This month I ll talk about two interesting relationships

Transcription

1 ndew Glassne s Noteboo icula Reasoning ndew Glassne Micosoft Reseach This month I ll tal about two inteesting elationships between cicles and lines. The fist topic is petty simple but cool, and I suspect thee s a compute gaphics application out thee that can use this to un faste o bette. The second topic is Ptolemy s Theoem, which is a genealization of the tiangle inequality. I ll show how it can be used to deive the angle-addition fomulas (which I always foget). Then I ll extend last month s topic of eflection and show how Ptolemy s Theoem can be used to pove that Snell s Law and Femat s Pinciple of Least Time both lead to the same geomety of efaction. icula powes We ll begin with an impotant popety of lines and cicles that I haven t seen mentioned befoe in gaphics liteatue. Figue 1 sets the stage: We have a cicle with cente and adius, a point P not on the cicle, and a line s though P that intesects the cicle at points Q and R. We ll wite (P) fo the value of point P in the implicit fomula fo the cicle. I ll demonstate the emaable fact that (P) is the poduct of the distances PQ and PR. In symbols, (P) PQ PR. To see this, we only need to wite out the standad intesection of a ay and a cicle this is the familia algeba that appeas in evey ay-tacing pogam. I ll use vecto notation fo two easons. Fist, it s a lot less messy than witing out all the coefficients. Second well, the second eason is sneay, as you ll discove in a moment. Witing out the value of the cicle at an abitay point, its value () may be witten in vecto notation as () ( ) ( ) We ll wite the line s as a paameteized ay with oigin P and diection vecto V (Q P). This means that points on s ae given by vaious values of t in the expession s P + Vt. Plugging this into the cicle equation to find (P + Vt) and then gatheing tems of t, we find 0 (P + Vt) (P + Vt) 2 (P + Vt) + ( ) 2 0 t 2 (V V) + t[2v (P )] + [(P ) (P ) 2 ] 0 at 2 + bt + c The oots t 0 and t 1 of this quadatic equation ae the values of t, which geneate points whee the ay intesects the sphee. Let s find the poduct p of the two oots this will come in handy in a moment. Witing d fo the disciminant d b 2 4ac, we find b d b d b d p t t + a a a 2 2 If we plug in the value fo d, expand, and simplify, we find 2 2 b ( b 4ac) c p 2 4a a This is a nice geneal elationship to eep in mind fo quadatic equations. Now bac to business. Let s assume that ou diection vecto V is nomalized, so V (Q P)/ Q P. This means that the values of t coesponding to the oots of the equation ae exactly the same as the distances fom P to Q and P to R. That is, if t 0 and t 1 ae the two oots (and t 0is the lesse one), t 0 PQ, and t 1 PR. Since V 1, then a (V V) 1, and thus p c/a c. Using ou value of c fom the quadatic fomula above, 1 The cicle (,) and a point P. The line s though P intesects the cicle at Q and R. P Q R s p t 0t 1 c (P ) (P ) 2 which is simply (P), the value of point P in the cicle equation. So we have poven that the poduct of the distances is the same as the value of P in the cicle s equation that is, (P) PQ PR, as pomised. Notice that nowhee did we actually use the fact that we wee in 2. The vecto notation I used doesn t cae 104 Mach/pil 1998

2 2 cyclic quadilateal. 3 Point is placed so that. how many dimensions we e in. That s the sneay eason I used vectos I was woing with lines and cicles in the plane, but the esult is equally tue fo lines and sphees in space. Ptolemy s Theoem Ou second topic this month is Ptolemy s Theoem. laudius Ptolemy ( ~87~150) was an astonome, mathematician, and geogaphe who lived in lexandia, Geece. He wote an ealy atlas emaable fo the fact that not only did he descibe the places listed, but he also included thei latitude and longitude. His othe majo wo was a set of boos called The Mathematical ollection. This was late tanslated into abic unde the title l Magiste, o The Geatest. This title was late coupted, and the boos ae now often called the lmagest. The lmagest consisted of 13 boos giving ealy algoithms fo 2 geomety as well as an astonomical system fo the motion of the stas and planets. The Eath and moon had thei obits centeed on the Eath, while eveything else evolved aound those centes. Such geocentic systems emained popula until the openican Revolution in the sixteenth centuy. Ptolemy also gave an appoximate value of π as 377/ , which is accuate to one pat in ten thousand accuate enough to sail a ship fo a few wees and then find a pot by eye. One of the still-influential pieces in the lmagest is a poof of what we now call Ptolemy s Theoem. This can be consideed an extension of the tiangle inequality. Recall that the tiangle inequality states that fo any thee points,, and in the plane, +. Ptolemy s Theoem gives a simila esult fo cyclic quadilateals. cyclic quadilateal is a figue ceated by fou unique points on a cicle. Figue 2 shows a geneic cyclic quadilateal, which I ve labeled counteclocwise as,,,. Ptolemy s Theoem says that if you multiply the lengths of opposite sides and add the poducts, this will equal the poduct of the diagonals. In fact, equality only holds if the points lie on a cicle (just as the tiangle inequality is only equal when the points ae colinea). We will stic with the cyclic equality vesion hee. To pove Ptolemy s Theoem, we ll ceate an additional point on line, such that, as in Figue 3. Without loss of geneality, we will assume that lies close to than the intesection of and. We will find two sets of simila tiangles to detemine the elations that lead to the theoem. We will use unsigned distances, so that. Futhemoe, to educe clutte I will leave the vetical bas off of the distance measues. Unless I specifically efe to a pai of lettes as something lie ac, will mean. Fist, obseve that and ae both inscibed angles that include ac, as in Figue 4. s we saw last time, this means that these two angles ae equal. Thus, as Figue 5 shows, tiangles and ae simila, since both have two common angles. ecause they e simila, we can wite o 4 oth the inscibed and include ac. Theefoe they e equal. Fo the second set of tiangles, note that by constuction. dding to both, we have + +, o, as shown in Figue 6. Futhemoe, both and ae inscibed angles including ac, which means they e also equal, as shown in Figue 7. nd. Thus tiangles and ae simila. Figue 8 shows them side-by-side. We can wite a simila pai of atios as last time: 5 Simila tiangles and. IEEE ompute Gaphics and pplications 105

