . Then, for : . Therefore, by the definition of the ruler function we can define as as the coordinate function of a vertical line (the function can be proved to be injective).

Case 2: If and , i.e. , where . Then, . This implies that a ruler function for this type of line is .

The model is called the Euclidean metric geometry.

The only main axiom left to make a complete model of the Euclidean geometry is the axiom of coordinate functions of bundles. Therefore, a measure on angles is needed. If and are two intersecting lines at a point , with points , and , then a measure of the angle satisfying the axioms of this measure is defined as:

where , , and . Moreover, .

A remark with this formula is that it uses the inverse of a trigonometric function which is not defined yet. Also, might be interpreted as a vector which is not defined either. The dot operation is also familiar and can be interpreted as a vector dot product. So, far we have not defined these terms as they are conventionally defined. We just wanted to show that such a measure exists for the geometric model; hence we have a mode of the Euclidean geometry in which points are interpreted as pairs of real numbers. In this geometry, vectors can be defined, i.e. then is a vector which is a member of a vector space, here when has the structure of a vector space. The function and its inverse need the definition of a circle to get defined. Although a circle (as per its notion) can be defined in this geometry, there is a pure algebraic way to define as per chapter 5 in Ref 3.

The motivation to this geometry which can also be expanded to and the above formulations is the concept of coordinates of points in the physical space geometry. This notion is born as the result of the notion of coordinate systems explained in next section.

# 2- Coordinate systems—

Let represents the physical Euclidean geometric space consisting of points, lines, planes, line segments, vectors, etc.

2-1 Cartesian coordinate system

2-2 Other coordinate systems

[1] Sharipov R.A. – Foundations of geometry for university students and high-school students.

[2] BIRKHOFF’S AXIOMS FOR SPACE GEOMETRY.

[3] Richard S. Millman – Geometry. A Metric Approach with Models.