Interactive geometry software (IGS) or dynamic geometry environments (DGEs) are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. In most IGS, one starts construction by putting a few points and using them to define new objects such as lines, circles or other points. After some construction is done, one can move the points one started with and see how the construction changes.
The earliest IGS was the Geometric Supposer, which was developed in the early 1980s.[1] This was soon followed by Cabri in 1986 and The Geometer's Sketchpad.
Comparison
There are three main types of computer environments for studying school geometry: supposers[vague], dynamic geometry environments (DGEs) and Logo-based programs.[2] Most are DGEs: software that allows the user to manipulate ("drag") the geometric object into different shapes or positions. The main example of a supposer is the Geometric Supposer, which does not have draggable objects, but allows students to study pre-defined shapes. Nearly all of the following programs are DGEs. For a related, comparative physical example of these algorithms, see Lenart Sphere.
License and platform
The following table provides a first comparison of the different software according to their license and platform.
The following table provides a more detailed comparison:
Software
Calculations
Macros
Loci
Animations
Scripting
Assignments
LaTeX export
Web export
Multilingual
Proofs
Extra
Cabri II Plus
Yes
Yes
Yes
Yes
Yes
Yes (with plug-in)
No
Yes
Yes
Yes (on relations)
Available on TI Calculator
Calques 3D
Yes
Yes
Yes
Yes
No
No
No
No
Yes (FRA ENG DEU ESP PTG)
Yes (on relations)
Experimental connection with some CAS
CaR
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
?
CaRMetal
Yes (recursive)
Yes
Yes
Yes (multiple)
Yes (JavaScript)
Yes
Yes
Yes
Yes
Yes (probabilistic)
Amodality, folder system, the Monkey
Cinderella
Yes
Yes
Yes
Yes
Yes
Yes
Yes (PDF)
Yes
Yes
Probabilistic
Several geometries, Physics simulations
Ganja.js
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
No
No
2D and 3D, projective and conformal, Geometric Algebra.
GCLC
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
Yes
Readable proofs, support for 3D
GeoGebra
Yes
Yes
Yes
Yes
Yes (JavaScript)
No
Yes (PSTricks & PGF/TikZ)
Yes
Yes (55 languages)
Yes
CAS, HTML5 Export (from version 4.2) 3D & Automatic Proof (from version 5.0)
Geometria
Yes
No
Yes
Yes
No
Yes
No
Yes
Yes
No
Two-role (teacher, student) model
Geometrix
Yes
No
Yes
Yes
No
Yes
No
No
Yes
Yes
Interactive proof, diagram checking, teacher/student models, labels with dynamic placeholders
Geometry Expressions
Yes
No
Yes
Yes
No
No
Yes
Yes (Interactive HTML5/JS Apps)
Yes
No
Symbolic calculations, which can be copied as input for CAS, TeX, and source code in 21 formats/languages. Functions. Arcs on any function or curve. Website for exported HTML5 Canvas and JavaScript Interactive Apps (Euclid's Muse).
GeoNext
Yes
No
No
Yes
?
?
No
?
Yes
No
Available as a web app
Géoplan-Géospace
Yes
Yes
Yes
Yes
Yes
No
No
Yes (activeX)
Yes
Yes
Sequences, 2D & 3D, human readable file format
GeoProof
Yes
No
No
No
No
No
Yes
No
No
Yes
Automatic formal proofs
GEUP
Yes
Yes
Yes
Yes
Yes
No
?
No
Yes
No
CAD functionality through CADGEUP
iGeom
Yes
Yes
Yes
No
Yes
Yes
No
Yes
Yes
Probabilist
Recurrent scripts
Kig
Yes
Yes
Yes
No
Yes (Python)
No
Yes (PSTricks)
No
Yes
No
Labels with dynamic placeholders
Live Geometry
Yes
Yes
Yes
Yes
No
No
No
No
No
No
Includes player.
Sarit2d
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
Available on web
Sketchpad
Yes
Yes
Yes
Yes
Yes
No
?
Yes (limited)
Yes
No
Functions & function plots, symbolic differentiation, mathematical notation
Tabula
Yes
Yes
Yes
Yes
No
No
No
No
No
No
Folding, cutting, taping, marker, and working instrument models.
