Exercises in Klein Model Constructions

\begin{document} last edited 26apr15 \maketitle These problems involve constructions and annotated figures, which can be realized by GGB or GEX, and also by hand using a compass and transparent plastic 30-60-90 triangle. \begin{itemize} \item[Problem 0:] The solution to the lab exercise on F13 may be deduced from the Doubling figure . \item[Problem 1:] Recall how we defined perpendiculars in the Klein model using the polar of a secant in the unit circle. Given two k-parallel lines that do not meet, even on the unit circle, demonstrate how to construct the single common k-perpendicular. (This again shows that non-Euclidean parallel lines have at most one common perpendicular. Recall how we originall proved this using Lambert quadrilaterals.) \item[Problem 2:] Demonstrate experimentally, by submitting a .ggb or a .gex construction, that being perpendicular to each other is a reflexive relation for lines. In other words, show that, given a line $m$, that if line $k$ is constructed perpendicular to $m$, then $m$ must also be the line perpendicular to $k$ at $(km)$. (Comment: This is difficult to prove rigorously in classical Euclidean geometry without the developing the theory of circular secants.) \item[Problem 3:] Demonstrate experimentally, by submitting a .ggb or a .gex construction, that two perpendiculars to a line are in perspective from a point on the Euclidean extension of the given line. Hint: you can continue your construction from Problem 2 to solve this problem. (Comment: this is even more difficult to prove synthetically. \item[Problem 4:] Recall the N-construction we did in class. Generalize the N-construction to the case of two intersecting line $(AB)$ and $(CD)$. (Careful, $A,B,C$ are k-points inside the Klein disk, and $D$ is an ideal point on the unit circle. This the \textit{X-construction}, and it also also uses two V-constructions to move the given segment $AB$ first to a helper line, and then to the given ray. Hint: The intermediate helper line still connects $D$ to $5$. Unless $C$ is where the two lines cross, $(AC)$ still becomes the common base line for both V-constructions. \item[Problem 5:] Solve the previous problem for the special case that $A=C=(AB)(CD)$. Note that his is the case where we want to use the segment $AB$ as the radius of a k-circle centered at $A$, as we discussed in lab. \item[Problem 6:] This special X-construction in Problem 5 is simpler for doubling a segment. Use for helper line one that crosses at the endpoint in which direction you plan to double the segment. \item[Problem 7:] Demonstrate how to use the X-doubling construction in Problem 6 to that show that the point found in the Halving Construction really is the midpoint of the segment. Hint: Use just one of the helper lines to make an X with the given line the segment resides on. \end{itemize} \end{document}