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Interactive Screen Tools for Measurements
on Images of Physical Phenomena
T. Romanovskis, University of Hamburg/Germany,
On leave from Latvia University
S.Ignatjeva, University of Daugavpils/Latvia
New analogue and digital screen tools for interactive Macromedia Flash
animations are offered. The analogue tools - a ruler and protractor - can be
moved over the screen by parallel translation and rotation with accuracy
sufficient for teaching purposes. The digital tools allow to measure distances
between two arbitrary points and angles between three points. The third one –
Minkowsky’s sausages – allow to find the fractal dimension of fractal curves.
All the screen tools may be used with self-made as well as commercial images,
for example, the “360 Color Images of Physical Phenomena” (Ztek®Co). The
measurements are illustrated by problems in optics, thermodynamics, and
fractal geometry of real objects.
From photogrammetry to on-screen measurements for everyone
Photography and video have become an indispensable part of modern times. A special
science – photogrammetry studies assessment of the size and dimensions of an object
from its image. So far it has been the field of a narrow group of professionals working for
architects, cartographers, and military services. Today, as a result digitalisation of
photography, determining dimensions of objects from images has become a general
practice, to mention determining distances and areas on topographic and city maps [1, 2],
inspection of hole diameters of printed circuits as examples, counting cells in biology,
measuring angles in textiles [3], determining the size of uterus from ultrasonic images
during pregnancy [4], and measuring angles and distances of movements in sports [5].
Video-analysis software for measurements on computer screen
Of course, education must follow the changes of the everyday technological practices. It
means that high-school and university students must learn to make measurements on
digital images. Presently two kinds of software are offered for screen measurements in
physics.
For example, the software of dynamic geometry, such as Cabri Geometre II, may be used
to measure the length of straight lines, arcs, areas, and angles in ideal geometric objects.
The software for video-analysis may be used to measure co-ordinates on video-images to
obtain them as functions of time. In Germany alone there are several programs for videoanalysis (Galileo, ViMPS DIVA, David) [6,7,8,9]. Visual databases Videopoint (USA)
[10], World-Im-Motion (USA) [11], and Multimedia Motion II (England) [12] are popular
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in English speaking countries. An interesting development of video-analysis software is
presented by “The Humanized Physics Project”: the video-sequence is mounted into
Flash animation allowing to make measurements on the computer screen without any
other software [13] since only a browser software and free Shockwave Plugin is
necessary to watch the video and to make measurements.
Most programs for video-analysis offer export of measured data for processing in Excel
media. The Coach 5 Video software [14] offers independent powerful mathematical tools
for data processing: filtering, smoothing, function fit, graphing, finding the slope of
tangent to and area under curves.
Importance of measurement in advancement of physical thinking
Galileo was the first to propose a model of advancement of rational thinking. He
formulated four stages of physical reflection: acquisition of facts, modelling, drawing
conclusions, and verifying the latter by experiment [15]. These stages can be recognised in
writings of many physicists. R. Feynman describes advancement of physical thinking [16]: “..first
we observe an effect, then we measure it and list in a table; then we try to find the rule (model,
T.R.) …, However, the importance of a powerful principle is that it predicts new things… and we
perform an experiment and measure … “
2. MODELS
3. CONCLUSIONS
1. FACTS
4. EXPERIMENT
Fig. 1. Four stages of physical reflection by Galileo and Feynman [15].
We can see that measurement is encountered in two essential stages of physical thinking:
in obtaining facts and in verification of predictions derived from our models.
Thus, necessity of measuring skills is important to both – practices of information
technologies and rational thinking. Simple classical tools – ruler and protractor are used
in school and everyday praxis to master measuring skills. The widely used programs for
video-analysis offer sophisticated digital measurement techniques. There is a deficiency
of intermediate tools for simple measurements on the computer screen. An evidence of
that is, e.g., the popular CD ROM of “Color images of physical phenomena” with 360
photos [17]. Practically measurements are assigned to each one of the images. However,
no tools are offered assuming that students may use drawing software that show the coordinates in pixels. For that reason we have developed animation the user of which may
move on the screen conventional ruler and protractor familiar from everyday practice.