Tabulae
Yes
Yes
Yes
No
No
No
No
Yes
Yes
No
Collaborative sessions over the internet.
Cabri 3D
Yes
No
No
Yes
No
No
No
Yes (limited)
Yes
No
?
Archimedes Geo3D
Yes
Yes
Yes
Yes
No
No
No
No
No (Eng De Fr)
No
Intersection of Loci
GEUP 3D
Yes
Yes
Yes
Yes
Yes
No
No
No
Yes
No
CAD functionality through CADGEUP
Netpad
Yes
Yes
Yes
Yes
No
No
No
Yes
No
Yes
Base on Web
Software
Calculations
Macros
Loci
Animations
Scripting
Assignments
LaTeX export
Web export
Multilingual
Proofs
Extra
Macros
Features related to macro constructions: (TODO)
Software
Allows recursity
Allows saving
Cabri II Plus
Yes
Yes
Calques 3D
No
Yes
GCLC
No
No
GeoGebra
Yes
Yes
Géoplan-Géospace
Yes
Yes
GEUP
Yes
Yes
iGeom
Yes
Yes
Kig
?
Yes
KSEG
Yes
Yes
Sketchpad (GSP)
Yes (via Iteration)
Yes
Loci
Loci features related to IGS: (TODO)
Software
Take a point of a locus
Intersection of two loci
Cabri II Plus
Yes
Yes
Calques 3D
No
No
CaR
Yes
Yes
GeoGebra
Yes
No
Géoplan-Géospace
Yes
No
GEUP
Yes
Yes
iGeom
Yes
No
Kig
Yes
No
Sketchpad (GSP)
Yes
No
NetPad
Yes
Yes
Proof
We detail here the proof related features. (TODO)
Software
Interactive proofs
Automatic proofs
Probabilistic proofs
Cabri II Plus
Feedback for
No
Yes in Cabri I
Cinderella
No
Using external CAS
Yes
GCLC
No
Yes
No
GeoGebra
Yes
Yes
No
Geometrix
Yes
Yes
No
Géoplan-Géospace
No
No
Yes
GeoProof
Yes
Yes
No
iGeom
No
No
Yes
Jeometry
No
Yes
No
NetPad
Yes
Yes
?
Measurements and calculation
Measurement and calculation features related to IGS: (TODO)
Software
Arbitrary precision
Arithmetic expressions
Trigonometric functions
If
Object existence test
Cabri
Yes
Yes
Yes
Yes
No
Calques 3D
No
Yes
Yes
No
No
C.a.R.
No
Yes
Yes
Yes
Yes
GCLC
No
Yes
Yes
Yes
Yes
GeoGebra
No
Yes
Yes
Yes
Yes (JavaScript)
Geometria
No
Yes
Yes
No
No
Géoplan-Géospace
No
Yes
Yes
Yes (μ function)
No
GeoProof
Yes
Yes
Yes
Yes
No
Geometrix
No
Yes
Yes
Yes
No
iGeom
No
Yes
Yes
No
No
NetPad
Yes
Yes
Yes
Yes
No
Graphics export formats
Software
PNG
BMP
TIFF
GIF
SWF
SVG
EMF
Fig
Postscript
Pdf
LaTeX/Eukleides
LaTeX/Pstricks
LaTeX/PGF/TikZ
Asymptote
Calques 3D
No
No
No
No
No
No
No
Yes
No
No
No
No
No
No
C.a.R.
Yes
No
?
?
?
Yes
No
Yes
Yes
No
No
Yes
?
?
Cinderella
Yes
Yes
?
?
?
?
?
?
No
Yes
?
?
?
?
GCLC
No
Yes
No
No
No
Yes
No
No
Yes
No
No
Yes
Yes
No
GeoGebra
Yes
No
No
Yes (animated)
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
Yes
Geometry Expressions
Yes
Yes
Yes
Yes (animated)
No
No
Yes
No
Yes
No
?
?
?
?
GeoProof
Yes
No
?
?
?
Yes
No
No
No
No
Yes
No
?
?
Kig
Yes
Yes
?
?
?
Yes
No
Yes
Yes
Yes
No
Yes
?
?
KmPlot
Yes
Yes
?
?
?
Yes
?
?
?
?
?
?
?
?
KSEG
Yes
Yes
?
