What’s new in Cabri Geometry II Plus? (Available Jan 2003)
Expressions
Cabri Geometry II Plus introduces a new type of object: the expression. An expression is a
text representing a mathematical expression depending on variables. For example, “3*x2*y+1” is an expression, depending on variables “x” and “y”. All the operators of the
calculator are accepted in expressions. Expressions are entered like text or numbers using the
“expression” tool.
An expression can then be applied, using the “Apply expression” tool. To apply an
expression, the user shall select a number on the worksheet for each variable of the
expression. After all variables are bounded to a number, a new number is produced,
containing the value of the expression.
Expressions can be used to graph functions of the type y = f(x), or to explore scalar or vector
fields defined on the plane, in conjunction with a locus from a point on a grid (see below).
In the example below, we define the expression x^3/5-x^2-x+4, and apply it with variable “x”
being the abscissa of a point m on the x axis. Then we transfer the result (displayed under the
equation) on the y axis and build the corresponding curve point c. The curve is obtained as the
locus of c when the point m varies on the x axis.
Locus from a point on a grid
A construction involving a point on a grid can now be used to generate a locus, the discrete
set of all objects obtained for all positions of the moving point on the grid.
Such a locus can be used to explore scalar or vector fields.
In the example below, we introduce a cubic polynomial P in x and y as an expression, and
evaluate it using the coordinates of a point m on a grid. Then we bound it (using the max
operator) and use it as the radius of a circle with centre m. The set of all circles is obtained by
taking the locus of the circle as m varies on the grid. This gives a good representation of the
cubic curve P(x,y) = 0.
Equations of algebraic curves
The “Equation and coordinates” tool has been extended to loci. The equations are determined
numerically, as real algebraic curves of degree 6 or less. If the locus does not match this class
of curve, the equation can’t be determined. Since computations are carried on standard 64-bit
floating point numbers, numerical errors are likely to appear for curves of high degrees.
In the example, we show the equation of a deltoid (hypocycloid with three cusps) inscribed in
the unit circle.
Locus intersection
Intersection of objects are now extended to loci.
The example shows a cardioid, locus of point p when m varies on the circle. We can draw a
small circle centred at p, and take its intersections with the locus, obtaining a good
approximation of the tangent and normal lines at p.
Smooth locus rendering
The rendering of loci has been automated, to obtain a full display/print precision without user
intervention.
Smart lines
Cabri Geometry II Plus now offers the possibility to draw only the interesting parts of lines.
Lines are internally infinite, but generally, only a small part of the line is needed for the
understanding of a figure, and the remaining parts are useless “noise”, disturbing the
perception. Such “smart lines” are truncated after the last point on the line in both directions,
and the ends can be moved freely. The user has the choice to add an arrow at the ends of the
truncated line.
The example shows the construction of a Simson line for a triangle. Using smart lines
simplifies significantly the reading of the figure.
Enhanced colour palette
Cabri Geometry II Plus introduces an extended colour palette. The graphics engine has been
rewritten to internally work on 16 million colours. The application palette now provides a
selection of 36 distinguishable colours and 5 levels of gray (see image below).
Colour mixing
The fill colours of non-opaque surfaces are mixed on screen, applying subtractive colour
synthesis. More precisely, the 8-bit red, green, and blue components of the colour are
combined using a bit-wise AND.
The example shows the mixing of the three primary colours of subtractive synthesis: cyan,
magenta, and yellow, generating red, green, blue, and black.
Dynamic pen and fill colours
Any object can receive an arbitrary pen and fill RGB colour obtained from a numeric value on
the worksheet. The three red, green, and blue components are assigned independently.
This dynamic colouring may be used for optics simulation, to represent the different light
wavelengths. It may also find applications in the exploration of scalar and vector fields.
In the example, we defined a scalar field from a polynomial in x and y. Then we animated a
small filled circle, with dynamic colours corresponding to the values of the field at its centre.
The colour component assignment is accessed through the contextual menu (right mouse
button) of the object to colour.
Name everything
Cabri Geometry II Plus allows the unrestricted naming of all objects.
Rich text edit
