We will here explain how to create a construction that will autmatically create the image on a circle through an owner defined complex transformation.

Create a new figure with icon and ask for an orthonormal frame.

Use icon to create two free points then icon to create the circle with center the first point an going through the second one.

With tool create a point linked to this circle et name this point M with toll .

With icon create the measure of the complex affix of point M in our frame. This affix is represented as*Aff(M,O,I,J)*.

Use menu item *Calculus - New complex calculus - Complex calculus* (keyboard shortcut**Ctrl + Maj + E**) to create a complex calculus named *z* with formula *Aff(M,O,I,J)* (you can use button **Values**.

Use menu item *Calculus - New complex calculus - Complex function* (keyboard shortcut**Ctrl + g**) to create a complex function named *f* defined by *f*(*t*) = *t*+1/*t*.

Menu item *Calculus - New complex calculus - Complex calculus* (keyboard shortcut **Ctrl + E**) will allow create a complex number named *z’* with formula *f*(*z*).

Now create the point of affix *z’* with tool .

Name this new point *M’* with icon .

Now we will create the point locus of point *M’* generated through the positions of point *M* on the circle it is linked to.

In the color palette activate the red color.

For this use icon . The indication line at the bottom of the window asks first to click on the point tarces of which will generate the point locus : click on point *M*. Then you are asked to click on the point positions of which will generate the locus : click on *M’*. A dialog box pops up. Ask for 300 points and set the checkbox **Closed locus** selected then validate. The point locus appears.

It is now time to create our construction.

We must first indicate the sources object of our construction.

Let’s start with the numerical sources objects.

Use menu item *Constructions - Sources elements choice - numerical*. A dialog box pops up. Our two source numerical sources elements are function *f* and frame (O,I,J).

Click on *f* then on button **Insert** (or double-click on *f* as shown underneath) then do the same with frame (O,I,J).

We must now choose our graphical sources objects. For this use menu item *Constructions - Sources elements choice - Graphical*. Click on the circle.

Let’s choose now the final objects of our construction.

We will start with numerical fianl objects with menu item *Constructions - Final elements choice - Graphical*.

You can only click on objects created exclusively with sources objects. Click on *M*, on *M*’ and on the point locus.

Now we have to choose the numerical final objects with menu item*Constructions - Final elements choici >> Numerical*. Select *z* and *z’* (use Ctrl key) as shown underneath then validate.

We now have to finalize our construction and save it in a file.

For this use menu item *Constructions - Finish current construction* then fill in the dialog bow as shown underneath :

Now we will save our construction on a disk file in the directory of your choice via menu item *Constructions - Save construction of figure to file* it is better to keep the same name for the construction and the file).

Now let’s show how to use our new construction in another figure.

Use menu item *File - New figure with - Frame with vectors* and ask for an orthonormal frame (choose for example *u* et *v* for the vectors names).

Use menu item *Calculus - New complex calculus - Complex function* (keyboard shortcut **Ctrl + g**) to create a complex function named *g* defined by *g*(*t*) = *t*+1/*t^2*.

With tool create a point linked to the x-axis the create the circle with center O and through this linked point.

Let’s now use our construction : Use menu item *Constructions - Implement construction from file*. A dialog box pops up. Open the directory where you saved your construction file , click on the file then click **Open**.

A dialog box pops up for the choice of sources elements. In the left listbox, select number 1. Il the right lits, click on *g*as shown underneath.

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Now in the left list box, select number 2. In the right list, select frame (O,I,J).

The indication line now asks to click on the circle.

Then you are asked if you wish to rename the final numerical objects . Here this is not necessary.

The image of the circle appears (it is a point locus) . With tool capture the point linked to the x-axis.

Pressing key **F6** you will see that final numerical objects are now available named *z* et *z’* which are the complex affixs of the point linked to the circle ans it’s image.

Here is underneath the figure animated through MathGraph32 JavaScript library :