?
?
No
No
?
No
No
No
?
?
Geometrix
No
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
?
?
iGeom
No
No
No
Yes
No
No
No
No
Yes
No
No
No
?
?
Object attributes
Software
Color
Filled/not filled
Width
Transparency
Shown/hidden
Layer
Shape of points
Type of line
Cabri
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes
Calques 3D
Yes
Yes
Yes
No
Yes
Yes
Yes
Yes
C.a.R.
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
GCLC
Yes
Yes
Yes
No
Yes
Yes
Yes
Yes
GeoGebra
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Geometria
Yes
Yes
No
Yes
Yes
Yes
No
Yes
Geometry Expressions
Yes
Yes
Yes
Yes
Yes
Yes
No (but size)
Yes
Géoplan-Géospace
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes
Kig
Yes
Yes
Yes
No
Yes
No
Yes
Yes
GeoProof
Yes
No
Yes
No
Yes
Yes
Yes
Yes
Geometrix
Yes
Yes
Yes
No
Yes
No
Yes
Yes
GEUP
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes
iGeom
Yes
Yes
Yes
No
Yes
No
No
Yes
Sketchpad
Yes
Yes
Yes
Yes
Yes
?
Yes
Yes
NetPad
Yes
Yes
Yes
Yes
Yes
Yes
No (but size)
Yes
2D programs
C.a.R.
C.a.R. is a free GPL analog of The Geometer's Sketchpad (GSP), written in Java.
Cabri
Cabri Cabri was developed by the French school of mathematics education in Grenoble (Laborde, 1993)
CaRMetal
CaRMetal is a free GPL software written in Java. Derived from C.a.R., it provides a different user interface.
Cinderella
Cinderella, written in Java, is very different from The Geometer's Sketchpad. The later version Cinderella.2 also includes a physics simulation engine and a scripting language. Also, it now[when?] supports macros, line segments, calculations, arbitrary functions, plots, etc. Full documentation is available online.
Dr. Geo is a GPL interactive software intended for younger students (7-15). The later version, Dr. GeoII,[4] is a complete rewrite of Dr. Geo, for the Squeak/Smalltalk environment.
GCLC
GCLC[5] is a dynamic geometry tool for visualizing and teaching geometry, and for producing mathematical illustrations. In GCLC, figures are described rather than drawn. This approach stresses the fact that geometrical constructions are abstract, formal procedures and not figures. A concrete figure can be generated on the basis of the abstract description. There are several output formats, including LaTeX, LaTeX/PStricks, LaTeX/Tikz, SVG and PostScript. There is a built-in geometry theorem prover (based on the area method). GCLC is available for Windows and Linux. WinGCLC is a Windows version of GCLC with a graphical interface that provides a range of additional functionalities. GCLC is open source software (licence CC BY-ND).
GeoGebra
GeoGebra is software that combines geometry, algebra and calculus for mathematics education in schools and universities. It is available free of charge for non-commercial users.[6]
License: open source under GPL license (free of charge)
Languages: 55
Geometry: points, lines, all conic sections, vectors, parametric curves, locus lines
Algebra: direct input of inequalities, implicit polynomials, linear and quadratic equations; calculations with numbers, points and vectors
Calculus: direct input of functions (including piecewise-defined); intersections and roots of functions; symbolic derivatives and integrals (built-in CAS); sliders as parameters
Parametric graphs: Yes
Implicit polynomials: Yes
Web export: all constructions exportable as web pages as a Java applet
Macros: usable both as tools with the mouse and as commands in the input field
Animation: Yes
Spreadsheet: Yes, the cells can contain any GeoGebra object (numbers, points, functions etc.)
Dynamic text: Yes (including LaTeX)
Platforms: Mac OS, Unix/Linux, Windows (any platform that supports Java 1.5 or later)
Continuity: uses a heuristic 'near-to-approach' to avoid jumping objects
GeoKone.NET
GeoKone.NET[7] is an interactive recursive natural geometry (or "sacred geometry") generator that runs in a web browser. GeoKone allows the user to create geometric figures using naturalistic rules of recursive copying, such as the golden ratio.
Geolog
Geolog[8] is a logic programming language for finitary geometric logic.