Farther we have developed digital tools to measure the distance between two points and
the angle between three points or two rays drawn from a common origin. A third tool is
an “inch-measure” – a rod of changeable length called Minkowsky’s sausages. It may be
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used to determine the fractal dimensions of curves.
The analogue ruler and protractor on computer screen
Since the high-school students are familiar with the use of ruler and protractor, we have
chosen the latter as the basic tools for on-screen measurements. The tools are either
dragged over the screen by mouse or moved by pressing cursor keys. The “Up” and
“Down” keys are used to turn the tools: protractor around the centre and the ruler around
its starting point. By pressing on the “Shift” key turning and moving proceeds by smaller
steps. It has to be noticed that some complementary tool is necessary for parallel
displacement of real ruler or protractor for which reason handling computer tools is easier
compared to desk tools. The disadvantage is that divisions of the computer ruler may not
be equal. However, the error is only a few pixels.
Fig. 2. The analogue and digital tools for measuring distance and angle in Flash
animations on the computer screen.
Examples in optics: refraction and diffraction
“Refraction of light” is an easy chapter to illustrate the use of the tools. The laser beam
hits the block surface at 60 degrees (Fig. 3). Turning the protractor by 180 degrees we
take reading of the refraction angle - 35 degrees. The ratio sin60/sin35=1.5 is in good
agreement with the refraction index of organic glass. With the ruler on that image we
may measure the length of a line of 7 holes and find out that under the glass the distance
is longer than on the table. It makes ground for a discussion of how to explain that a
parallel-plane block functions as magnifier. Measuring is made quickly and accurately
enough to use the tools in lectures as well as in laboratory classes.
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Fig. 3. Using protractor on computer screen for measuring the angles of incidence and
refraction. Photo made by J.Harja from Latvia university.
In the popular photo collection of “Color Images of Physical Phenomena” (CIPP) [17]
over 60 images are related to ray and wave optics. Our tools may be successfully used
with all of them. Let us consider two problems of the CIPP collection: one of geometrical
and one of wave optics.
Photo A12 is complemented with the following description and suggestions. “Three coins
of the same size are photographed from above, coin A being at the bottom of a circular
cylinder filled with water, coin B being put in the air at a height which is equal to 1/4 of
the depth of the water. Coin C is on a table at the same level with the bottom of the
water. In spite of coin A being at the same level as coin C, A is seen to be larger than C.
Coin B is seen to be of equal size with A. This indicates that the image of A is formed at
the same level as B. The picture shows that the apparent depth of a body in water, if it is
looked down upon vertically, is 3/4 of the actual depth. Measure the sizes of the three
coins and use those sizes to determine the index of refraction of the water.”
Using our tools we find that diameter of the coin image in water is 24 mm and in air – 18
mm. Since the picture is taken from the same height and the angles are small, refraction
index of water may be calculated from the size ratio of the coin images: 24/18=1.33. The
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value is consistent with the known index of refraction of water.
Fig. 4. Image A12 from “Color images of physical phenomena”, Ztek®Co [17].
Photo K14 is supplied with the following description and assignment. “Parallel rays of
lights incident normally upon a diffraction grating. Lessons suggestions. Use l=650 nm
for (red) light to estimate the diffraction grating slit spacing.”
Measuring the angle between the central and the red beams we obtain 23 degrees.
Therefrom we may calculate the diffraction grating slit spacing: 650/sin23=1664 [nm] or
6000 lines per cm.
Fig. 5. Image K14 from “Color images of physical phenomena”, Ztek®Co [17].
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Statistical physics: distribution functions
Distribution functions is one of the core topics in thermodynamics and statistical physics.
