Interactive Geometry Illustrations
The greatest value of a picture is that it forces us to notice what we never expected to see. - John Tukey
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This page contains a number of links to pages designed to give an interactive illustration of some of the theorems covered in class. These were developed using Geogebra.
Another excellent resource is David Joyce's interactive version of Euclid's Elements.
The links below will take you to the theorems. I will (slowly!) update this page as I code more theorems. Suggestions are welcome!
Euclidean Geometry
Pictures related to tutorial and homework problems
Drawing in two-point perspective
If you project a grid of rectangles from one plane to another, the result will be a grid of quadrilaterals. The page drawing a grid of rectangles in two-point perspective explains how to draw this picture using a ruler. To see how the construction works, you can also read the proof using Desargues' theorem
If you project a circle from one plane onto another, then the resulting figure will be a conic section. It is natural to ask how to precisely construct this conic (i.e. how to construct the focal points). The page How to draw a circle in perspective explains how to do this for the case where the conic section is an ellipse.
If you are interested in perspective drawing, then you may be interested in watching a talk about the Mathematics of Sidewalk Illusions.
Constructing conics
The links below will take you to interactive pictures explaining various constructions of the ellipse and hyperbola.
Inversive Geometry
You can create your own constructions and try out the theorems from class in the Inversive Geometry Interactive Website.
Click here to see an illustration of the theorem that inversion preserves orthogonal circles.
The following link will take you to an explanation of the three different constructions of the inverse that we studied in class.
Here is an example of Steiner's Porism.
Click here for an interactive picture of the inverse of a pencil of parallel lines.
Click here for an interactive picture explaining Question 6 on Tutorial 9 (constructing Poncelet quadrilaterals).
Spherical Geometry
Spherical geometry can be visualised in the \(xy\) plane, using stereographic projection through the north pole. The following pages explain this further.
Once we stereographically project the sphere onto the \(xy\) plane, we can then use a ruler and compass to construct lines and circles, and then start carrying out spherical analogs of other Euclidean constructions, such as equilateral triangles, perpendicular bisectors, etc. The following pages contain exercises to carry out these constructions.
Hyperbolic Geometry
Here is an example of a rigid motion in the Poincaré disk.
In Tutorial 11 you have to do some ruler and compass constructions in the Poincare disk model of hyperbolic geometry. Below are links to interactive pictures of these constructions.
If you have finished these and would like more of a challenge, then try the constructions below. If you really understand the Euclidean constructions then you will find these easy.
Below are some illustrations of more theorems in hyperbolic geometry.
Further material beyond what we have covered in the course
The page on constructing cubics contains some more interesting constructions in the plane extending what we have seen for conics.
The following links contain interactive pictures explaining
Other useful external resources
For hyperbolic geometry, another excellent resource is the interactive website Hyperbolic Geometry in the Poincare Disk.
The Poncelet-Steiner theorem says that if you are given a single circle and its centre, then any points constructible with a ruler and compass are also constructible with a ruler alone. The Poncelet-Steiner workbook explains how this works with the aid of Geogebra illustrations.
