We are not licensed to use the trial software with a class. This does not preclude setting a homework task and this article has been designed with that thought in mind.
A screen dump of this project has been used on the cover of this edition. You will notice that the learning environment is visually appealing to students even though surfaces could be more translucent so that hidden structures become a bit clearer. Using Cabri 3D is just as intuitive as using the familiar 2D Cabri Geometry.
In this exercise we will build a 3D model to find the intersections of a plane with a cone pair. Start with the default natural view which consists of a section of a horizontal plane supporting a set of vectors along the three axes. Toolboxes are arranged across the top of the screen and we will use tools from the first six sets which are listed in the order: Manipulation, Points, Curves, Surfaces, Relative Constructions and Transformations.
Start by selecting a Plane from the Surfaces toolbox and then click on the vectors OY and OZ. This will construct the YOZ plane. Select the Point tool and place points at A, B, C and K on the plane YOZ in that order. To label the points, simply select the point and start typing.
By embedding the points in the plane YOZ we ensure that they are all co-planar. From the Curves toolbox select the Vector tool, click on A and then B to form the vector AB. Form the vector AC in the same way. Click on the YOZ plane and type Control with M to hide the plane from view. Be careful not to delete the plane or you will lose all the structures defined using it.
From the Relative Constructions toolbox select the Parallel tool and draw line KJ through K parallel to the vector AB and another line through K parallel to the vector AC. Select the Translation tool from the Transformation toolbox and translate the point K by using the vector AC to form a new point L. Then translate the point L to form another point M. Thus KM should be parallel to AC and twice as long. I could not find an Enlargement tool but this is a messy way of achieving the same thing. We can control the length and direction of KM by moving the point C.
From the Curves toolbox, use the Circle tool to construct a circle through the point M about the line KJ. Use the Cone tool from the Surface toolbox to draw a cone using this circle and the point K. Click on the cone and then go to the Transformation toolbox and select Central Symmetry to "reflect" the cone in the point K. I teach this transformation as an enlargement of the cone about K with a scale factor of negative one.
From the Curves toolbox select Intersection Curve and then click on the cone and then on the visible plane. The intersection will appear as a thin ellipse. You may like to choose a bolder colour and a thicker "curve radius".
By moving the point B you can re-orientate the line of symmetry of the cones and by moving the point C you can change the size and shape of the cones. You can also move the point K to translate the cones. It is easy to demonstrate the ellipses and hyperbolas shown here and on the front cover, but I could not get a fine enough adjustment to get a circle or a parabola.
Eventually I made new models. In one I defined the line KJ along the vector OZ so that the intersection should have been a circle. However, even when forced, the software could only recognise the circle as a ellipse. When I forced KM to be parallel to the plane the intersection should have been a parabola but I found that I could not then form any intersection of the cone with the plane.
In later versions we might expect some ability to measure the figures we have constructed. For example, it would be useful to be able to compare the volumes of a cylinder and a cone with the same base and height.
However, we do not want students to have to do a lot of calculating to define their models. Many 3D packages require a higher level of mathematical sophistication to define their figures than the level we are trying to teach. By contrast, Cabri 3D is conceptually very simple to use and I imagine it would help students who have trouble visualising 3D models. I would love to use it with a class.
If you are still preparing next year's budget I recommend that you contact the AAMT office. There are substantial discounts for AAMT members. Prices for Cabri 3D are:
Single user Members $115.00 Others $143.75 2-10 users Members $295.00 Others $368.75 Site license Members $560.00 Others $700.00
There are bundles available with both Cabri II and Cabri 3D. I would also like to see a student version made available--at an affordable price point so that all students can have the advantage of being able to use the software at home.
Hartley Hyde * firstname.lastname@example.org * users.senet.com.au/~hhyde/CACTUS