Progress Report 2

Poles


Introduction and instructions:

Definition

A point $P$ is called a pole of a line $a$ if there are two lines $u$ and $v$, both perpendicular to $a$, that intersect at $P$. If $P$ is a pole for the line $a$, then $a$ is called a polar line for $P$.

In answering questions that ask you to either "support" or "justify" an assertion, provide a summary version of the proof. Your answer should make it clear how you would go about proving the assertion and include any key observations necessary for your proof.

In answering the following questions, do not assume any of the parallel postulates or the betweenness axioms. While the sphere is a useful model for this project, your arguments should only rely on the axioms on (and consequences of) Incidence, Perpendiculars and Motions for transformational geometry. The results of the the first progress report and developed in the lecture can also be used (hint!).


Questions

  1. Let $a$ be a line and $P$ be a pole of $a$. Justify: The line $b$ is perpendicular to $a$ if and only if $P$ is on $b$. (Hint: First show that if $P$ is a pole of $l$ and $m$ is perpendicular to $l$, then $P$ is on $m;$then show the converse.)

  2. Can a line have more than one pole? Support your answer.

  3. Give a complete proof of the following: If one line has a pole, then every line has a pole.

    [To prove this, it helps to consider two cases. Suppose that $l$ has pole $P$ and $m$ is an arbitrary line. a) First show that if $m$ is perpendicular to $l$, then $m$ has a pole. b) Next suppose that $m$ is not perpendicular to $l$. Construct a line $n$ which is perpendicular to $m$ and $l$ and show that $n$ has a pole. (Do not assume that you are in an elliptic plane at this step!) Now appeal to case a) to show that $m$ has a pole.]

  4. Justify the following assertion: Given lines $a$ and $b$ with poles $P $ and $Q$. Then $a$ and $b$ intersect at the pole of the line determined by $P$ and $Q$.

  5. Justify the following assertion: There is a line with a pole if and only if the plane is elliptic.

  6. One role of the quizzes at the end of each progress report and final report is to encourage the groups to make sure each group member has a good understanding of the entire report. Is the group quiz score a good indicator of how well the group has worked together on a project? What are some contributing factors in a low group quiz score?

A note: As part of the final report your group may be asked to define a reflection in GSP, NonEuclid and the Lenart sphere. This is was started in the first progress report. Now is a good time to start thinking about how using the notion of distance and perpendiculars one can define a reflection in each of these three models.

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