Let us consider the distribution of the pumpkinseed size to introduce the Gaussian
distribution, which may be easily applied in theoretical consideration of errors. The
scanner photography (Fig. 6) shows 100 pumpkinseeds. Measuring the size of
pumpkinseeds may be assigned as a laboratory work, a small project, or team work for a
group of students. In such way we reach the processing of experimental data and arrive to
the distribution function in ten minutes. In the simplest case the size of seeds is
approximated to an integer value and the histogram is obtained to calculate the mean
value, deviation, and the Gauss function.
Fig.6. Scanned image of 100 pumpkinseeds. Hystogram and Gaussian distribution of the
size of 100 measured seeds: N ( x)  (1 /( 2 ) exp(( x  m) 2 /( 2 2 )) x , m=12.6
mm, =1.7 mm, x=1.4 .
Obtaining the distribution of velocities is a more interesting case. D. Easton [18]
introduces the following concept. Photography of the motion of disks on air table is taken
with exposure of 0.5 s. The picture shows traces. Length of the trace is proportional to the
speed. Students can use Interactive Digital Screen Tools to collect velocity data for a
large number of randomly moving pucks through a cooperative approach. If each student
determines the velocities of a small number of pucks by measuring their blurred paths in
a photograph, the data can be collated to create a velocity distribution for a much larger
sample. Programs like MathCad or Excel can be used to sort their data and to create a
histogram or to determine the parameters in the velocity distribution function
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f (v)  (2 Nv / v rms
) exp( (v / v rms ) 2 )v .
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Fig. 7. Traces of the pucks on air table during 0.5 s. The diameter of the pucks is 9.1 cm.
The measured trace of 49 l.u.=38 cm gives the velocity of 0.19 m/s. The
hystogram
and
distribution
of
the
velocities
on
air
table
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f (v)  (2 Nv / v rms ) exp( (v / v rms ) )v , where N=179, vrms=19.6, v=6.
Interdisciplinary relations: scaling laws and fractal dimensions
Mathematics, biology, medicine, and geography are related by the modern topic of fractal
structures and fractal dimensions. An object is specified as a fractal object if the scaling
law relating its area or volume to a characteristic length is not an integer.
Let us consider the simple question of scaling law in the first place. We know that in the
set of similar ideal geometric objects a precise relationship exists between the linear
dimension l and the circumference, the area, and the volume: the circumference is
proportional to l1, the area – to l2, and the volume – to l3. Biologists before long have
found that power index 3 does not describe the volume (mass) of humans and animals,
that cross section of bones is proportional to power 3/2 of the length. The term fractal
comes from fractional dimension.
H.D.Sheets and J.C.Lauffenburger [19] show that the topic may be introduced with
measurements of shells and nuts. They find that the volume of freshwater clams is a
power function of length with index 2.6. K. Zembowska and M. Kuzma [20] report that
samples of fractal dimensions may be found on geographic maps as fractal dimensions of
rivers.
We offer convenient computer screen tools for quick acquisition of data and the most
essential part of data processing – finding the fractal dimension. Let us start with a simple
problem on a classical topic – the scaling law of a set of similar objects. Let us take
matryoshkas that have originated in Japan and become a toy of the Russian cult of today.
The image (Fig. 8) provides data on the mass of a series of matryoshkas, the students
have to measure their height with the movable ruler. Processing of the data by least
square method yields the following relation between mass and height: m=al b, where
a=3.5*10-4 and b=2.5. In case of matryoshkas the value of the index has an explanation.
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If matryoshkas were bold the m  l 3 scaling law would be expected. If they had the same
thickness of shells, the mass would be proportional to height square. In reality the fractal
dimension is close to that of clam shells – b=2.6 [19].
Fig. 8. Matryoshkas represent a set of fractal objects the mass in which is proportional to
power 2.5 of height.
