Inversions: SketchPad Files

This is one of my favourite bits! I'm a recent convert to Inversions and have written some scripts to help me to try to find the best spot to place the Circle of Inversion on the Shape to be Inverted. One of the reasons Inversions are useful is that they can transform a problem having difficult geometry into one with much simplified geometry. Once the problem can be solved in its inverted form then it is a relatively straight forward matter to transform the inverted answer back to the original geometry. See the Introductory and Second Example SketchPad files below and their associated explanatory notes in the two Word files.

Notes:
(1). On opening any of these scripts enclosed in the above pages you may not actually see them! - What you may see instead is a blank Sketch File probably called "Sketch01.gsp". If you are unfamiliar with SketchPad all that you need to do is to click on the "Restore" Button in the top right of the active Sketch file Window Toolbar.
(2). All of these filenames in this page are LONG! - Therefore, if you want to preserve these (Because Version 3.* of SketchPad does not use long filenames) you must either RIGHT-CLICK on the file links within the above directories and then choose "Save Target As...". This allows you to save the file directly to a disk/drive or after using a LEFT-CLICK select the "Save this file to disk" Option Box rather than selecting the "Open this file from its current location". If you choose this last option then Geometer's SketchPad will automatically open - if you have it! - but then after viewing the file if you then try to save it you will not be able to use a long filename.

Scripts

Each of these scripts puts a random point on the original object and inverts this so that its image point is placed on to the final inverted form. This is particularly useful because with complicated geometries the connection between the object and its inverted image can be awkward to spot at first but by dragging either of these extra points we can see how the associated point also moves.
Once you have the hang of them and are looking at more complex problems it is useful to have open all of the scripts given below at the same time as investigating your problem under consideration. It is then a simple matter to invert all manner of basic shapes that are contained within the problem. Then skill at placing and resizing the Circle of Inversion to simplify the inverted geometry may be rapidly acquired. See the Introductory and Second Example SketchPad files below, they are animated and show how a sensible positioning of the Circle of Inversion make both problems almost trivial.

D - Inversion of a Circle.gss
Here the black Circle is the one to be inverted, the green Circle is the Circle of Inversion and its position and size can be altered simply by dragging as in the second figure and the red Circle is the inverted image (Though it is not always circular - when is it not?).
Notice that this script cannot be used on an Arc but the script given below can.


D - Inversion of a Line.gss
Here the black Line is the object to be inverted. The green Circle is the Circle of Inversion and as above its position and size can be altered simply by dragging. The script works for a finite line and for an infinite Line.


D - Inversion of a Point.gss
Here A is the original Point and A' its inverted image.


D - Inversion of an Arc.gss
Notice that this script cannot be used on a circle but instead the script given above can.

Sketches

D - An Introductory Example to Inversions.gsp
If the square is of width a, what are the radii of the two internal circles - Here the Circle of Inversion is very poorly placed and does nothing to simplify the geometry of the original object. Look at the file to see a demonstration of a better choice!


D - Inversion Example 1 Notes.doc (158KB) (PDF 160KB)
This is a Word file with some background notes on Inversion Theory - as I understand it so far! - and a fairly full explanation of the solution.

D - A Second Example to Inversions.gsp
If the square is of width a, what is the radii of the 'Trapped' circle - Here, as with the case above, the Circle of Inversion is very poorly placed and does nothing to simplify the geometry of the original object. Again, look at the file to see a demonstration of a better choice!


D - Inversion Example 2 Notes.doc (135KB) (PDF 148KB)
This is a Word file with some background notes on Inversion Theory - as I understand it so far! - and a fairly full explanation of the solution.

D - A Third Example to Inversions.gsp
The problem here is to simplify the Inverted Geometry by resizing and repositioning the Centre of Inversion so as to facilitate the calculation of the radii and centre positions of the 'Trapped' Internal Circles in the given Arch structure. Again, look at the file to see a demonstration of a better choice!


D - Inversion Example 3 Notes.doc (325KB) (PDF 267KB)
This is a Word file with some background notes on Inversion Theory - as I understand it so far! - and a very detailed explanation of the generalized solution to this none trivial problem.

D - Chessboard Inverted in a Circle.gsp
I just wanted to see if I could do this one - Its really only use as decoration now because I've hidden all of the random points, etc...Still pretty though!


D - Circle(s) to Two Circles and a Tangent Line.gsp
This is a good file on which to experiment. You can use a Circle Tangency script to add further Circles to the original figure and then use the Inversion of a Circle script above to see where the new circle is inverted to - Try to work it out before you try this though!



D - Inversion of a Circle.gsp
This is an animated sketch that can be experimented with to investigated this type of inversion.


D - Inversion of a Line.gsp
This is an animated sketch that can be experimented with to investigated this type of inversion.


D - Inversion of a Modified Steiner Ring.gsp
This is another file with which to experiment by dragging the random points, and by dragging or resizing the Circle of Inversion.


D - Steiner 8-Ring.gsp
A cosmetic file that is animated but does not show any theory. Not one of mine but from Chuan, et al. Geometric Construction. A collection of advanced constructions, construction challenges and exercises, with downloadable sketches and JavaSketchpad illustrations.