1 VOL. 77, NO. 4, OTOR Whee the amea Was KTHRN McL. YRS JMS M. HNL Smith ollege Nothampton, M How many times have you seen something like this? Then Now Souces: outesy of The ostonian Society/Ol State House Simon lay/hysalis mages On the left is a pictue of the Massachusetts Statehouse in oston, taken about On the ight is a pictue taken in They appea in oston Then an Now  an ae meant to show us how the builing an its setting have change, but the effect is iminishe because the camea was not in the same place fo both photogaphs. How ha is it to etemine the exact location of the photogaphe fom infomation in a photogaph? The poblem of unestaning the elative positions of image an object is actively stuie by compute scientists. n Kanatani , it is pat of computational pojective geomety. The specific task of locating the camea fom the photogaph is calle camea calibation. n Kanatani s book the pocess is quite involve an technical. n a mathematical pape publishe late, gga  tackles the same poblem. He poves that the task can be one, but the technique is similaly complex an the pape oes not eive a pactical metho o fomula. n this pape, we pesent a metho an a fomula fo locating the position of the photogaphe. Ou basic esult is the following: PROPOSTON. f a pictue of a ectangula soli taken by a vetically-hel pinhole camea has measuements (on the photogaph) of a, b, c,, an e, a b c e
2 252 MTHMTS MZN then the camea was positione c (b c) + e(b a) to the left of in the iection fom to an ae (b c) + e(b a) in font of point,whee an ae on-site measuements. The poof is base on high-school plane geomety an the basic pinciples of pojective geomety taught in a beginning awing class. ackgoun Ou assumption is that the camea is a pinhole camea with the film in a vetical plane (plane pepenicula to the goun). Une these cicumstances, the image on the film is the same as if we pojecte the thee-imensional wol onto a plane, what we ll call the, using staight lines to the viewe s. film = pinhole The only iffeence is that with a pinhole camea, the image appeas on the film upsie own. We ll nee a few elementay facts about this pojection: () The images of lines that ae paallel to the goun an to one anothe, but not paallel to the, meet at a single point in the. This point is calle the vanishing point of the collection of paallel lines.
3 VOL. 77, NO. 4, OTOR magine a collection of planes, each passing though the an one of the paallel lines. Then the planes intesect in a line that meets the at the vanishing point. ll such vanishing points lie on a single hoizontal line calle the hoizon line. () Lines in the eal wol that ae paallel to each othe an also paallel to the ae paallel when pojecte onto the. o Sale om this it follows that eal hoizontal lines ae pojecte to hoizontal lines. () lso, atios along lines paallel to the ae peseve when pojecte to the. n the iagam below, this means that X/Y = x/y. X Y o Sale SOLD! x y inally, (D) Lines on the goun connecting an object to the photogaphe appea as vetical lines on the. gain, imagine a plane containing the of the photogaphe an the line to the photogaphe.
4 254 MTHMTS MZN photogaphe That plane is vetical an intesects the in a vetical line. conveseof(d) is also tue: lines in the goun plane whose images ae vetical connect to the photogaphe. Ou metho iven the tools above, we pesent a simple metho fo etemining the location of the photogaphe. We stat with a photogaph of John M. eene Hall at Smith ollege, taken aoun 1935 by ga Scott. Since the builing is a complex soli, we pick a ectangula soli on it whose cones ae easy to locate. Souce: Histoic Nothampton, Nothampton, Massachusetts We ll call this outline the schematic pictue.
5 VOL. 77, NO. 4, OTOR a b c e The schematic coespons to the aeial view below, whee is the font of the builing an P is the location of the photogaphe. D J P Ou goal is to compute the istances an J. We ll compute the computation of J can be one symmetically. Ou poceue is to expess in tems of the five measuements a, b, c,, ane in the. ssuming we can measue on site, we can then multiply this times the atio to fin. To make the poof easie to view, we will show ou wok on a schematic with shape angles: a b c e We begin by extening an in the schematic pictue to etemine the location of the left vanishing point, V.
6 256 MTHMTS MZN Next, notice that P in the aeial view is paallel to, hence by act (), inthe schematic pictue it passes though V. lso, since it is a line to the photogaphe, by act (D) it is vetical in the schematic pictue. Thus point is the intesection of this vetical with the extension of. V Now we a a hoizontal line though paallel to the an exten P an D to meet it. n the aeial view, it looks like: D K L J P y act (), this line is also hoizontal in the schematic. The aeial view line L is paallel to an P, so it too passes though V. V K L om K L in the aeial view we have = K L. om act (), this popotion is equal to the atio of istances /s.
7 VOL. 77, NO. 4, OTOR V K s L To fin /s, we a two moe hoizontal lines, N an the hoizon line VH, then focus on the lowe half of the esulting figue. V N K a' b' c' H J e s L om VLK VN we have + s b = + e c, fom which we can eive: om VJ VH we have s = c b + b e c. =, fom which we can eive: = b b a b a. These togethe give us s = c b b a b b + b b a e c = b b a c b + b e ea c. We pomise to expess this atio in tems of a, b, c,,ane. We can accomplish that by one moe application of simila tiangles: We have x a a' b' b e c c' y a b = x x + = a b, an c b = y y + e = c b,
8 258 MTHMTS MZN an so giving us = s = a a = b b = c c b b c b b b + b b b e e b a b b c b = c (b c) + e(b a). The coesponing fomula fo J/ can be foun symmetically: J = ae (b c) + e(b a). This completes the poof of the poposition. The last step in locating the position of the camea is fining its height. This is accomplishe in a pimitive way by noting whee the hoizon line cuts acoss the pictue. The height of the camea is the height of this line as it appeas against the builing in the pictue. Souce: Histoic Nothampton, Nothampton, Massachusetts onclusion The close ageement of the two pictues illustates the poposition. Then Now Souce: Histoic Nothampton, Nothampton, Massachusetts
9 VOL. 77, NO. 4, OTOR Thee ae poblems, though, in applying the poposition. t may be ifficult to fin an appopiate pat of a builing to analyze. t can be ifficult to measue the builing. t can be ifficult to measue the photogaph. inally, locating the spot compute by the poposition, is not easy without equipment. onsieing these poblems, the close ageement of the pictues of John M. eene Hall might be consiee goo luck. We use a high-esolution scan on the achive photogaph b was measue at 470 pixels. ven so, if b wee measue just one pixel less, the compute location of the photogaphe changes by almost two feet (because of the stategic location of b in the enominato of the fomula). RRNS 1. M. H. gga, Pinhole cameas, pespective, an pojective geomety, me. Math. Monthly 105:7 (1998), Kenichi Kanatani, eometic omputation fo Machine Vision, laenon Pess, Oxfo, lizabeth McNulty, oston Then an Now, Thune ay Pess, Poof Without Wos: xtema of the unction a cos t + b sin t y (cos t, sin t) 1 ax + by = 0 x 1 a cos t + b sin t / a 2 + b 2 1 a 2 + b 2 a cos t + b sin t a 2 + b 2 M. HSSN,M.YT, ND H. TMOOR NSTTUT OR DVND STUDS N S SNS, P. O. OX , V ZN,ZNJN 45195, RN
9. Intoduction CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS In this chapte we show how Keple s laws can be deived fom Newton s laws of motion and gavitation, and consevation of angula momentum, and
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