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.
b2h2
Definition of the
case
of
rectangular
triangle,
J yz  
J yz  yzdF
72
product of inertia
O
is
in
the
center
of
mass
F
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.
To make our calculations more convenient we introduce a parameter
S 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’.
ξ χ
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
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
3
 
1 
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 – daria_kazachkova@yahoo.com
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.