Methods / ti-cas.org
Geometrical application with complex numbers

Objectives
Show how to fight against the automatic simplification of a complex expression, and how to carry out simple transformations with CABRI.
Public
Students
Concepts
Basic transformations, Euclidean Division.
Author
Olivier Miclo

The study of the complex numbers is rich. We will be interested in this article to the geometrical transformations, like complex homotheties, rotations, similarities and inversions. The symbolic computations will be carried out with a TI-89 …and you will be able to see how to transform a line to a circle (!) with the fabulous CabriII software.

* Translation: The specific transformation associated with z® Z=z+b (bÎ C, b=a +ib ) is a translation of vector , of affix B.

Yes … this screen comes from a TI-89!!!

Example with the image by Z=z-i of the line of equation y=x-1 (passing by A(0,-1) and B(2,1)).

* Rotation : The specific transformation associated with z® Z= is a rotation R(o,q )

The opposite screen shows the image of the line y=x-1 by Z=z*i, which is a rotation from angle +p /2 compared to the point O.

* Homothety: The specific transformation associated with z® Z=k* b B is a homothety of center O and ratio K.

Example with the image by Z=3*z of y=x-1

* Direct plane similarity : The specific transformation associated with z® Z= is a direct plane similarity of center O, of ratio r , angle q [2p ]. One notes . The similarity associates two transformations: Homothety and Rotation. For a Rotation or Similarity, it is thus necessary to convert a complex expression into polar format. Let us notice all the same that the " Rotation " example is a similarity

The opposite screen shows the image by Z=3*i*z of our line.

It's a similarity (rotation + homothety)

* Complex inversion :

The image of a line passing by O (O excluded) is a line. A point M(x,y) placed on this line has for image M(1/x,1/y).


The image of a line not passing by O is a circle which passes by the origin, this one being excluded.


The image by Z=1/z of this line passing by A(0,-1) and B(1,0) is a circle of diameter [ CD ] with C(0,1) and D([1,0)

We created the segment [ CD ], the center of the circle (medium of [ CD ]), then the circle. We checked if the circle passes by O (property) …although O is excluded from the circle …a drug please !!!

 

An invariant point is a point which, after transformation did not moved..

It is thus necessary to solve .

.

To avoid the automatic simplification of a complex expression by z=a+i*b, let us replace i by variable i

 

So that the equality is checked, it is necessary that the numerator is null:

The two invariants points are A(-1 ;1) and B(1 ;1)

getNum & part allows us to easily extract the numerator of this expression, and zeros enables us to obtain the roots of this last expression (by change i by i)

Show that if z=i*y then Z=i*Y

While using one obtains . So

The TI-89 can carry out this type of calculation easily:

Put Z in the form

One carries out the Euclidean division of i× z+2 par z-i

PropFrac enables us to obtain the desired form … but it will be necessary to replace i by i in order to fight against automatic simplification a+i*b

Let us use this decomposition to build image per T of the line x-y-1=0.

- Line y=x-1 passing by A(0,-1) and B(2,1)

  • - Translation of vector

After having created a unit vector , we used "TRANSLATION" tool by showing successively the line, then the vector.

  • - complex Inversion
  • Complex inversion of the line passing by C(0,-2) and D(2,0) is a circle of diameter [ EF ], with E(0,1/2), F(1/2,0)
  • We have created the objet " 0.5 ", then carried out a transfert of measurement on the two axes (points E and F). "MIDPOINT" allowed us to create the center of the circle (medium of [ EF ])

  • - Translation of vector

After having created a second unit vector , we have used " TRANSLATION " tool by showing successively the circle , then the vector.

This article shows that the TI-89 is an exceptional tool : the student can rock between two environments, CAS and geometry.

The geometry of TI-89.92 Plus, with CABRI, is a powerful tool for investigation and of discovered of a broad variety of models, relations and connections in mathematics.

CABRI-GEOMETRIE allows the students to follow the advance of the mathematical discovery through demonstrations while exploring and by learning the principles.

Also see in GEOMETRY heading how to carry out a macro-construction allowing to obtain the complex inversion of a line.