Minkowsky’s sausage is the new tool we have developed to determine the fractal
dimension of linear objects, such as coastlines and rivers. One-dimensional fractal
structures, e. g., coastlines, are formed as a result of competition between determined and
undetermined processes. As a sample object we shall consider the “coastline” of a patch
dropping water has left on canvas (canopy over bed) hung at 4 corners. The determined
component is due to gravitation, the undetermined – to diffusion. In our animation the
length of segments [fractions] of the ruler may be chosen equal to s=1, ½, ¼, 1/8, 1/16,
1/32. The initial default length is 100 pixels but the user may define other length.
Measuring a straight line of length L with Minkowsky’s sausage of length s we may
calculate the number of sausages on the line N(s) = L/s. The length of a fractal curve
between any two points at straight distance L is infinite. It means that by measuring the
curve with straight-line segments of length s the total length L increases faster than s
decreases. This may be expressed as N(s)=(L/s)D=(1/s)DLD where D>1 is the fractal
dimension. That allows to determine D by covering the fractal curve with Minkowsky’s
sausages of different length.
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Fig. 9. Margin of a patch left on canvas by combined determined (gravitational)
movement and diffusion (undetermined motion) of water. The margin is covered
with 9 Minkowsky’s sausages of length ½.
Results of measuring the marginal line of the patch with Minkowsky’s sausages we may
put into the following table.
Length of segment, s
Number of segments, N(s)
1
4
1/2
9
1/4
20
1/8
47
1/16
107
From the table it is seen that the number of segments necessary to cover the curve
increases more than twice as the length of the segments is halved.
Fitting of the data to a power function yields the following relation between the number
of segments and the segment length N(s)=3.5 (1/s)1.2 . It means that the fractal dimension
of our “coastline” is D=1.2.
The access to interactive multimedia problems and animations described above are free at
http://www.liis.lv/fizika/. Other animations are available in the “Physik multimedial”
mediabase at the University of Hamburg. It comprises over half a thousand multimedia
materials (applets, animations, pictures, video, software, and illustrated databases on
commercial multimedia collections) to support lectures and laboratory classes in physics
for non-physicists. The physics media base is part of the e-teaching and e-learning system
“Physik multimedial”. The users of Mediabase, Exercise server, and Courses server
services of e-teaching and e-learning system must register (free of charge) at
http://www.physik-multimedial.de as teachers.
The present study has been partly made (T.Romanovskis) within the “Physik
multimedial” framework of the Association of North German Universities.
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References
1. http://www.ifag.de/GI/Produkte_nat/top200/flaeche.htm
2. http://cityguide-deutschland.com/doku/de/5.html
3. http://www.sltec.de
4. http://w3.abdn.ac.uk/webcal/applet_templates/linear_measurement.html
5. http://www.simi.de
6. http://we1x01.physik.uni-wuerzburg.de/~pkrahmer/home/galileo.html
7. http://www.physik.uni-mainz.de/lehramt/ViMPS/
8. http://www.physik.uni-augsburg.de/did/
9. http://www.physik.uni-muenchen.de/ didaktik/Computer/DAVID/ david.htm
10. http://www.lsw.com/videopoint/
11. http://members.aol.com/raacc/wimcover.htm
12. http://www.csmedia.demon.co.uk
13. http://www.doane.edu/hpp/Flash-VidDat/index.html
14. http://www.cma.science.uva.nl
15. T.Romanovskis. Models - the basics of physical thinking. Conclusions for
multimedia. First International GIREP seminar 2001: “Developing Formal Thinking
in Physics.”, FORUM, Udine/Italy, 2002, 215-219.
16. R.Feynman, R.Leighton, M.sands. The Feynman lectures on physics. Addison
Wesley.
17. Ztek®Co. http://www.ztek.com/
18. D.Easton. Solving a problem by analogy. The Physics Teacher 37,150(1999).
19. H.D.Sheets, J.C.Lauffenburger. Looking for Scaling Laws, or Physics with Nuts and
Shells. The Physics Teacher 37,376(1999).
20. K.Zembrowska, M.Kuźma. Some Exercises on Fractals for High School Students.
The Physics Teacher 40,470(2002).