The Toolbox

The Toolbox is a collection of Geometer's Sketchpad scripts especially created to do Poincaré geometry. The tools are organized into four subfolders: Construct, Measure, Transform and Auxilliary. The first three contain Poincaré analogues to some of the corresponding GSP menu items, with a few extras specific to Poincaré geometry. The last contains a few tools which may be useful for constructing your own Poincaré scripts. Please note that, since Geometer's Sketchpad is not optimized for Poincaré geometry (for example, the circle through three points simply disappears when the points become collinear, instead of becoming a line), some of these tools may not work when one of the objects involved is a Euclidean line. Simply move a point or two slightly and the tool will again do what it's supposed to.

You can download the Toolbox all at once, or you can download the tools individually from this page as you need them. If you download the Toolbox all at once, it will come already organized into subfolders. If you download the tools individually, you can create a Toolbox folder with subfolders yourself and use it to organize the tools as you download them. In either case, the Toolbox folder should be designated as the GSP Script Tools folder. (Under Preferences in the GSP Display menu, click on More . . . to get Advanced Preferences, then choose Toolbox as the Script Tools folder.) If you already have script tools that you want to use concurrently with the Poincaré tools, you can place the Toolbox inside your existing Script Tools folder; however the Poincaré tools will then be one menu level deeper and slightly less convenient to use.

The tools are designed to work with the "empty universe" sketch, which you can download by clicking on wherever you see it. As described on the Points page, this sketch contains several specially labeled objects used by the Toolbox: these labels should not be altered. To use each tool, select it from the menu to the left of your sketch, then select in turn the objects requested in the Tool Status box at the bottom of your sketch.

Here is a inventory of the individual tools.

Construct Folder

• Angle bisector. Bisects the Poincaré angle given by three points, matched in the order initial-vertex-terminal.
• Asymptotic line. Constructs the line asymptotic to any line through the given edgepoint and through another point not on that line.
• Circle by centre + point. Constructs the Poincaré circle with the given centre and through the given point.
• Common perpendicular. Constructs the Poincaré line perpendicular to two divegent lines (intersecting or asymptotic lines do not have a common perpendicular).
• Line. Constructs the Poincaré line through two points.
• Midpoint. Constructs the midpoint of the Poincaré segment given by its endpoints.
• Perpendicular line. Constructs the line through a point perpendicular to another line given by two points on it.
• Ray. Constructs the Poincaré ray from the initial point through the second point.
• Segment. Constructs the Poincaré segment joining point one and point two.

Measure Folder

• Angle. Measures the Poincaré angle given by three points, matched in the order initial-vertex-terminal.
• Circle area. [to come]
• Circle circumference [to come]
• Distance. Measures the Poincaré distance between two points.
• Triangle area [to come]

Transform Folder

• Transform. Transforms any point to its image under the basic Poincaré motion , given the point and the point to be transformed. Can be combined with a (Euclidean) rotation to give a general direct Poincaré motion. To produce an indirect Poincaré motion, start with a reflection in the origin; then do a direct motion.
• Translate [to come]
• Horolate [to come]
• Reflect. Poincaré reflects a given point across a line. (NOTE: If you can't select the line, try selecting and then deselecting the yellow interior of the edge circle, or hide it altogether.)
• Rotate. Poincaré rotates the first point about the second through the measured angle.

Auxilliary Folder

• Invert. Inverts the matched point in the edge circle.
• Line base. Constructs the Möbius circle perpendicular to the edge, through two Poincaré points, the centre of this circle, its points of intersection with the edge, and the inverses of the two given points in the edge.