Lab 3 -DWEG Model

Math 487 Lab 3 - 1/21

The next assignment will include certain constructions from this lab.

Background

Dr. Whatif's Euclidean Geometry (DWEG)

We will investigate 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.

DWEG Lab

The goal is to build up some tools for DWEG constructions and some example sketches of DWEG figures.

A. The Basics - Starting by downloading a file

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.

To start you off, download the file dweglab.gsp at this link.
The file contains one page and 3 tools. Notice that the tools use automatic matching. To keep this going, you should make new sketches by adding pages to your document in this way. This ensures the name of the point O stays exactly the same ("O-removed") in each sketch to insure the automatic match.

File->Document Options->Add Page -> Duplicate Page 1

Step 1 below is done in the sample file. There are two DWEG line tools. One includes the E-center of the E-circle in case you need it in constructions.

[This is already done in the file.] 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 tool D-line AB that takes as givens, the points O, A, B. (There are two D-line tools in the file.)

Construct a segment in DWEG. Given D-points A and B, construct the E-circle through O, A, B. Then let the E- angle bisector of angle AOB intersect the circle at point F. Then the E-arc through points AFB is the D-segment AB. You can hide F and the angle bisector. Make a D-Segment AB tool. Learn how to auto-match point O by reading in Sketchpad Help>Advanced Topics>Advanced Tool Topics>Automatically Matching a Given Object.

Construct a rectangle in DWEG. In a new sketch, construct m = D-line AB and construct D-lines a and b through A and B perpendicular to m. Then for any D-point E on b, construct the D-line through E perpendicular to b. Finally, intersect D-lines to construct point F to complete the set of vertices of DWEG rectangle ABEF. Construct the segments that are the sides to finish the figure.

Given a D-line m and a D-point C not on m, construct the unique D-line through C perpendicular to m. In a sketch construct a D-line m through A and B and a D-point C not on m. Construct a D-line n through C that is perpendicular to m. Make a tool D-perp ABC that automatches the point O-removed and takes as the other givens the points A, B, and C.

EXTRA CREDIT for later: It is possible to construct a tool that will also work when C is on m.

Given a D-line m and a D-point C not on m, construct the unique D-line through C parallel to m. In a sketch construct a D-line AB. Then construct D-line p through C which is D-parallel to m. Make a tool D-parallel ABC that takes as givens the points O (automatched) and A, B, C.

B. DWEG Lab Activity. D-Circles from D-reflection

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.

Background: Traditional Euclidean Case: circle from line reflections

Suggestion: You can either (1) just read and think about this, or (2) you can use physical mirrors or (3) you can use GSP to figure out how this works in the Euclidean case that you are going to mimic in the DWEG model.

To see how this goes, first try the concept out with standard Euclidean geometry using Sketchpad. To be clear, do this on the side for insight; this is not a D-constuction. We will do the D-construction 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?
Why? Explain why E-segments AB and AB' will be always E-congruent.
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.

DWEG model case: circle from line reflections (an experiment)

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 m = D-line AC. D-reflect B across m to get B'.
Then trace B' as C moves and A is fixed, 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 E-circle does the trace appear to be? How could you construct it.

A tool to construct the D-circle as an object. There is a tool in the sample file that will construct the D-circle through B with D-center A. You will be asked on the next assignment to explain and carry out such a construction, but you can use the tool for the next tasks for now.

C. DWEG Lab Activity. Using the D-circle tool.

(1) Constructing a D-Equilateral Triangle and a D-Square

Since you have tools that draw D-lines and D-circles, you can mimic the Euclidean constructions to construct:

Given any two D-points A and B, construct a DWEG equilateral triangle with side D-segment AB.
Given any two D-points A and B, construct a DWEG square with side D-segment AB.

(2) Taking Equal Steps with a D-compass

Let A and B be D-points. Construct p = D-line AB and the circle with center A through B.
Now mark equal steps on p by drawing a sequence of circles. For the first step, construct the D-circle with D- center B through A. Let C be the points of intersection of the circle with P so that AC is a diameter.
Next, construct the D-circle with center C through B and intersect the circle with p to get a new point D. Then the points A, B, C, D are equally spaced.
Keep going for a few more points to see what equally spaced points look like in the D-model.

(3) Taking Equal Steps with line Reflections

Let A and B be D-points. Construct p = D-line AB and the two lines a and b through A and B orthogonal to p.
Reflect A in b to get C. Reflect B in a to get B'; then reflect B' in b to get D. Then A, B, C, D should be equally spaced in the DWEG geometry.
Remember the link between double reflection and translation. This explains the even spacing. We consider the "marks" on the line p to be lines of a ruler on p.