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Cabri 3D workshopJean-Jacques DAHANjjdahan@wanadoo.fr

Historically, the roots of plane dynamic geometry lie in the works of the great geometers of

the french tradition (Clairaut for example). It is why a tool like Cabri allows our students to

have a dynamic approach of Mathematics in harmony with the technics of these geometers. Atthe opposite side, it does not exist a culture of 3D geometry practice and that is why the

arrival of Cabri 3D with its potentialities very close to the ones of Cabri 2 Plus and its

ergonomy can change radically the way of approaching this geometry.

The use of such a software, linked to the perception of the real world around us must help the

teacher to communicate better in a domain where tools of communication really miss.

It must also give to students the pleasure of 3D creativity in an environment similar to the

ones used in 2D video games. I think of texture that give this realistic impression

The principal aim of this worshop is to show with two particular examples that it is possible to

begin very easily with Cabri 3D and at the same time to get model of its use in the classroom.

We begin for those who have not attended the worshop with a quick presentation of the

toolbar.

0. INTRODUCTION TO CABRI 3D

0.1. The worksheet

When opening Cabri 3D, we get a page where a portion of the horizontal plane is represented

with a rectangular system of axis (really 3 vectors having the same origin in this plane). The

toolbar is similar to the one of Cabri 2 in the aspect with some rather different tools in the

contents. Perspective is a central perspective with a point of view that can be changed at every

moment with a continuous right click (other different perspectives can be chosen in the

preferences). This possibility gives an non static approach of the objects in space allowing the

manipulator to obtain the best point of view.

Very good explanations are provided in the Help ; this help contains perfect tutorials

prepared with turbo Demo (they are movies showing step by step how to solve very classical

problems : roof theorem, intersection of a cube and a perpendicular plane to one of its

diagonal...).

BACK

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0.2. The tools in the toolbar

These tools can be reached in the toolbar showed below :

Here are the tools accessible under the icons (the tenth one has been added on the released

version as the underlined tools Arc, MeasurementTransfer and Trace) :

1 2 3 4 5

ManipulationRedefinition

PointIntersection Point(s)

LineSegment

Ray

Vector

Circle

Arc

Conic

Intersection Curve

PlanePolygon

Triangle

Half Plane

Sector

Cylinder

Cone

Sphere

PerpendicularParallel

Perpendicular bisector

Midpoint

Vector Sum

MeasurementTransfer

Trace

6 7 8 9

Central Symetry

Half-Turn

Reflection

Translation

Rotation

EquilatralTriangle

Square

Regular Pentagon

Regular Hexagon

Regular Octogon

Regular Dcagon

Regular Dodcagon

Pentagram

Ttrahedron

XYZ Box

Prism

Pyramid

Convex Polyhedron

Open Polyhedron

Cut Polyhedron

Regulat Tetrahedron

CubeRegular Octahedron

Regular Dodecahedron

Regular Icosahedron

And the brand new tenth icon with numbers: 10

Distance

Length

Area

Volume

Angle

Coord. & equationCalculator

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0.3. Some basic constructions

0.3.1. Points

With the tool Point , we can create a point

on the given horizontal plane if the cursor is

located on this plane (in this case the point

cannot be dragged out of the plane apartusing the tool Redefinition).

A point can be yet created wherever we want

out of this plane: for that we must click on the

shift button.

When a point has been created by this

method, as soon as we drag it, the grid of the

given plane appears, really only a part of thisgrid. Four arrows appear indicating the four

directions where the point can be dragged in a

parrallel plane to the given one.

A point created in space can be dragged

vertically in clicking on the shift button ; in

this case, only two arrows appear indicating

the two vertical directions to drag.

0.3.2. Lines

If we use the tool Perpendicular , when

approaching the cursor from the horizontal

given plane, this plane blinks because it is

recognized by Cabri which propose to createa perpendicular line to this plane.

After clicking, Cabri proposes immediately a

point where we want this perpendicular to

pass through. Before accepting it is possible

to drag the line until the chosen position.

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If we drag and reach a position out of the

rectangle representing the given plane, still

with the tool Perpendicular Line activated,

Cabri still propose to create a perpendicularline but passing through a point defined in

space located by default on the given plane.

0.3.3. Circles and some objects in space

0.3.3.1. Circles

There are two different ways to create circles

(as showed on the rightside) :

By a plane approach, in three steps: click

first on the plane which will contain thecircle, , click after on the center point of this

circle and at last click on a point to define the

radius.

By a 3D approach, in two steps: click first

on a line or a segment which is the axis of the

circle and after on a point of the circle.

0.3.3.2. Cubes and polyhedrons

We need three steps to create a cube: first

click to choose the plane which will contain

the first side of the cube, second click tochoose the center of this side and third click

to choose one vertex of the expected side.

This construction is similar for other

polyhedrons.

0.3.3.3. Rotations

This cube can be transformed by a rotation :first click to choose the axis of the rotation,

second click to choose the object we want to

rotate and the two last clicks for two points,

the second being the rotated one with our

rotation. Cabri will display texts like

rotation of this cube , around this axis,

mapping this point (here the intersection

point between the displayed circle and the

visible horizontal plane) and on to this point

(on circle) (here the thick point). This

transformation needs four clicks.

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0.3.3.4. Planes and intersections with planes

Here we have created two perpendicular lines

to the given plane and a plane perpendicular

to one of them through one of its point.

We create the second intersection point of this

plane with the second perpendicular line. We

create the segment showed at the right

handside. With a right click, we obtain a

menu containing the tool Mask-Show .We

use this tool to hide the upper plane.

