The next assignment will be to turn in certain constructions from this lab. Be sure to check on the assignment sheet as you work through the lab.
We will investigate an example of a model for Euclidean geometry which is not the standard (x,y) plane. In Marta Sved's book it is called Dr. Whatif's Euclidean Geometry. We call it DWEG for short.
Here is one description of the model: Choose a point in the ordinary plane and label it O. This is the point that we are going to remove.
The points in DWEG are the points of the plane excluding O [but including I, the point at infinity]. We will call the DWEG points D-points.
The lines in DWEG are either ordinary Euclidean circles through O or Euclidean lines through O. Since O is the point that is not there in the DWEG plane, it is not included as a D-point in any DWEG-line. [But the point at infinity is considered as one of the points of any DWEG-line that is also a Euclidean line]. We will call the DWEG lines D-lines.
The angles between lines in DWEG are the usual angles between circles or between lines and circles in Euclidean geometry. By perpendicular D-lines we mean Euclidean lines or circles that are perpendicular or orthogonal
In the constructions we will call the usual points of Euclidean geometry E-points, E-lines and E-circles to distinguish them from the objects in the DWEG model.
Now here is the drill. Use what you know about constructing Euclidean lines and circles to carry out constructions of Euclidean geometry that will demonstrate basic properties of this model.
Two D-points determine a unique D-line. Given two DWEG-points A and B construct a DWEG-line through A and B. (In other words, construct a Euclidean circle through A, B and O. In this Sketchpad exercise you can leave off the special case when the D-line is a Euclidean line. Make a script DlineAB that takes as givens, the points O, A, B.
There is a unique D-line through a given D-point that is perpendicular to a given D-line. In a sketch construct a D-line m through A and B and a D-point C, construct a D-line n through C that is perpendicular to m. (Note two possible cases: C is on m and C is not on m. Can you handle both with one sketch?) Make a script DperpABC that takes as givens the points O, A, B, C.
Given a D-line m and a D-point C not on m, there is a unique D-line through C parallel to m. In a sketch construct a D-line AB. The construct D-line p through C which is D-parallel to m. Make a script DparallelABC that takes as givens the points O, A, B, C.
Construct a rectangle in DWEG. Construct a rectangle by constructing two parallel lines and two lines perpendicular to the parallels. Save this figure as Drect.gsp.
Construct a segment in DWEG. Given D-points A and B, construct the E-circle through O, A, B. Then let the angle bisector of angle AOB intersect the circle at point F. Then the arc through points AFB is the D-segment AB.
We don't yet know how to measure distance in DWEG, but we do know how to reflect across a line in DWEG. We declare D-line reflection across mirror D-line m to be the same as circle inversion in Euclidean geometry if the D-line m is an E-circle or line reflection if the D-line m is an E-line. Now we will see how to produce the points of a circle with center A through B just by using line reflections.
To see how this goes, first try the concept out with standard Euclidean geometry using Sketchpad. To be clear, this is not a D-constuction. We will do that next. In a new sketch, draw points A and B and also line AC. Reflect B across line AC to get B'. Now trace B' as you drag C. What does the trace look like? Can we explain why? The line AC will always be a line through A and thus segment AB will be congruent to AB'. This means that if we take AB as radius, then AB' is always on the circle with center A through B. What's more, for any point G on the circle, there is a position of C so that line AC reflects B to G, so the trace of B' in principle traces out the whole Euclidean circle.
Important note: The E- circle through B with center A is orthogonal to all the E-lines through A.
Now we can mirror this circle construction in DWEG to find what a D-circle should look like. Take a D-point A and a D-point B. Construct a D-line m through A and a D-point C. D-reflect B across m to get B'. Then trace B' to see the locus of all such reflections. Does it appear to be an E-circle? Does the E-center of the E-circle appear to be A? What kind of E-circle does the trace appear to be?
Construct the D-circle as an object. Since A is not the E-center, we have to construct the E-center first, we can find 3 points on the D-circle and construct the E-circle through the 3 points. To do this, construct another D-line n through A and reflect B across this line to get B''. Now construct the E-circle through B, B', and B''.
To make a D-circle script, this construction is not so good, because the script will use the points O, A, B as givens (which is appropriate) but also the two points defining m and n (so it will have 5 givens when it should have 3). To reduce the givens to O, A, and B only, you will have to use what you have learned about orthogonal circles.
Explain why the D-circle through B with D-center A is orthogonal to all the D-lines through A.
If the previous statement is true, explain why the D-circle through B with D-center A is an Apollonian circle. (What two points define the Apollonian circle?)
Use the previous observation and your earlier construction work with Apollonian circles to make a script DcircAB.
First get the idea with a traditional Euclidean figure.
Now do the same for a DWEG figure to construct a ruler on a DWEG line p from D-points A and B on p and D-lines a and b..
Since you have made scripts that draw lines and circles, you have the Euclidean construction tools. Use these tools to construct these figures. If you are unsure what to do, carry out the standard Euclidean construction first and make note of each step. The circle tool is the compass and the line tool is the straightedge.