NonEuclid uses a bounded, two-dimensional, model (the Poincaré Model) of a particular Non-Euclidean Geometry called Hyperbolic Geometry. The large empty circle that appears when you first start NonEuclid is called the "

The following steps lead you through the construction of the Hyperbolic Geometry Triangle.

- Select the "Draw Line Segment" option from the "Constructions" Menu. This will cause the "Draw Line Segment" Dialog to appear.
- Move the mouse inside the boundary circle.
- Click the mouse somewhere inside the boundary circle. This will cause a point to be plotted. After plotting this first point, notice that as you move the mouse inside the boundary circle "Length = " followed by a number is printed in a text box within the "Draw Line Segment" dialog box. This length is the distance from the first point you plotted to the current location of the mouse.
- Click the mouse in a second place inside the boundary circle. This will cause a second point to be plotted and a straight line to be drawn between the two points.
- Click the mouse on one of the endpoints of your new line segment. Then move the mouse to a third point and click again. This will cause a second straight line segment to be drawn. Two sides of your triangle are now complete.
- Construct the third side by clicking on the two open endpoints. Your First Hyperbolic Geometry Triangle is now complete.
- You can now measure angles and the length of the sides of your triangle by selecting the "Measure Triangle" command from the "Measurements" Menu. Notice that the sum of the three angles of your triangle is LESS THEN 180°.

**More Exercises for Getting Started:**

Now that you have successfully constructed and measured a triangle,
it would be a good idea to get a "feel" for this strange geometry.
Construct lots of lines and see if you can notice any patterns. Some
straight, Hyperbolic Lines appear to be very curved, and others appear
almost perfectly straight. Can you predict which pairs of points
will determine straight looking straight, hyperbolic lines, and which will
produce curved looking straight, hyperbolic lines? Do straight, hyperbolic
lines appear curve toward the center, toward the boundary or sometimes
toward the center and sometimes toward the boundary? Given
two points, try to predict the approximate path of straight, hyperbolic
line that passes through them.

Another good way to use NonEuclid is to open your regular high school geometry book to the section called "Constructions With a Straight Edge and Compass". Try to duplicate these constructions in NonEuclid. Some of them will work perfectly (but look quit odd), and others will totally fail. Try to figure out why.

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Copyright©: Joel Castellanos, 1994-2002