Instructions for Realize applet

Drawing area

You can draw an edge connecting any two vertices by pressing the mouse button when the cursor is over the square marking a vertex, and releasing it when the cursor is over another square. If the edge you draw intersects a previously drawn edge, the old edge is deleted. When the polygon is completely triangulated, the applet will reposition each point so that the polygon triangulation is a Delaunay triangulation.

You can zoom in by pressing the 'i' key, zoom out with the 'o' key, and centre the view on the current mouse position with the 'c' key. Depending on your platform, you might have to click the mouse button inside the drawing area to get keyboard focus in the drawing area.

Sides

Lets you specify the number of sides of the polygon. Enter a number and press Return. Removes any edges you might have drawn. You can't have more than 99 sides because there is only room to type two digits.

Make

If you are too lazy to draw the edges yourself, you can click on one of these buttons to get a predefined triangulation.

Fan

Connects all the vertices to vertex 0

Zigzag

The interior edges form a zigzag path from one end of the polygon to the other.

Ring

Connect every other vertex by a diagonal, then repeat in the remaining interior polygon. This is also known as a "bushy" triangulation.

Random

Choose one possible triangulation at random. Each triangulation is equally likely to be chosen.

Show

These check boxes allow you to control what is displayed in the drawing area

Circumcircles

Shows the circumcircle of each triangle. You can verify that no vertex is inside any circumcircle.

Angles

This shows the value of the angle at each corner of each triangle. They are in arbitrary units. You can verify that the sum of the angles in each triangle is the same (s, say). And that the sum of all the angles adjacent to a vertex is less than s (this means that the polygon is convex). And that the sum of the two angles opposite to an interior edge is less than s (this means that the triangulation is Delaunay).

Triangles

Shows the interior edges.

Vertices

Draws little squares to mark the position of each vertex.

Parentheses

There is a one-to-one mapping between balanced strings of parentheses and polygon triangulations. Whenever you specify a polygon triangulation, either by drawing it or clicking on one of the buttons, the corresponding parenthesis string is displayed here. You can also type in any balanced parenthesis string and see the corresponding polygon.