Math 487 Lab 6 - 2/15
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 dweglab.gsp.
- 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 takes as the
givens (besides O-removed) the points A, B, and C (you can make it
automatch O-removed if you learned how above)..
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
This is explained
on this web page on Constructing Circles in
several geometries.
D. DWEG Lab Activity. Using the D-circle tool
This will be introduced
and discussed in morning 445 class.
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.