Exercise on Drawing a Checkerboard Floor

\begin{document} \maketitle \section{Introduction} The simple observation that the diagonals of a checkerboard are parallel leads to an efficient construction of a checkeboard in perspective. It is done by sequential use of the edge and diagonal vanishing points. The above is an example of applying a genral principle which is elaborated in Lesson P8 motivating the method of drawing a circle and measuring a box. The principle is to infer a perspective construction of a figure by observing geometric facts of the figure in reality. For a plane figure, one studyies a drawing of the figure in a plane, as if seen from (far) above the plane. This is called an \textit{orthographic projection}, or more colloquially, an orthographic view. \subsection{Not Using KSEG} For this exercise you do not need KSEG at all, you could do it with ruler alone. It can also be done efficiently in Paint/iPaint because only straight lines need to be drawn through points. \section{Exercise} Propagate a given convex quadrilateral into a 5x5 checkerboard design of your choice. \subsection{Hint:} Start with an arbitrary convex quadrilateral and construct the horizon with its vanishing points as in the previous exercise. Now imagine laying a tiled floor by constructing an adjacent square by drawing its diagonal first! \subsection{Discussion:} You may wonder about the purpose of such an easy exercise. It constitutes one of the three different methods for laying out a square (or rectangular) grid in perspective discussed in the lessons. Which to apply in a praticular drawing depends on convenience, but also on how accurate the drawing needs to be. For the record, the three methods are these: \begin{itemize} \item Using the diagonal vanishing points of construction lines. \\ \item Using local \textit{division} and \textit{multiplication} multiplications based on proportions. \\ \item Using the perspective frame. \\ \end{itemize} The first is illustrated by this exercise. The second is best understood by an example. It is used with a vengeance in Lesson P9 on constructing perspective circles.The third is explained in detail and used in Lesson P7 on measuring the proportions of a box given in perspective. \section{Example} Given convex quadrilateral $ABCD$. We construct the right and left vanishing points. We want the midpoint of the side \textit{AD}. For this, we first find the center of the square in perspective. Note that in the orthographic view the diagonals cross at the center $M$. Therefore the diagonals in perspective locate the center in perspective (blue lines). In the orthographic view, the line through the center and parallel to the sides crosses at the desired $N$. In perspective, locate $N=(MLvp)(AD)$ (still blue.) Finally, in the orthographic view, the (orange) line $(CN)$ crosses the line $(AB)$ forming a right triangle in the ratio of 1:2. Therefore the base seqment $A'A$ duplicatex $AB$, and erecting a parallel to $(AD)$ completes the duplicated cube $A'ADD'$. The same construction in perspective achieves the same end, in perspective. \end{document}