Each character of a text box can now receive specific font and attributes. Object names,
numbers, and equations can be dynamically inserted in a text box, and are updated when the
figure is manipulated.
Contextual menus
Pressing the right mouse button over an object, or over an empty part of the worksheet
activates a contextual menu.
For an object, the contextual menu allows a rapid selection of its attributes, and displays the
tool used to construct the object, along with a list of the input objects for this tool.
The example below shows how a point was constructed. The object under the mouse arrow is
highlighted (here, the triangle) while browsing the list of construction objects.
Setting the pen and fill colour components is done using the “Variable colour” and “Variable
fill colour” entries of the contextual menu (see the item on dynamic pen and fill colours).
The opacity of objects is also accessed from this menu. Opaque objects don’t mix colours.
Setting a texture for a four-sided polygon is accessed from this menu (see the item on
textures).
The contextual menu obtained by right-clicking on an empty worksheet space allows to set a
background colour or image for the whole worksheet.
More attributes
The attribute bar (displayed by selecting “Show attributes” in the “Options” menu) provides
more options. It can control the pen, fill and text colours, text size, and all other object
attributes, including smart line behaviour.
Improved ambiguity resolution
When the user clicks on more than one acceptable object for the current tool, an ambiguity
menu is displayed, allowing choice between the acceptable objects. While browsing this
menu, the corresponding object is now highlighted in the figure, making it possible to
distinguish between unnamed objects of the same type.
The example shows what happens when selecting an implicit intersection point on the three
perpendicular bisectors of the sides of a triangle.
Textures
Rectangular images can be attached to points, segments, and convex quadrilaterals. The
images are files in BMP, GIF, or JPG formats. Attaching an image to an object is done
through the contextual menu for the object (right mouse button).
The first example shows two images attached to a planet and its satellite on an elliptic orbit.
The second example shows a connection rod image linked to a segment of the figure. This can
be animated to visualise the motion of mechanical parts inside an engine.
Session recorder
This feature was implemented for use by researchers in didactics. It allows to record and print
the evolution of the construction, in order to study the behaviour of pupils solving a problem
using Cabri Geometry II Plus.
British English Extended Manual
A new 124 page manual in British English, edited by Sandra Hoath, is provided.
Cabri Geometry II Plus, and other support materials for Cabri, are available from ChartwellYorke Ltd., 114 High Street, Belmont Village, Bolton, Lancashire, BL7 8AL, tel 01204
811001, fax 01204 811008, info@chartwellyorke.com, www.chartwellyorke.com.
‘Using Cabri-Geometry II in Key Stage 3’ by Jack Jackson
Geometry is a dynamic subject, best understood when we discover it for ourselves
with a dynamic interactive tool. Cabri Geometry II software is the perfect tool for
exploring geometry dynamically. This set of worksheets shows how to use Cabri
to explore and understand KS3 school geometry.
Each worksheet is in two parts: the first part provides systematic instructions to set
up a robust screen that will demonstrate particular geometrical properties. The
second part is an investigation that will lead to the students discovering the
geometrical properties for themselves.
They are designed to be used in a variety of ways.

Used by teachers to set up a screen, which can be saved on a network
so the student can access it, and so start the investigation immediately.

Used by students to set up their own screens so that when they start
investigating they understand the geometrical properties of the screen.

Used by teachers to set up a demonstration on an interactive whiteboard.
Many of the activities provide opportunities for students to make the first steps
towards developing geometrical proof, in the four-step process of: convince
yourself; convince a friend; convince a pen-friend; convince your maths teacher.
These activities will help with the crucial first step of students understanding
the geometry before starting on a recording process that will eventually lead to
a written proof.
The activities cover all of the specific examples referred to in the UK's Key
Stage 3 National Strategy Framework for Teaching Mathematics Years 7, 8
and 9, and much more besides. References to objectives in the Key Stage 3
Framework can be found in the last section of the book.
ISBN 1-904506-00-3, £20, Published by Chartwell-Yorke
Available exclusively from Chartwell-Yorke Ltd., 114 High Street, Belmont Village, Bolton,
Lancashire, BL7 8AL, tel 01204 811001, fax 01204 811008, info@chartwellyorke.com,
www.chartwellyorke.com.