Lab 4 -DWEG Model

Math 487 Lab 4 - 1/26

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. 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.

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.
D-Reflection in a D-line m is either inversion in the circle m if the D-line is an E-circle and reflection in the line m if the D-line is an E-line.
Two figures are D-congruent if there is a sequence of D-reflections so that the composition of the reflections maps one figure to the other.

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. You should do your work in this file.

Now here is the drill. Use what you know about using Euclidean lines and circles to carry out constructions of Euclidean geometry to construct objects in the D-model and demonstrate basic properties of this model.

To start you off, download the file dweglab2005.gsp at this link.
The file contains
- Two tools for constructing D-lines that use automatching of the "removed" point O.
- A tool for constructing the E-circle through 3 points.
- A tool for inverting a point in a circle.
- Separate pages for each part of this lab set up for you to do your work.

If you wish to add pages, to keep the auto-match working, you need to keep the same full name for O. One way to do this is to add pages to your document by duplicating another page. 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

Line: [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.)

Segment: 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.

Triangle: Draw 3 D-points A, B, C and use your segment tool to construct the D-triangle with these vertices. Drag the points around to see what the triangles look like.

Perpendicular D-line: Given a D-line m through A and B and a D-point C not on m, construct the unique D-line through C 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.

Note: It is possible to construct a tool that will also work when C is on m using the radical axis but you need not do this now..

Parallel D-line: 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.
Rectangle: Construct a rectangle in DWEG given two vertices A and B.
- 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. Make the segments thick so that they stand out.

B. DWEG Lab Activity. D-perpendicular bisector and midpoint

We don't yet know how to measure distance in DWEG, but it is still possible to construct the perpendicular bisector of a segment AB as the D-line that D-reflects A to B.

Draw D-points A and B and construct D-line m that reflects A to B. This is the D-perpendicular bisector (or midline) of AB.
Intersect m with D-line AB to construct the midpoint M of AB.
Make D-tools for the perpendicular bisector and the midpoint.

C. 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 D-reflect across a line in DWEG to move a D-segment to a congruent D-segment.

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

As was explained in class, in the usual Euclidean plane one can trace the points of the circle through B with center A by reflecting B in all lines through A. To see this in action, download this Euclidean GSP sketch.

DWEG model case: Now mimic these steps in the in the DWEG model.

Draw points A and B and also a D-line AP. Reflect B across line AP to get B'.
Now trace B' as you drag P. What object does the trace look like?
Now based on this observation, find a CONSTRUCTION that will construct this object from A and B (and O).
Explain why this object is really the trace, as it appears to be. You should be thinking about inversion and about orthogonal circles, pencils, etc. Note: the D-lines AP are all diameters of the D-circles.

Use your construction to make a tool to construct the D-circle as an object.

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

If you have succeeded in making a D-circle tool, use your tool. Otherwise, you can download one here to continue with the lab.

1. Construct a D-Equilateral Triangle

Since you have tools that draw D-lines and D-circles, you can mimic Euclidean constructions.

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

2. Constructing 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 square with side D-segment AB.

E. Extras to think about (this will come up in class later; there is probably no time to carry them out in lab today).

Taking Equal Steps with a D-compass

Let A and B be D-points. Construct p = D-line AB.
Now mark equal steps on p by drawing a sequence of D-congruent circles. For the first step, construct the D-circle c1 with D-center B through A. Let C be the point of intersection of the circle with p so that AC is a D-diameter.
Next, construct the D-circle c2 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 with a circle c3 through C with center D for a few more points to see what equally spaced points look like in the D-model.

Taking Equal Steps with line Reflections and D-translations

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.
For any point Q, D-reflect Q in a to get Q' and then D-reflect Q' in b to get Q''. Then Q'' should be a translation T(Q) of Q.
Check this by D-reflecting line a in b to get a'. Then D-reflect a and b in a' to get b' and a''. If you drag Q to a, then Q'' should be on a'. If you drag Q to be on b, then Q'' should be on b', etc.
Also, q = D-line QQ'' should be D-parallel to p.

Pencils of D-lines

In the Euclidean plane, there are two kinds of pencils of lines. A point pencil is the set of lines concurrent at a point P. A parallel pencil is the set of lines parallel to a line p.
In the D-model, construct a D-point pencil of D-lines. Viewing this as a set of E-circles, what kind of set of circles is it?
In the D-model, construct a pencil of D-lines D-parallel to a D-line p. Viewing this as a set of E-circles, what kind of set of circles is it?
In the Euclidean case, the composition of three line reflections by lines in a pencil is actually a single reflection by a line in the same pencil. Using the geometry of circles, can one explain this in the D-model?