connecting past and future - ucke

This is the original - long - version of the plenary lecture which was held at the International Conference 'Turning the Challenge intoOpportunities' August 19-23, 1999, Guilin/China

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Connecting past and future

Christian UckePhysics Department E 20/ Technical University Munich/Germany

[email protected]

Abstract: A huge number of excellent didactical approaches are known to exist in physics education;these treasures should not be forgotten but revived and combined with new developments and findings.Anamorphic pictures, known for more than 300 years, are fine examples involving some geometricaloptics, some not too difficult mathematics and a lot of cultural background. These ideas can be ex-plained in a completely traditional way, using conservative mathematical methods to calculate theanamorphic distortions, but today's powerful computer tools open additional new opportunities throughmodern raytracing methods from digitalized pictures. These enable the users to see almost at once theirown pictures distorted and thus become motivated to vary parameters and to experiment for themselves.I will present this theme along with other examples and experiments and will discuss the physics andinterdisciplinary background

My goal is to present some ideas which will hopefully be fruitful for the users. Amongmy ‘users’ in Germany are physicists who would like to have some physics recreation inbetween normal physics work, teachers who can use some of these ideas for their lessonsand students or pupils who can do experiments themselves and learn by doing. Thisgroup of users is a very broad spectrum. I will present some of my reflections and showyou some experiments.

Let me start with the so-called anamorphic images or anamorphoses.

Anamorphic images are distorted images.You all know the fun children have in muse-ums or at fairs when they see themselvesdistorted in big concave or convex mirrors.There are many types of anamorphic images.I will describe some of these here.

Probably the first known anamorphosis isfrom Leonardo da Vinci and shows the faceof a child. If you view this image at a veryshallow angle, you will recognize it. If youscan this image and have it in a digitalizedform in your computer, you can ‘work’ withit. With many paint programs you can des-kew and resize the image in all angles anddirections and get a ‘normal’ image. When Ilook at this child’s face, it could also be aChinese child.


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Here I have distorted some words in Latin letters.Try to recognize what is written here. It is pos-sible but not clear at the first look. However, ifyou view this image at a very shallow angle, youwill recognize it. First this view reveals ‘Guilin1999’ at the bottom; then the view perpendicularto the first view reveals ‘anamorphotic’.

And here I have tried the same with Chinese cha-racters. Please excuse my insufficient efforts withChinese characters. But nevertheless it seems thatthis type of anamorphosis is also appropiate forChinese letters. A Chinese colleague in my de-partment said he was able to recognize at oncewhat is written here. For non Chinese readers thecharacters mean 'Guilin 1999' and perpendicular'International Conference'

This type of simple perspective anamorphic di-stortion can be find in many places: in paintings,wall paintings, drawings, underground stations oron the streets.

A very famous example is the painting ‘The Am-bassadors’ by Holbein (1533), which shows twopersons in a normal view. But there is a strangeelement. You can recognize what that is if youlook at the painting in a very shallow angle. Youcan also deskew this part with some better pain-ting programs like Photoshop or Paint Shop Pro.If you do that, you see at once that this strangepart is a skull.

This type of anamorphosis was closely relatedwith the development of perspective in the fif-teenth and sixteenth centuries. And of courseyou can describe this in mathematical terms as arelatively simple form of linear transformations.


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The so-called cylindrical anamorphoses were not de-veloped until quite a time later. The pictures drawnappear correctly when observed with a cylindrical mir-ror (or conical or prismatic mirror).

Mainly artists used this form of anamorphic pictures,not only in Europe but also in Chinese culture, andthey used it without thinking in physics or mathema-tical terms. I will show here only one example of aChinese anamorphosis from the sixteenth century. Ihave not found another one. The anamorphic pictureis turned upside down. And here you see the soluti-on: a pair of lovers. As far as I know, it is still notclear whether cylindrical anamorphoses were develo-ped independently in China and Europe or whetherone culture influenced the other. Since the Chineseculture has a special relationship to the topic mirror,it is at least probable that the Chinese developedanamorphic pictures independently. This is an inte-resting question.

With anamorphic images you can hide the content. Ifyou know the trick of how to resolve the image, youcan see them normally. Therefore these images havea certain magic aspect. And this is fascinating, notonly for children.

Look at this image from the Hunga-rian artist István Orosz. You see alandscape where some people seemto be looking at a shipwreck. Youcan guess that there must be some-thing more because I am showingthis in connection with anamorphicimages. Put a cylindrical mirror onthe moon (or sun?) and suddenly aface appears. The man is, by theway, Jules Verne.


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In Europe the French friar Niceron investigated the optics of anamorphic imaging syste-matically in the sixteenth century. This drawing about how to construct an anamorphicpicture originates from him. It was a first attempt and is not quite correct with regard tothe optics and mathematics. But that is not important, because the images which wereconstructed in this way are not very sensitive to small deviations.