3 ndew Glassne s Noteboo 6 Equal angles and. 7 ngles and ae both inscibed and include ac, so they e equal. 8 Simila tiangles and. 2 sin 2 cos a 2 sin b o. Now we ll add these two equalities togethe: + + ( + ) + + This last line is Ptolemy s Theoem. s pomised ealie, it tells us that fo a cyclic quadilateal, the poduct of the lengths of the diagonals equals the sum of the poducts of the lengths of opposite sides. ngle addition To get some expeience with Ptolemy s Theoem, we can deive the fomula fo finding the sine angle-addition fomula sin (α+β) sin α cos β+cos α sin β. We ll stat with a cylic quadilateal of adius, as shown in Figue 9, with α and β. Points and lie at opposite ends of a diamete, thus 2. Recall that the law of sines tells us that fo a tiangle, a/sin b/sin c/sin 2. We use this law to find 2 sin, whee angle α+β. Note that angles and ae both ight angles, because they e inscibed in the semicicles made by diamete thus we can label them with the sines and cosines as shown. Now we just wite out Ptolemy s Theoem, eplacing each length with its value fom Figue 9, and simplify: + 2 sin (α+β) 2 2 cos β 2 sin α+2 cos α 2 sin β sin (α+β) cos β sin α+cos α sin β 2 cos 9 cyclic quadilateal whee is a diamete of length 2. which is the standad fomula fo sin (α+β). I love it when these things wo out so nicely. The othe vaiations, fo example, cos (α β), can all be found by plugging in standad tig identities. Now we can move on to a moe challenging application. Fom Snell to Femat via Ptolemy Refaction is an impotant pat of ou visual wold. Evey tanspaent object bends light to some degee as that light passes though it. This phenomenon gives ise to eveything fom multicoloed pisms to the ability of ou eyes to focus at diffeent distances. In Januay 1998 I wote about poving the law of mio eflection by using the mathematical technique of eflection. The two had a lot in common, which pobably wasn t too supising. Now I ll show how to use Ptolemy s Theoem to pove that Snell s Law (an algebaic elationship) leads to Femat s Pinciple of Least Time (a physical hypothesis). The law of specula eflection tells us what happens to a ay of light when it passes fom one medium to anothe. When light passes though the bounday, o inteface, fom an incident medium i (say, ai) to a tansmitted medium t (say, glass), its speed v changes fom v i to v t. This causes the light ay to bend, as illustated in Figue 10. We won t discuss the mechanics of that bending hee; it s coveed in detail in most optics and compute gaphics texts. People geneally use two laws to compute the tansmitted angle θ t: Femat s Pinciple of Least Time and 106 Mach/pil 1998

