First, let me separate those transformations which preserve the object shapes, their relative positions and proportions. Affine Transformations.
Change the objects at left. They are transformed to the point of view of P, at right. Notes: [list] [*]This is measuring in an ordinary way. Say I draw a triangle on a sheet of paper, walk outside and hold it up against the window so you can see it. I can move the paper up and down, left and right, rotate it, and (holding the paper flat), walk away from the window with it... ... none of this changes the triangle drawn on the paper. [*]Zoom (red circle) affects only the right panel. In other words, if the player examines objects more closely, it does not affect the setup (left). [*]The transformations of an arbitrary number of objects can be done simultaneously. (Alas, Javascript is unthreaded.) [*]Pick up O'. P and the green circle follow. To obtain this crude object behavior, I use vectors and script to get related points to push each other around. If you know of a better way, leave a comment! _________________ The Tangent Circle Problem: [list] [*]1. Tangent along the rim: solve for k [*]2a. Initial position: [url]http://www.geogebratube.org/material/show/id/58360[/url] [*]2b. Tangent to equal circles: [url]http://www.geogebratube.org/material/show/id/58455[/url] [*]3a. Four mutually tangent & exterior circles (Apollonius): [url]http://www.geogebratube.org/material/show/id/58189 [/url] [*]3b. Vector reduction: [url]http://www.geogebratube.org/material/show/id/58461[/url] [/list] [list] [*][b] →Affine Transformation[/b] [*]Reflection: Line about a Circle [url]http://www.geogebratube.org/material/show/id/58522[/url] [*]Reflection: Circle about a Circle [url]http://www.geogebratube.org/material/show/id/58185[/url] [*]Circle Inversion: The Metric Space [url]http://www.geogebratube.org/material/show/id/60132[/url] [/list] Iteration: [list] [*]Sequences 1: Formation [url]http://www.geogebratube.org/material/show/id/58896[/url] [*]Sequence 1: Formation [url]http://www.geogebratube.org/material/show/id/59816[/url] [*]Sequence 1: Iteration 1 [url]http://www.geogebratube.org/material/show/id/59828[/url] [*]Example of equivalent projections: [url]http://www.geogebratube.org/material/show/id/65754[/url] [*]Final Diagram: [url]http://www.geogebratube.org/material/show/id/65755[/url] [/list]