Chromogeometry and relativistic conics

Chromogeometry and relativisti c conics N J Wildb erger Sc ho ol of Mathematics and Statistics UNSW Sydney 2052 Australia This pap er shows ho w a recent reformulation of the basics o f classic al ge- ometry and trigonometry reveals a three-fold s ymmetry b etw een Euc lide a n a nd non-Euclidean (relativistic) planar geometr ies. W e apply this chr omo ge o metry to lo ok at co nics in a new lig h t. Pythagoras, area and quadrance T o measure a line segmen t in the plane, the ancient Greeks measured the ar e a of a squar e c onstru cte d on it . Algebra ically , the paralle lo gram formed by a vector v = − − − → A 1 A 2 = ( a, b ) and its p erp endicular B ( v ) = ( − b , a ) has ar ea Q = det  a b − b a  = a 2 + b 2 . (1) The Gr eeks re ferred to building s quares as ‘quadratur e’, and so we say that Q is the quadrance of the vector v , or the quadr a nce Q ( A 1 , A 2 ) betw een A 1 and A 2 . This no tio n ma kes sense ov er a ny field. A A 1 2 A 3 25 5 20 Figure 1: Pythagor as’ theorem: 5 + 20 = 25 If Q 1 = Q ( A 2 , A 3 ), Q 2 = Q ( A 1 , A 3 ) a nd Q 3 = Q ( A 1 , A 2 ) a re the qua d- rances of a tr ia ngle A 1 A 2 A 3 , then Pythago ras’ theor em and its converse can together be stated as: A 1 A 3 is p erp endicular to A 2 A 3 pr e ci sely when Q 1 + Q 2 = Q 3 . 1 Figure 1 shows an example where Q 1 = 5 , Q 2 = 20 and Q 3 = 25 . As indicated for the larg e s quare, these ar e as ma y also b e calculated by subdivisio n and (translational) rear rangement, followed by counting cells. There is a sister theorem—the T riple quad formula —that Euclid did not know, but which is fundamen tal for r ational trigonometry , intro duced in [2]: A 1 A 3 is p ar al lel to A 2 A 3 pr e ci sely when ( Q 1 + Q 2 + Q 3 ) 2 = 2  Q 2 1 + Q 2 2 + Q 2 3  . Figure 2 shows an e xample where Q 1 = 5 , Q 2 = 20 and Q 3 = 45 . A A 1 2 A 3 20 45 5 y x Figure 2: T riple quad formula: (5 + 20 + 45) 2 = 2  5 2 + 20 2 + 45 2  In terms of side lengths d 1 = √ Q 1 , d 2 = √ Q 2 and d 3 = √ Q 3 , and the semi-p erimeter s = ( d 1 + d 2 + d 3 ) / 2 , o bs erve that ( Q 1 + Q 2 + Q 3 ) 2 − 2  Q 2 1 + Q 2 2 + Q 2 3  = 4 Q 1 Q 2 − ( Q 1 + Q 2 − Q 3 ) 2 = 4 d 2 1 d 2 2 −  d 2 1 + d 2 2 − d 2 3  2 =  2 d 1 d 2 −  d 2 1 + d 2 2 − d 2 3   2 d 1 d 2 +  d 2 1 + d 2 2 − d 2 3  =  d 2 3 − ( d 1 − d 2 ) 2   ( d 1 + d 2 ) 2 − d 2 3  = ( d 3 − d 1 + d 2 ) ( d 3 + d 1 − d 2 ) ( d 1 + d 2 − d 3 ) ( d 1 + d 2 + d 3 ) = 16 ( s − d 1 ) ( s − d 2 ) ( s − d 3 ) s. Thu s Hero n’s formula in the usual form area = p s ( s − d 1 ) ( s − d 2 ) ( s − d 3 ) may b e resta ted in terms of quadr ances as 16 area 2 = ( Q 1 + Q 2 + Q 3 ) 2 − 2  Q 2 1 + Q 2 2 + Q 2 3  ≡ A ( Q 1 , Q 2 , Q 3 ) . This more fundamental for m ulation deser ves to b e called Arc himedes’ theo- rem , since Arab sour ces indicate that Archimedes knew Heron’s formula. The T riple quad formula is the sp e cial case of Archimedes’ theo r em when the area is zero. The function A ( Q 1 , Q 2 , Q 3 ) will b e called Arc himedes’ function . 