CAS-aided Visualization in LaTeX documents for Mathematics Education ー Achievement of KETpic ー M. Kaneko, T. Abe, M. Sekiguchi, Y. Tadokoro, S. Yamashita, (Kisarazu National College of Tech.) & S. Takato (Toho Univ.)

Contents 1. What is KETpic ? 2. Simple demonstration of drawing pictures using KETpic 3. Other examples 4. Examples of space curves 5. Conclusion 6. Future works

What is KETpic?

KETpic is Macro Package for CAS (Maple, Mathematica etc.)

What can we do by using KETpic?

We can draw Fine Pictures in LaTeX documents We can draw Fine Pictures in LaTeX documents. (Fine = accurate & beautiful)

Usually many people use EPS. User’s document CAS EPS \includegraphics Compile User’s LaTeX text file Mathematical document The output (EPS) of CAS is inserted into the LaTeX text file by using command “includegraphics”.The above procedure fits in case of not only CAS but also various graphic softwares or Gnuplot． Also, the command of Maple and Mathematica which generates LaTeX file follows the above procedure.

BUT…

EPS is not satisfactory for us. The size of file is large. Graphics can not be corrected as we like. Mathematical expressions (e.g. legends, accessories, hatchings,...) are poor. What we want to do (drawing tangential line to a curve at a specified point, curves defined by implicit functions, etc. ) are not supported.

Solution So we have developed “KETpic”.

Though… Maple is NOT FREE KETpic for Maple has been accomplished (v1.7). This talk is concerned with this version mainly. Recently we have started transplanting KETpic to Mathematica． How to get KETpic? http://www.kisarazu.ac.jp/~masa/math/ (Of course FREE) Though… Maple is NOT FREE

Comparison with EPS KETpic User’s Tpic CAS document \input Compile User’s LaTeX Text file Mathematical document By using KETpic, graphical outputs of CAS are formatted to generate graphical codes (Tpic specials) and written on text files. We insert these files into LaTeX document by the command “input”.

It is a Fourier series approximation of the step function \pm 1 with period twice of pi.

A simple demonstration of drawing pictures by using KETpic An example from the convergence theorem of Fourier series.

Merit of KETpic (1) By using CAS, we can draw accurate curves easily in the LaTeX documents. The output of KETpic is just a text file, which is much smaller than EPS, JPEG, PDF etc. Therefore, it is easy to attach the output to E-mail.

Merit of KETpic (2) You can insert mathematical expressions at your favorite position in graphics, with the same quality as those in LaTeX text. You can use various accessories such as hatchings, tickmarks, legends, …, so that you can give students very expressive printed matter.

Other Examples Following examples are effective for math. education, and we actually use them in our classroom. chi^2 function 9-pointed circle polar coordinate asteroid tangential circles stability contour Japanese Mathematics (18C～19C) cycloid

Examples of space curves

This is the cross eyed view of the intersection of two cylinders.

These are the figures of regular icosahedron These are the figures of regular icosahedron. In the right figure, hidden line elimination of KETpic is used unlike the left. Skeletons like this are drawn automatically by using KETpic. You see the right figure is much more expressive than the left.

This is a picture used to educate spherical polar coordinate.

Mathematica version (1) Loading KETpic Get[StringJoin[“c:”,bksl[“ketpicmath.m”]]] Calculation of numerical data setwindow[{-2*Pi,2*Pi},{-2.3,2.3}]; f=(4/Pi)*Sum[Sin[(2*k-1)*x]/(2*k-1),{k=1,N}]; N=10; g=plotdata[{f,{x,XMIN,XMAX},PlotPoints->100}];

(3) Writing down the numerical data onto LaTeX file (“fig1.tex”) openfile[“e:\\Hungary/fig1.tex”]; openpicture["1cm“]: drwline[g]; closepicture[1]; closefile[];

This is a picture of chaos drawn with KETpic for Mathematica.

Mathematica Codes Putting 40000 points. Perhaps it is the delimitation of LaTeX and DVI.

Conclusion KETpic takes full advantage of CAS so that we can insert fine graphics into LaTeX document with reasonable efforts. KETpic has many merits which are effective for mathematics education. Especially it is useful when the outputs are used in math classroom as printed matter.

Future works Extension to other CAS’s Now on developing Mathematica version And other CAS’s, especially for free, such as Risa-Asir, Scilab, … Various and richer 3d-expression

KETpic is wide spread to the world. Please use KETpic! We hope KETpic is wide spread to the world.

Thank you for your attention. 感謝

Structure of user’s document Maple file (1) Loading KETpic > read `c:\\ketpicw.m`: Calculation of numerical data > setwindow(-2*Pi..2*Pi,-2.3..2.3): > f:=(4/Pi)*sum(sin((2*k-1)*x)/(2*k-1),k=1..N); > N:=10: > g:=plot(f,x=XMIN..XMAX,numpoints=100):

Writing down the numerical data onto LaTeX file (“fig1.tex”) > openfile(`e:\\Hungary/fig1.tex`): > openpicture("1cm"): > drwline(g): > closepicture(): > closefile():

LaTeX file (4) Generation of LaTeX file of graphic codes \begin{document} \input{fig1} \end{document} Then the output is the following picture:

This picture of cycloid can be drawn by using do-loop and the commands “rotate” and “translate” of Maple. We believe that the educational effect of this picture is not inferior to that of animation. Also this picture has the advantage that it can be used in the form of printed matter.

This picture of trochoid can be drawn by the same Maple program with only the coordinate of point substituted.

Inspiring students’ imagination While seeing these pictures, some student may put up a question “How will the picture become if ellipse rotate?” Then we can give them answer by using KETpic as in the next picture.

The roulette of the focus is called “Delauney curve”.

This is a trace of the circle which rotates twice around the given ellipse on the outer side of it.

In case of the rotation on the inner side of ellipse, the diameter of circle need to be small to a certain extent so that the rotation would not be interrupted.

This shows a delicate situation This shows a delicate situation. Whether safe or not safe is easily judged using the two diameters of ellipse and that of circle. Deduction of this safety condition is a good exercise for high school students.

This is a picture of 9-pointed circle This is a picture of 9-pointed circle. The dotted circle is drawn by automatic calculation of Maple.

This is a picture of circles which are tangent to each other and the two lines. To draw it, do-loop of Maple is used. Also hatching is done very easily.

This figure shows an example of the graph of function given by polar coordinate. Here do-loop of Maple is effectively used.

Closed chain circle The sum of the inverse for radii of dotted circles are the same as that of dashed circles.

This is the case that bold circles are put side by side This is the case that bold circles are put side by side. Then touching circle is turned upside down. This picture seems quite different from the previous one, but essentially the same.

These curves are graphs of chi^2 functions with various values of deviation.

These are graphs of functions obtained by integration of the above chi^2 functions. Without using CAS, it would be impossible to draw graphics like this.

Z=((x+1)^2+y^2)^(-1/2)+((x-1)^2+y^2)^(-1/2) This is contours of the Coulomb potential with two electric charges at (\pm 1,0). To draw this picture, “contourplot” is used.

The width and direction of hatchings are easily handled.

This figure is used to explain the stability of dynamical systems in economics classroom. You see that various kind of lines (real line, dotted line, dashed line,…) and arrowhead are very expressive