From a general point of view in developing the basis of differential geometry it is useful to consider the concept of a group and its connection to geometry.

It can be shown that by assuming the mappings:

T: x' _{i} = F _{i} (*x*_{1 }*, x _{2}*

**,***x*)

_{3}are one-to-one and defined on the whole space R ^{3 } i.e. mappings of the whole space R ^{3 }onto itself we can expect:

x' _{i} = F _{i} (*x*_{1 }*, x _{2}*

**,***x*), x'

_{3}_{j}= F'

_{i}(

*x'*

_{1 }*, x'*

_{2}

**,***x'*) (i, j = 1,2,3........)

_{3}To be effected successively, thereby enabling us to obtain a composite mapping (or product) of the mappings such that:

x' _{j} = {F _{j} (*x*_{1 }*, x _{2}*

**,***x*), F

_{3}_{2}(

*x*

_{1 }*, x*

_{2}

**,***x*), F

_{3}_{3}(

*x*

_{1 }*, x*

_{2}

**,***x*) }

_{3}Thus a set *G * of mappings: T: x'

_{i}= F

_{i}(

*x*

_{1 }*, x*

_{2}

**,***x*)

_{3}Is called a group of mappings or a transformation group if *G * has the following properties:

i) The identical mapping *x'*_{i }_{= }_{ x}_{i}* _{ }* (i= 1,2,3......) is contained in

**G**ii) The inverse T ^{-1 }mapping of any mapping T contained in *G* is also an element of

**G**iii) For any arbitrary pair of mappings contained in *G* the product of these mappings is also an element of

**G**Then the direct congruent transformations form a group. Related to the above there are three axioms of equivalence:

1) Every geometric configuration is equivalent to itself

2) If a configuration A is equivalent to a configuration B then B is also equivalent to A.

3) If a configuration A is equivalent to a configuration B and B is also equivalent to configuration C then A is also equivalent to C.

Of particular importance are invariants (denoted __ I__) corresponding to certain geometric objects which will remain unaffected by any mapping contained in

*. Then*

**G***is said to be an invariant of the group*

__I__*. For example the distance:*

**G**d = Ö å ^{3}_{ }_{b }_{=1}** **( y _{i} - x _{i}_{ }) ^{2}

Between two points ( y _{i} ) and ( x _{i}) of R ^{3} is an invariant with respect to the group of displacements. If related to some motion it will in general cause a change in the value of the coordinates, but d remains the same value. From the preceding we can consider some geometry which will be identical with a class of invariants for a certain transformation group. For example, affine geometry corresponds to the group of affine transformations:

__x__ _{i} = å ^{3}_{ }_{k }_{=1}** **a _{ik} x _{k} + b _{i }det (a _{ik} ) **≠**** ** 0

Of projective geometry to the group of projective transformations:

__x__ _{i} = å ^{3}_{ }_{b }_{=1}** ** b _{ik} x _{k} + b _{ik }/ å ^{3}_{ }_{b }_{=1}** **a _{ik} x _{k} + b _{i }

_{ }det (b _{ik} ) **≠**** ** 0

Now, since vector components are invariant with respect to translations, the basic transformation shown in Part 1 (see conditions (1), (2) of that post) corresponds to a transformation of the components of a vector which is of the form:

x' _{i} = å ^{3}_{ }_{k }_{=1}** **a _{ik} x _{k} , å ^{3}_{ }_{i }_{=1}** **a _{ik} a _{il} = d _{kl , }

_{ik}) = -1

^{3}) a vector having the origin of the coordinate system (

*x*

_{1 }*, x*

_{2}

**,***x*) as its initial point and a point Q as terminal point is called the position vector of

_{3}**Q**with respect to that particular coordinate system.

**Q**will have the same numerical value as the coordinates of the point Q. Given a fixed coordinate system, any point in space can be uniquely determined by a certain position vector. (N.B. The origin of the coordinate system will change if the coordinates are transformed according to that prescription defined for affine transformations)

**See Also:**

**And:**

**Suggested Problems**:^{3}find the coordinates of the position vector

**Q**and its length.

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