2 A A 1 2 A 3 13 25 26 Figure 3: 1 6 area 2 = (13 + 25 + 26) 2 − 2  13 2 + 25 2 + 26 2  = 11 5 6 In Fig ur e 3 the quadrances a re Q 1 = 13 , Q 2 = 25 and Q 3 = 26 , so 16 area 2 = 1156, giving a rea 2 = 289 / 4 and a rea = 17 / 2 . Irratio na l side lengths are no t needed to deter mine the area o f a r ational tria ng le, and in any case when we mov e to more g eneral geometries, we hav e no choice but to g ive up on distance and angle. Blue, red and green geometries Euclidean g eometry will here be called blue geome try . W e now in tro duce tw o relativistic geo metries, ca lled r e d and gr e en , which arise fro m E instein’s theory of r elativity . These rest on alter nate notions of p e r p endicularity , but they share the same underlying affine concept of a rea as blue g eometry , and indeed the same laws of ra tional trig onometry , as will b e explained sho rtly . A A 1 2 A 3 12 27 3 y x A A 1 2 A 3 -27 12 -15 x y Figure 4: Red Pythag o ras’ theorem and T riple qua d fo r mula Define the vector v = − − − → A 1 A 2 = ( a, b ) to be red p erp endicular to R ( v ) = ( b, a ) . This ma pping is e a sily visualized: it c o rresp onds to E uclidean reflectio n in a line of slo pe 1 o r − 1 . A red square is then a par a llelogra m with sides v 3 and R ( v ), and hence (sig ned) area Q ( r ) = det  a b b a  = a 2 − b 2 (2) which we call the red quadrance bet ween A 1 and A 2 . Figure 4 illustrates that bo th Pythagor as’ theorem and the T riple quad formula hold also using red quadrances and red p erp endicularity , where the ar eas of the r ed squa res can b e computed a s before by sub divisions, (trans la tional) r e arrang ement a nd co un ting cells—or by applying the a lgebraic formula for the red quadr ance. In a similar fashion the vector v = − − − → A 1 A 2 = ( a, b ) is green p erp e ndicular to G ( v ) = ( − a, b ). This cor resp onds to Euclidean reflectio n in a vertical or horizontal line. A green square is a par allelogr a m with sides v and R ( v ), and hence (signed) are a Q ( g ) = det  a b − a b  = 2 ab (3) which we call the green quadranc e b etw een A 1 and A 2 . A A 1 2 A 3 16 36 4 y x A A 1 2 A 3 18 24 -6 y x Figure 5: Gr een Pythago r as’ theorem and T riple quad formula Figure 5 shows P ythagor as’ theorem and the T riple quad formula in the green context. This version of relativistic geometr y cor resp onds to a basis of nu ll vectors in red geo metry . All three geometries can b e defined ov er a g eneral field, not of characteristic t wo. Spreads and rational trigonometry The thr ee quadr a tic forms Q ( b ) ( a, b ) = a 2 + b 2 Q ( r ) ( a, b ) = a 2 − b 2 Q ( g ) ( a, b ) = 2 ab hav e cor resp onding dot pro ducts ( a 1 , b 1 ) · b ( a 2 , b 2 ) ≡ a 1 a 2 + b 1 b 2 ( a 1 , b 1 ) · r ( a 2 , b 2 ) ≡ a 1 a 2 − b 1 b 2 ( a 1 , b 1 ) · g ( a 2 , b 2 ) ≡ a 1 b 2 + a 2 b 1 . 4 T ogether with a 1 b 2 − a 2 b 1 = 0 describing para llel vectors, these are the four simplest bilinear ex pr essions in the four v aria bles . In rationa l tr igonometry , one w ants to w or k over gener al fields, so the notion of angle is not av a ilable, but it is imp ortant to realize that the dot pro duct is not necess arily the b est replacement. Ins tead we in tro duce the rela ted notion of spr e ad b etw een tw o lines (not b etw een rays), whic h in the blue framework is the square of the sine of the angle b etw een the lines (there ar e actua