## GeoGebra Lab F5

20feb15 and updated same day

\begin{document} \maketitle \section{Prerequisites} It is presumed that you have watched the video on inversive geometry. We need the concept of the inverse of a point in the Cartesian plane in the unit circle. \section{Instructions} Put your name at the top of this sheet of paper and check off the items as you finish them in the lab. Submit this sheet of paper before your leave. I will check off the rest when I look at your work after F5. There are three GGB files to be submitted on M6. There is NO paper to be handed in. I will retrieve the three files from the Moodle and check them. You will receive feedback on the lab sheet you hand in. \begin{enumerate} \item Create a folder called ggbLabF5 (do not use spaces in any names of folders or files) and download a copy of ladderlemma.ggb

into this folder. \item Make a copy of this file for safekeeping. \item Open your working copy of the ladderlemma.ggb \end{enumerate} \subsection{Experiment 1} In this experiment you discover that hyperbolic segments as arcs of circles perpendicular to the unit circle. We will do this by constructing the circle through two points inside the unit circle which is its own inverse, and therefore perpendicular to the unit circle. Note, that we will distinguish between Euclidean and hyperbolic objects by attaching the letter H or h somewhere in the name for the Euclidean object that is an interpretion of a hyperbolic object. \begin{enumerate} \item Find $C' = inv(C)$ by reflecting $C$ in the unit circle. Use the reflection in a circle tool of GGB. \item Draw $\gamma=circ(B,C,C')$ through these three points. Observe that it is perpendicular to the unit circle. \item Show that inversions $B',H'$ are also on $\gamma$, which suggests that $\gamma$ is its own inverse. \item Verify that the lines $(HH'), (GG')$ etc are concurrent at the origin, illustrating that inversion preseves lines through the origin. Note such lines are necessarily perpendicular to the unit circle. \item For homework, investigate inverses of other circles and lines and write up a list of properties. For example, what is the inverse of a circle passing through the origin. Collect these in your yournal. You will be asked to submit this in your lab report due at a later time. \item Save your work with the name \textbf{ F5experiment1.ggb.} You will submit this file on the Moodle. \end{enumerate} \subsection{Experiment 2} In this experiment you will learn how to make new tool in GGB and save it for later use. The purpose of this exercise is learn how Campos extended GGB to the box of hyperbolic tools. \begin{enumerate} \item Start with a clean copy of GGB, one that \textit{ does not} have the Campos hyperbolic tools in it. \item Construct the unit circle, then hide the xy-axes and the point (1,0). But keep the origin, re-named as $O$ \item Choose two points labeled $P,Q$ inside the unit disk. \item Find the two points, labeled $P1, Q1$ (subscripts are clumsy in GGB), where the $circ(P,Q,Q')$ crosses the unit circle. \item Construct the circular $arc(P1,Q,Q1)$. This arc is the interpretation in this model of the full hyperbolic line $hline(PQ)$ \item Now follow the directions on constructing a new tool. Note it is under a new tool button. See Tool Elaboration at the bottom. \item Leave this file open to work on further, but save the tool as follows. \item Save this as the toolfile named \textit{hline}. It will appear in your folder ast \textbf{hline.ggt}. Note the suffix is not .ggb. You will submit this file on the Moodle. \item Continue working on the open file. Apply the tool you have made to other pairs of hyperbolic points. Note carefully, that the tool requires you to name more objects than the two points you are connecting. They come last. This is a flaw in GGB and I don't know how work around it. \item Continue to make two more tools. The $hray(P,Q)$ is easy to do because it is the Euclidean $arc(P,Q,Q1)$. \item To build the tool $hseg(P,Q)$ will require you to find the center of the circle $\gamma$ at the intersection of $perbis(P,Q)$ and $ perbis(P,P')$. \item To show that your work was successful, use your new tool to connect two further hyperbolic points with their hyperbolic segment. Label the points $P2,Q2$ \item Save this construction and name it \textbf{F5experiment2.ggb}. You will submit this file on the Moodle. \end{enumerate} \section{Elaboration} During the lab of F5 the following issues arose and were ammended. \subsection{Making and Using GGB Tools} The procedure I demonstrated in class to make the tool hline.ggt actually was correct. What threw me off was that when I chose the two axes, they did not "light up". GGB displays in bold suitable ojects you choose as feedback. There really was a tool-file named \textit{hline.ggt} in the Downloads. Also, I was puzzled by the fact the the tool you helped me make in lab asked for "line,line,point,point", instead of what I had observed earlier "point,point,point,point". Here is the reason. In order for GGB to apply a native or a user-made tool, it must have all the component inputs. Recall that to make the tool, we started with the axes, and then used the axes to make a unit circle by center and radial length. You can also do this by choosing the center and a point on the circle. This has to be specified, along with the last pair of points you want to connect with a hyperbolic line. The need for this additional information stems from the native reflection, which can use any circle as its mirror, not just the unit circle. So this is not a defect in GGB, just an inconvenience for us. Alexandre Campos did not make his hyperbolic tools in this way. He re-coded the XML file itself to specify the center (the Origin) and the mirror (Unit Circle) automatically. The conclusion is thus to use the tool-making ability of GGB not instead of Campos's tool set, but as an extension of it. However, for other models of non-Euclidean geometry, such as the Klein-Beltrami and the Upper Half Plane model, we will have to make our own tools because they are not part of Campos's tool set. \subsection{Assignment} It proved impractical to collect the handouts (a copy of the first two pages of this document) at the end of the lab since most of the work was assigned to be done at home. Therefore, four items are due on Monday M6: \begin{enumerate} \item The instructions handed out F5 (or a printed copy of it) with your name on it, and marked with completed steps. Submit these during class M6. These will be returned with feedback and a grade. \item The 3 files F5experiment1.ggb, hline.ggt, and F5experiment2.ggb submitted on the Moodle on M6. \item Additional reports written into your Journal. These will be evaluated at a later date, possibly by a question on the midterm. \end{enumerate} \end{document}