4 . Snell s Law. We encounteed Femat s Pinciple in the Januay 1988 column when we looed at eflection: It says that light taes the least amount of time to get fom one point to anothe. s in that column, we ll assume a wold whee light tavels in staight lines. Suppose the light tavels fom ai point to suface point S and then to glass point G. Femat s pinciple tells us that the time it taes to get fom to S to G must be a minimum. Given and G, ou job is to find S. Note that S is not simply on the line G. Remembe that we e looing fo the least time of flight, which is not necessaily the shotest path. We want to tavel less distance whee the mateial is dense, even if it means we tavel fathe in the moe aified medium. In tems of Figue 10, we can obseve that the time it taes light to get fom to S is the distance divided by the speed in the incident medium, o S/v i, and similaly the time fom S to G is SG/v t. Thus we want to minimize S/v i + SG/v t. In symbols, fo any othe $S on the suface, we want to show that the path $SG taes longe than SG: θ i S θ t G v 10 The basic geomety of efaction fom ai to suface to glass. Sˆ SG ˆ S SG + > + v v v v i t i t θ i S h famous geometical elationship, Snell s law tells us whee S is located. In tems of the geomety of Figue 10, Snell s Law says v t sin θ i v i sin θ t I will show that Snell s Law and Femat s Pinciple ae equivalent. Fist assume Snell s Law, and then Femat s Pinciple is automatically satisfied. Let s see how this wos. Figue 11 shows the basic setup. I ve dawn a cicle (,) though points, S, and G. I ve also dawn a vetical line v pependicula to the hoizontal inteface h, and found the point whee v intesects the cicle (,) below line h. Fist, we need to find the lengths and G. Note that S foms an inscibed angle in the cicle. Recall that in the column on specula eflection we showed that an inscibed angle of θ equals a cental angle of 2θ. Figue 12 shows how to find the length of a chod UW in a cicle (,). If V is the midpoint of UVW, then angle VW θ/2, and VW sin (θ/2), so UW 2VW 2 sin (θ/2). Retuning to, the inscibed angle S is π θ i, so the cental angle is 2(π θ i), and theefoe the length of the chod is 2 sin (2(π θ i)/2) 2 sin (π θ i) 2 sin θ i θ t θ/2 G U V W 11 The inscibed quadilateal GS. 12 Finding the length of a chod. y simila easoning, G 2 sin θ t. Witing out ou chod lengths and using Snell s Law, we find whee 2v i sin θ t 2v t sin θ i. Now we ll use Ptolemy s Theoem. Since, S, G, and v i sinθt 2 sinθi 2 all lie on a cicle, in that ode, we have vt vt G S S G + SG G v t sinθi 2 sinθt 2 vi vi Substituting ou lengths fo G and fom above, IEEE ompute Gaphics and pplications 107

5 ndew Glassne s Noteboo G S S SG S SG + + vi vt vi vt This is the citical esult. We ll use it in a moment to complete the poof. Fo a moment, imagine some othe point $S on the line h between the two media. Is it possible that this point, in violation of Snell s Law, could lead to a shote time of flight fom to G? Since S and ae both on v, which is pependicula to h, $S>S fo all $S S on h. Multiplying both sides by G, we get G $S > G S Now we use the esult fom above. Replacing both sides of the inequality with thei equivalent fomulas fom Ptolemy s Theoem and factoing out the common facto, we find Sˆ SG ˆ S SG + > + v v v v i t i t which was what we had hoped to find. So if we choose S in accodance with Snell s Law, then no othe point $S on the inteface h can fom a path $SG with a shote time of flight than SG. Thus we used Ptolemy s Theoem to show that if we assume Snell s Law, we have also satisfied Femat s Pinciple of Least Time fo efaction. Wapping up I m supised that I haven t seen moe on Ptolemy s Theoem in geneal it seems lie a vey useful tool. Thee ae often lots of ways to pove these useful 2 theoems, and afte I woed out one to my satisfaction, I hunted aound on the Net fo something bette. I found it! The nice little poof that I used in this column follows the poof given at poofs/ptolemy.html. The tiple-play using Ptolemy s Theoem fo efaction is based on the discussion by an Pedoe in Geomety (ove Publications, 1970). Thee must be lots of ways to use these bits of cicleand-line geomety to acceleate ay tacing, but I m not sue how. I d be happy to hea fom any eades who find good applications fo these geometical tidbits. fte I had sent this column to G& fo publication, I mentioned it to Jim linn, who emaed that he had ceated an animated vesion of Ptolemy s Theoeem and the angle-addition fomulas as pat of his wo on Poject Mathematics!. He bought in the tape the next day, and it coveed these topics in Jim s usual clea and entetaining style. I ecommend you eep an eye open fo Sines and osines, Pat III the next time Poject Mathematics! appeas on you local television. Reades may contact Glassne at Micosoft Reseach, glassne@micosoft.com. is online ompute Gaphics and pplications Online Featues Instant ccess Table of ontents ac Issues Full Text Seach Options Electonic Only - $26.00 Pape Only - $33.00 ombination - $43.00 IEEE ompute Society Los Vaqueos icle Los lamitos, Toll-Fee S.OOS Phone: The full text of ompute Gaphics and pplications is available to ompute Society membes who have an online subsciption and an e-account. Not a subscibe? Ty a sample (in dobe cobat 3.0 fomat): Thee ae 16 othe optional magazines and tansactions subsciptions also available online. Online subsciption access to ompute Gaphics and pplications is a benefit eseved fo ompute Society membes only. 108 Mach/pil 1998