Mathematically this is a transformation from a cartesian (rectangular) system to a polarcoordinate system. Artists don't normally use this construction system. They paint free-hand.

This structure from Niceron has been reproduced almost unmodified since then. You canfind it in many books about anamorphic pictures. For instance in this small ‘experimental

book’ that is sold suc-cessfully in Germany andFrance. Or in other books(Magic Mirror) by theHungarian inventor andauthor Moscovich. Mo-scovich also clearly ex-plains the law of reflectionfor plane and curved mir-rors.


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In the internet there are several examples of how to use anamorphic art. They focusmainly on three subject areas: art, math and physics. In this way they show the connecti-on between them.

As I have already mentioned, the simple polar coordinate system is quite sufficient butnot completely correct.

There is a simple experiment on howyou can create a correct anamorphicgrid. Take a laserpointer and make aconstruction where a cylindrical lensis fixed in front of the laserpointer (Ihave taken a small piece from a per-spex rod from a curtain). The result isa straight laser line. The reflectedlaser line will hit the plane and thusdirectly mark one part of the ana-morphic grid. Obtaining a completegrid in this way means quite some

work. This is, by the way, principally the raytracing method that I will explain later morein detail.

If you compare this grid with the polar coordinate system, you will see some specificdifferences especially in the corner regions. The correct grid does not consist of circles!And straight lines in the original object are not transformed into straight lines in the pla-ne.

I will now try to go deeper into the mathematics andphysics of the anamorphic transformation.

Here are two drawings about different possibilitiesof viewing or constructing anamorphic grids. Thefirst is the normal one where the observer is at afinite distance from the cylindrical mirror. But thiscase is mathematically complicated. As you can see,even the vertical straight lines from the quadraticobject grid are not transformed into straight lines inthe anamorphic grid. One reflected ray is drawn inred.

In the other case the observer is at an infinite distan-ce. In this case the vertical straight lines from theobject grid are also transformed into straight lines inthe anamorphic grid. This case is mathematicallyeasier.


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The difference between these two cases can be seen clearly. You can also see now whysmall deviations in the anamorphic grid will have an even smaller influence on the image.The whole anamorphic image is reduced into the image you see.

Now we are looking at the cylinderfrom above. Here are the incident andthe reflected rays. The construction issimple with the law of reflection, whichmeans that the angle between the inci-dent ray and the perpendicular to thecylinder surface must be the same asthe angle between the perpendicularand the reflected ray. Additionally, thedistance between the reflection point onthe cylinder surface and the point youwant to calculate must be the same asthe distance between the reflectionpoint and the equivalent point in theobject grid. With elementary mathema-tics (principally sine and cosine rule)you get the following equations for thepolar angle and the radius. And this isnot a linear and not even a quadratictransformation. One must be carefulwith the definition and handling of thepolar angle and with the sign conventi-ons for angles.

Some other interesting properties canalso be seen. If you extend all the re-flected rays backwards and then drawthe tangential curve, you receive exactlythe caustic that is known from the re-flection in a convex mirror. This curveis a cycloid. A simple mechanical con-struction for the anamorphic grid resultsfrom this. Cut out this cycloid so thatyou get a cycloid with a thickness ofperhaps 1cm, take a piece of string andattach one end onto the cusp. At theother end attach a pencil. Now you candraw the curved anamorphic grid linesdirectly by moving the pencil with thestring tautly. Centuries ago machineswere even developed to draw anamor-phic pictures automatically.


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With the formulas it is clear that you can calculate any anamorphic picture pixel by pixelwith a computer. I call this the classical way of calculation. Programs for that have beenwritten. I did some myself about 15 years ago. Another program under DOS was writ-ten by the German student Juergen Bergauer in the Institute of Theoretical Mechanics atthe University of Kaiserslautern in Germany, but the program is not easy to handle. Ihave looked for other easily available programs but I have not found anything in the in-ternet or any other reference in literature. If any of you know a source, please let meknow.

Another method of calculating anamorphic images is the so-called raytracing. In the caseof raytracing anamorphic images, a virtual slide projector with the non-distorted, originalimage is assumed. From the eye a ray goes through the original image pixel by pixel andhits the mirror, where it is reflected and then hits the plane or another surface in a pixel.The color of the pixel is the same as in the image. The mathematics is prinicpally similarto the classical method.

The German student Friedel Ulrichdeveloped a program for anamor-phic images with the raytracing me-thod as a thesis during the college-level at his German Gymnasium inPfaffenhofen in Bavaria. He evenwon a prize in the German competi-tion ‘Young Researchers’. Thestructure of his program can be seenfrom this transparency. A camera, amirror object (cylinder, cone orsphere) and a plane can be selectedand positioned. A digitalized image,e.g. from a digital camera, can beinserted. The program calculates theanamorphic image.