On the right we have constructed a cylinder as

the product of a circle with a vector, a prism

as the product of a polygon and a vector, a

cone as the product of a circle and its summit

and a sphere with its center and one of its

point.

Here we have created a plane defined by a

segment a point (a thick point on the given

plane). It was possible after that to define and

display the intersections of this plane with the

solids we have created (apart from the prism).

We dont pretend this introduction to be complete. Our aim was only that the reader becomes

quickly familiar with this new environment. A lot of possibilities have not been presented and

especially the tool Animation . A very good thing to do is to have a look on the animation

of Claude, the little character created by Kate Mackrell of Queens University in Kingston

(Ontario, Canada). Claudes animations will show you the huge potentialities of Cabri 3D in

modelisation. As I know, there does not exit at this moment a competitor of Cabri 3D able todo such wonderful things.

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1. CONIC SECTIONS AND SQUARE FUNCTION

1.1.To discover the 3D geometric definition of conics

We have constructed a figure that allows us to generate the intersection of a cone (displayed

completly with its two parts) and a plane rotating around an horizontal line. We can notice

that Cabri recognize two sorts of intersections : ellipses (intersection with only one part of thecone) and hyperbolas (intersections with the two parts of the cone).

Another construction can generate the intersection of a cone with a plane tangent to one of its

generatrix ; in this case Cabri recognize as expected a parabola :

This activity lets the students understand very quickly in changing the point of view that

ellipses, hyperbolas and parabolas are curves generated by the intersection of a cone and aplane.

1.2. To understand that the curve of the square function is a parabola as defined

previously in space

We have first modelised the blackboard in the front plane coloured in black. We have

constructed lines to get the grid of this plane. For that we have used, vectors, translations and

symetries. At last we have constructed the conic passing through the 5 points (-2 ; 4), (-1 ; 1),

(0 ; 0), (1 ; 1), et (2 ; 4). Cabri recognizes this curve as a parabola.

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What is possible to do and what has been done is to find out a cone whose intersection with

the blackboard plane is the conic previously displayed.

The red thick point that can be dragged along a vertical line. This vertical line is chosen to be

a generatrix of the cone we are searching for. Dragging this point will modify the opening of

the cone we have constructed.

On the two figures below, we can see that the constructed cone cut the blackboard in a curve

that can be changed to reach progressively the parabola displayed at the beginning.

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At last we obtain a position wher the intersection curve between the blackboard and the cone

is superimposed to the representative curve of the square function y = x.

This presentation justifies in fine the name of parabola given to the curve of the square

function.

2.

TWO WAYS TO APPROACH NETS OF POLYHEDRONS

2.1. The direct way

First we construct a cube directly with the tool Cube . A click on this cube with the tool

Open Polyhedron provides immediatly the dynamic net of this cube.

The net can be opened and seen from all points of view.

A right click on the net opens a menu where

we can choose to edit the classical net in a

plane. You can see what we get below

2. A dynamic way using knowledge and generating knowledge

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We obtain the first vertical side of the cube in rotating the inferior one around one edge of this

inferior one. We obtain the other sides in using a rotation around a vertical axis having an

angle given by two consecutive points of the initial square.

We can see on the right that we

have applied to the horizontal

square the rotation mapping thecenter of the horizontal square onto

a new point on a special circle. This

circle has been defined around a

segment which is a side of the

horizontal square and passing

through its centre.

To modelize the opening of the top, we have created this one as the image of the square with

big holes with a rotation defined by the angle of the two displayed segments (these are

obtained as translated of the two displayed vectors).

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To open or to close the cube is possible in dragging the center of the square with big holes.

We obtain a similar result as the one we had qwith the tool Open Polyhedron .

We have shown that the same technique could be used in staring with a regular pentagon toobtain the dynamic net of a dodecahedron (Schumans construction).

In the last figure, we have created the intersection point of two lines which are therefore

secant for Cabri. We are at a level which is the Cabri proof (the validation at this stage is a

particular validation typical of information tools). The construction ends in creating the

symetric pentagons of the five first ones with respect to the special intersection point.

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The dodecahedron is finally obtained in closing the net, in dragging the pilot point located on

the initial circle.

The second figure above has been obtained in dragging the pilot point onto the inside. At the

beginning of the operation we can get the impression that the dodecahedron will collapse butat the end we obtain this wonderful stared polyhedron. It is clearly possible to understand that

a Cabri figure is more special than than a paper and pencil figure.

The last figure shows the interior of the

dodecahedron in emptying one side of this

solid.

3. CLAUDES ADVENTURES

3.1. Claudes caroussel

This animation uses symetries and

rotations with all the solids availables

in Cabri 3D. The polyhedrons aregoing up and down and Claude is

saying hello with his arm.

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3.2. The rowing Claude

The modelisation realised by Kate

Mackrell in this file is absolutely

wonderful.

3.3. Claude is biking

Here we can see Claude cycling like

on a real bike.

3.4. Claude slides on a Mbius strip

With this file we have two problems of

construction.

How to construct the strip ?

How to realise the animation of Claude along

this strip ?

3. CONCLUSION

A completly new approach of space geometry is possible with this software. This paper tries

to give the best information about it. Teachers must know that using this software will change

their one approach of 3D geometry as well the approach of their students. Problems can besolved in such an environment in an experimental way, that was not possible before. New

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problems can be generated by this environment. 3D geometry can be approached like a game

which is motivating for students.

http://www.chartwellyorke.com/cabri3d/introtocabri3d.htm

http://www.cabri.com/v2/pages/fr/products_cabri3d_tutorials.php