Geometry Expressions
Geometry Expressions[9] does symbolic geometry. It uses real symbolic inputs and returns real and symbolic outputs. It emphasises use with a computer algebra system (CAS), as well as exporting and sharing via interactive HTML5, Lua and OS X dashboard widget apps.
GRACE (The Graphical Ruler And Compass Editor) is an analog of The Geometer's Sketchpad (GSP), written in Java.
Jeometry
Jeometry is a dynamic geometry applet.
Kig
Kig is a free (GPL) analog of The Geometer's Sketchpad (GSP) for KDE, but more calculus-oriented. It is a part of the KDE Edutainment Project.
KmPlot
KmPlot is a mathematical function plotter released under the free GPL license. Includes a powerful parser and precision printing in correct scale. Simultaneously plot multiple functions and combine function terms to build new functions. Supports functions with parameters and functions in polar coordinates. Several grid modes are available. Features include:
powerful mathematical parser
precise metric printing
different plot types (functions, parametric, polar)
trace mode: cross-hair following plot, coordinates shown in the status bar
zooming support
ability to draw the 1st and 2nd derivative and the integral of a plot function
support user-defined constants and parameter values
various tools for plot functions: find minimum/maximum point, get y-value and draw the area between the function and the y-axis
KSEG
KSEG is a free (GPL) analog of The Geometer's Sketchpad (GSP) with some unique features. This software can handle heavy, complex constructions in Euclidean geometry.
Macros: Yes. Editable and with support for recursion
Java-applet: No
Animation: No
Locus: Yes, but no direct way to place a point on a locus.
Assignments: No
Measurement/calculations: Yes (the calculator is a bit strange)
Platform: Unix/Linux, Windows, Mac OS (any platform that supports Qt)
Proofs: No
Extra: Editable
Live Geometry
Live Geometry is a free CodePlex project that lets you create interactive ruler and compass constructions and experiment with them. It is written in Silverlight 4 and C# 4.0 (Visual Studio 2010). The core engine is a flexible and extensible framework that allows easy addition of new figure types and features. The project has two front-ends: WPF and Silverlight, which both share the common DynamicGeometry library.
TracenPoche
TracenPoche is a completely Adobe Flash program. It is available in English, Spanish, and French.
3D programs
This section needs expansion. You can help by adding to it. (November 2013)
Archimedes Geo3D is a shareware program designed for 3D geometric constructions. It extends traditional ruler and compass constructions into 3D space, allowing users to work with elements such as points, lines, circles, planes, spheres, vectors, and loci. This software is compatible with Windows, macOS, and Linux platforms.
Euler 3D is a program that allows you to create and manipulate your own polyhedrons. It has a number of facilities: transformations, animations, creating duals, import/export VRML, etc.
All these programs can be divided into two category: deterministic and continuous. GeoGebra can be deterministic or continuous (one can change it in preferences).
All constructions in the deterministic programs (GSP, Cabri, Kseg and most of others) are completely determined by the given points but the result of some constructions can jump or behave unexpectedly when a given point is moved.
On the contrary, some constructions in continuous programs (so far only Cinderella and GeoGebra), depend on the number of hidden parameters and in such a way that moving a given point produces a continuous motion of the construction, as a result, if the point is moved back to the original position the result of construction might be different.
Here is a test to check whether a particular program is continuous:
Construct the orthocenter of triangle and three midpoints (say A', B' C' ) between vertices and orthocenter.
This is the nine-point circle, it intersects each side of the original triangle at two points: the base of altitude and midpoint. Construct an intersection of one side with the circle at midpoint now move opposite vertex of the original triangle, if the constructed point does not move when base of altitude moves through it that probably means that your program is continuous.
Although it is possible to make a deterministic program which behaves continuously in this and similar simple examples, in general it can be proved that no program can be continuous and deterministic at the same time.[12]
↑ Schwartz; Yerushalmy and Wilson (1993). The Geometric Supposer: What is it a Case of?. Hillsdale, NJ: Lawrence Erlbaum Associates.
↑ Battista, M.T. (2007). "The Development of Geometric and Spatial Thinking". In Lester, Jr., F.K. (ed.). Second Handbook of Research on Mathematics Teaching and Learning. Charlotte, NC: Information Age and the National Council of Teachers of Mathematics. pp.843–903.
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