Solution 7 Ipho 2012

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Solution

While this problem can be solved both geometrically and numerically, the geometric solution is

definitely much more elegant (shorter, and provides more insight to the geometry of image

formation). On the other hand, all the purely geometrical solutions were similar, so the award

for the best solution is this time distributed evenly between all these seven purely geometrical

solutions which were submitted before the appropriate hints were published (see below). Out of

these seven, six were suitable for presenting on this web-page. According to the rules, the best

solutions are published without additional bonus factor. Here, however, the divided bonus

becomes small ( ), and there is a need to differentiate between the published and

non-published awarded solutions. Therefore, there published best solutions received this time

also the publication bonus of 1.1.

Let us start with the solution of Nadezhda Vartanian , which provides a short, yet clearenough motivation and explanation of the geometrical construction.


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Exactly the same construction is provided by Jakub afin , who shows how to draw everythingusing only a ruler and a compass (as is typically done in the case of mathematical construction

tasks). His solution is best viewed using his .pdf file , because changing pages yields an

animation effect. His explanation is as follows.
http://www.ipho2012.ee/wp-content/uploads/03_safin_c.pdfhttp://www.ipho2012.ee/wp-content/uploads/03_safin_c.pdf


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The next solution with exactly the same construction is provided by Ilie Popanu . He found anice program for easy drawing the required geometrical elements with high precision, geogebra .

Owing to this, he was able to reconstruct the original square with a high precision; here we

provide only his drawing (the method has been already explained well enough, see above).
http://www.geogebra.org/http://www.geogebra.org/


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Among the best solution award winners, there was one more solution with the same

construction as above, sent by Andrew Zhao; however, his presentation style is somewhat

sketchy, not suitable for publication.

It appears that the centre of the lens can be found as an intersection of circles, different from

those shown above. Previously, the circles were drawn around diameters; SzabAttila and Ion Toloaca drew these around segments with a central angle of 90 degrees.

Attila's solution is provided below, Ion's solution can be downloaded in the pdf-format
http://www.ipho2012.ee/wp-content/uploads/09_ion_x.pdfhttp://www.ipho2012.ee/wp-content/uploads/09_ion_x.pdf


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Last but not least the solution of Nikita Sopenko , which is actually the simplest one (andthus my favourite) as it substitutes one circle with a straight line.


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Next, let us consider some solutions which involve calculations. All these are relatively long and

therefore are provided only as downloadable files. First, AleksandraVasileva performed calculations only to motivate certain geometrical constructions (so, nonumerical measurements and numerical calculations are needed). Second, Adrian NugrahaUtama constructed one circle and was thus very close to a fully geometrical solution .Third, Petar Tadic measured angles and using these values calculated two angles required fordetermining the position of the centre of the lens . Fourth, Selver Pepi measured angles andcalculated the focal distance together with the tangential offset of the centre of the lens; the

highlight of his solution (in the form of a slideshow) is the error analysis .

Jaan Kalda, Academic Committee of IPhO-2012
http://www.ipho2012.ee/wp-content/uploads/05_vasileva.pdfhttp://www.ipho2012.ee/wp-content/uploads/05_vasileva.pdfhttp://www.ipho2012.ee/wp-content/uploads/06_adrian_y.ziphttp://www.ipho2012.ee/wp-content/uploads/10_petar_y.pdfhttp://www.ipho2012.ee/wp-content/uploads/10_petar_y.pdfhttp://www.ipho2012.ee/wp-content/uploads/07_selver_y.ppthttp://www.ipho2012.ee/wp-content/uploads/07_selver_y.ppthttp://www.ipho2012.ee/wp-content/uploads/05_vasileva.pdfhttp://www.ipho2012.ee/wp-content/uploads/06_adrian_y.ziphttp://www.ipho2012.ee/wp-content/uploads/10_petar_y.pdfhttp://www.ipho2012.ee/wp-content/uploads/10_petar_y.pdfhttp://www.ipho2012.ee/wp-content/uploads/07_selver_y.ppthttp://www.ipho2012.ee/wp-content/uploads/07_selver_y.ppt

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Solution 7 Ipho 2012