Both programs can be downloaded from the internet (URL: http://www.e20.physik.tu-muenchen.de/~cucke/ftp/anamorph/). There are short help-files in English. Both me-thods, classical calculation and modern raytracing, need a lot of computer power andtherefore are not very quick, especially for high image resolution. With a Pentium400MHz processor you need several minutes.

My last example - a cylindrical anamorpho-sis - was calculated with this raytracinganamorphic program. I have to remark thatanamorphic images are normally turnedupside down, which makes it even moredifficult to recognize the content. The firstimage is very strongly distorted. I believenobody can recognize which person is por-trayed here. The second image is not di-


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storted as strongly. Now youmay already guess who it is.Finally here is the solution:Prof. Luo XingKai. Pleaseexcuse me for taking yourpicture from the internet.

I want to cite one author here. In the publication ‘Anamorphosis’ (Mathematical Gazette76 (1992), 208-221) Philip Hickin wrote about the use of anamorphosis in school clas-ses: When I gave this activity (drawing cone anamorphosis) for homework I had the bestresponse I have ever had from a class.

When you think about what I have told you so far, not much physics has been involved.This is an important point. I believe physics teachers must embed their lessons much mo-re into other areas of more interest for the students, even if this means that ‘real’ physicstakes only a minor part of the whole lecture. This is especially important for studentswho don’t have physics as their principal interest.

Several difficulties will occur, strongly depending upon the level of teaching: A physicsteacher is probably familiar with some non physics topics, but only up to a certain level.On a higher level the physics teacher cannot know these topics as well as an expert. Sohe should be more an organizer of the themes and the students. He must cooperate inter-disciplinarily with colleagues.

German physics teachers who I have encouraged to engage in the topic of anamorphicimages have told me that they were successful with this approach.

The cone that rolls uphill An "antigravity" experiment

This is an old and well-known experiment. The principle issimple, but the performance is surprising. There is also alittle magic for non-scientists.

A roller is made by uniting two congruent cones by theirbases. It is placed on two inclined tracks meeting at an acuteangle. The tracks have an inclination angle ß to the horizon-tal. The doublecone will roll uphill under certain conditions.The center of gravity rolls downwards, of course


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This experiment is old, as you can see from thispicture. For friends of Latin there is the originaldescription. A friend of mine who is a physicsteacher talked with his colleague for Latin, andthen combined physics and Latin and had thepupils translate this small text. With some his-torical remarks you can make a small projectout of this experiment.

Here is the title page of the book by the Dutchauthor Gravesande, where this drawing comesfrom.

If α is the angle between the tracks and γ is thecone-angle, the condition for uprolling is

tanß < tan(α/2)·tan(γ/2)

The proof of this condition is not very compli-cated but a little bit lengthy (see Ucke, C.,Becker, J.: Roll Kegel roll!, Physik in unsererZeit 28, (1997), 161-163; in German). Youhave to look carefully at the relations betweenthe different angles.


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Here are two otherpossibilities to realizethe experiment. Withthis special doubletrack you get an oscil-lating double cone.And in this way youcan realize the experi-ment with a diabolo.

In this way the ex-periment is a nice hands-on experiment which can be used in classes. It shows the princi-ple clearly, and it is easy to construct.

I want to mention an interesting prop-erty of the double cone. Railway engi-neers know it. If you set the cone notperpendicular to the center line of thetracks but a little obliquely and let itroll, the cone will wobble at first butthen stabilize the line of the movementto the center line as you can see here.This also happens with parallel tracks.This behavior is very important. Thewheels of railways also have a doublecone profile but with a very small coneangle. The wheels will stabilize them-selves. But this is dependent upon thevelocity (and other parameters). It cancause instability at high velocities. Itwould be an interesting and probablycomplicated problem to treat this withthe aidof theoretical mechanics.

What about the quantitative acceleration and movement of the center of gravity of thedouble cone? This is also a somewhat complicated problem and appropiate for studentsof theoretical physics or engineering. This problem has been investigated in the Austrianuniversity of Graz but not yet published.

The following game is fascinating, not only for children. And the background is the sameas with the uphill-rolling double cone. My wife has suggested the name Sisyphus for thisgame because you always have to try the same frustrating procedure again. Sisyphus isan unfortunate man from Greek mythology who was punished by the old gods to keep onrolling the same stone up a steep hill.


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This arrangement can be built out of two billiard queues and the billiard ball or even sim-ply with two straws, paper clips and a marble.

Conclusion

These examples show clearly the actuality of long-known and proved physics experi-ments. They can be used on very different levels, from kindergarten to university, some-time amusing, sometime serious. There are even some problems which have not beensolved or published. By using the power of modern technical aids (e.g. computers, videocamera) you can find new approaches. There are strong connections with other discipli-nes, which is probably a very important point nowadays. Teachers are encouraged tocooperate with colleagues in other disciplines. Intercultural questions are also involved.I know other examples where similar approaches can be used and that it is thereforefruitful to look for them. This is an everlasting challenge.

connecting past and future - ucke

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