Virtual Weighing and Dissection of Real-Life Flat Figures

Virtual Weighing and Dissection of Real-Life Flat Figures (Student’s Research Project) Kazachkova Darya National Aerospace University 'Kharkiv Aviation Institute' Tetyana Ignatova Institute for Low Temperature Physics NAS of Ukraine Kharkiv, Ukraine In present project we study different ways to dissect a triangle into parts with the areas being in the certain ratio. My inquiry started when our class learned products of inertia* and to solve the problem we were “cutting” flat figures with a pair of perpendicular crossed straight lines. In fact, “dissecting” lines were the axes of Cartesian coordinate system, its origin O positioned onto the figure’s center of mass. The sign of inertia products is negative when the bigger part of the figure is in even quadrants, and vice versa. For right triangles this comparison of areas is easily performed by eye – or at least our tutor says so. Intrigued by this property, I came up with an idea of the more general problem to compare ‘even’ and ‘odd’ parts of an arbitrary triangle dissected by two mutually perpendicular straight lines. This project involves a variety of computer methods, including the technique of virtual measurement of simulated objects. ______*Products of inertia To explain the meaning and the value of this parameter here is the figure. b 2h 2 Definition of the J  yzdF case of rectangular triangle, J   yz  yz 72 product of inertia F O is in the center of mass Sections in the even quadrants are marked grey. If their area Seven is smaller than the area of ‘odd’ segments of the triangle, then Jyz is negative and vice versa.

We started with dissection of rectangular triangle by crossed lines parallel to its legs b and h. Rigorous formulas connecting areas of dissected parts with the position of the O point were calculated and plotted in OriginPro 7.5. Figure on the right presents some graphical results. When the point O travels along the red curve the triangle is divided into the parts marked white and grey that are equal by areas. When we performed calculations we set the position of cutting center with coefficients μ and k (see the figure below). And it was only the simplest case and the beginning of interesting research.

The position of cutting center. The second step was to study isosceles triangles. It was not so simple because formulas were different for cases whether the cutting center is to the right or to the left of the triangle’s height.

Dissection of triangles in arbitrary proportion. Parameter χ. S To make our calculations more convenient we introduce a parameter even   , that sets the ratio between S the sum of triangle’s areas that got to the even parts Seven and an aggregate area of triangle S. Notice that 0    1 . There are some figures below to plot the ‘dissecting center’ O locus for different values of χ. We must notice that one should not take into considerations the curves segments outside the triangles. For such cases we have performed separate calculations (below).

χ =1/2 χ =1/10 Less symmetrical figures. Cutting ‘from outside’.

The next step was examining arbitrary triangles. Those we obtained by transformation of isosceles ones introducing the parameter ξ that makes the base leg asymmetric. In other words we take one of the angles and changing ξ stretch it in or out relatively to the other angle.

For cases when cutting lines intercross outside the triangle we performed additional calculations. We divided the triangle’s surroundings into six rectangular and two triangular zones (corner cases are trivial) and here are the formulas for some principal ones: 1. k  1 0    1 6.   1 0  k  1

  0 ( 1)  0 k 1    1 ( 1)

Virtual computer experiment. The climax goal of the project was to find the cutting center’s locus for any orientation of the triangle. For that computer simulator was very practical. A technique of virtual measurement developed by my mentor Tetyana Ignatova [1, 2] proved to be highly efficient. Results could be checked by comparison with Origin-plotted graphs of earlier calculated orientations of ‘cutting axes’.

Asymptotes. But one of the most exciting results was neither planned nor expected at all: that is the asymptotes of O locus curves. In the case of dissecting isosceles triangle we noticed that when χ =0.5 (areas of ‘even’ and ‘odd’ parts are equal) we have straight-lined locus and simple ξ χ correlation. That is why after calculation the asymptote formulas we learned that they are present for any kind of triangle. The figure presents asymptotes for different values of χ and ξ parameters. Asymptotic behavior is offered in a few instances below 2 1       3    1 3

Length of the cutting lines. After finishing the research on positioning the cutting center we wanted to check which incision is the shortest for any given χ. Computer program of virtual measurement developed by my mentor gave us the most descriptive answer. There you set χ, h and b, move the cutting center along the calculated locus and in the right sub-screen observe live how the length of cutting lines is changing (plotted in blue 4 times shorter). On the screenshot figures below you can see how the length of the cutting line changes.

Cutting and weighing of cardboard triangles. Our computer results were interesting but we could not help trying to check the results in another, more convincing way. That is why we took a sheet of cardboard and cut triangles of arbitrary shape. Then we dissected them into four parts according to our calculations. The last step was to weigh cross-lying fragments with digital scales and the results didn’t disappoint us! In the figure χ =0.5

For more detail please contact Darya Kazachkova – [email protected]

Bibliography

[1] A.Kazachkov, T.Ignatova, A.Zholobenko. Virtual Optical Illusions for Creative Learning. In: C.P.Constantinou, Z.C.Zacharia (eds), Computer Based Learning in Science. International Conference Proceedings. Volume I: New Technologies and Their Applications in Education, Nicosia, Cyprus, 2003, pp.25-29. [2] T.Ignatova, A. Kazachkov, I.Szczyrba. Virtual and Hands-On Experiments in Statics: Balance Properties of Asymmetrical Bodies. G.Planinsic, A.Mohoric (eds), Informal Learning and Public Understanding of Physics, 3rd International GIREP Seminar 2005. Selected Contributions, University of Ljubljana, Ljubljana, Slovenia, p.244-249.