lly many such angles, but the square of the sine is the s ame for all). The blue, red and gree n spreads b etw een lines l 1 and l 2 with equations a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 are resp ectively the num b ers s ( b ) ( l 1 , l 2 ) = ( a 1 b 2 − a 2 b 1 ) 2 ( a 2 1 + b 2 1 ) ( a 2 2 + b 2 2 ) s ( r ) ( l 1 , l 2 ) = − ( a 1 b 2 − a 2 b 1 ) 2 ( a 2 1 − b 2 1 ) ( a 2 2 − b 2 2 ) s ( g ) ( l 1 , l 2 ) = − ( a 1 b 2 − a 2 b 1 ) 2 4 a 1 b 1 a 2 b 2 . These quantities are undefined when the denominators are ze r o. The negativ e signs in front of s ( r ) and s ( g ) insure that, for eac h of the co lo urs, the spr e ad at any of the thr e e vertic es of a right t riangle (one with two sides p erp endicular) is the quotient of the opp osite quadr anc e by the hyp otenuse qu adr anc e . See [3] for a pro of of this, and other facts ab out ra tional trigonometry , in a wider context. In Figure 1 the sprea ds a t A 1 and A 2 are 1 / 5 and 4 / 5 resp ectively , in the left diagram of Figure 4 the spreads at A 1 and A 2 are − 4 / 5 and 9 / 5 resp ectively , and in the left diag ram of Fig ure 5 the spreads at A 1 and A 2 are − 1 / 3 and 4 / 3 resp ectively . In each case the spr ead at the right vertex is 1, and the other tw o spreads sum to 1 . A A A A A A 1 1 1 2 2 2 A A A 3 3 3 289 650 289 168 -289 249 289 325 -289 35 289 288 289 338 289 120 289 120 13 -5 -12 25 -7 24 26 24 10 y x Blue R e d Gr e en Figure 6: Blue, red and gr een quadrances and sprea ds Figure 6 allows you to compa r e the v arious quadrances and spre ads of a fixed triang le in ea ch of the three geometries. No te the c ommon num era tors of 5 the spreads, arising because A ( Q 1 , Q 2 , Q 3 ) = ± 4 × 289 is up to sign the same in each g eometry , with a plus sign in the blue situation and a neg ative sign in the red and gre e n o nes. Aside from Pythag oras’ theorem and the T riple quad formula, the main laws of ratio na l tr igonometry a r e: for a triangle with qua drances Q 1 , Q 2 and Q 3 , and spreads s 1 , s 2 and s 3 : s 1 Q 1 = s 2 Q 2 = s 3 Q 3 (Spread law) ( Q 1 + Q 2 − Q 3 ) 2 = 4 Q 1 Q 2 (1 − s 3 ) (Cross law) ( s 1 + s 2 + s 3 ) 2 = 2  s 2 1 + s 2 2 + s 2 3  + 4 s 1 s 2 s 3 (T riple spr ead formula). As shown in [2], these laws ar e derived using only Pythago ras’ theo rem and the T riple q uad formula. Since these latter tw o results hold in all three ge o metries, the Spread law, Cro s s law and T riple sprea d for mu la also hold in all three geometries. F or any points A 1 and A 2 the sq uare of Q ( b ) ( A 1 , A 2 ) is the s um of the squares of Q ( r ) ( A 1 , A 2 ) and Q ( g ) ( A 1 , A 2 ), and for a ny lines l 1 and l 2 1 s ( b ) ( l 1 , l 2 ) + 1 s ( r ) ( l 1 , l 2 ) + 1 s ( g ) ( l 1 , l 2 ) = 2 . The fir st statement follows from the Pythag orean triple iden tity  x 2 + y 2  2 =  x 2 − y 2  2 + (2 xy ) 2 while the latter follows fro m the identit y  a 2 1 + b 2 1   a 2 2 + b 2 2  −  a 2 1 − b 2 1   a 2 2 − b 2 2  − 4 a 1 b 1 a 2 b 2 = 2 ( a 1 b 2 − a 2 b 1 ) 2 . So in Figure 6 there a re three linked (signed) Pythagorean triples, namely (13 , − 5 , − 1 2) , (25 , − 7 , 2 4) and (26 , 2 4 , 10 ) , and three triples of harmonica lly related sprea ds . - 5 5 1 0 1 5 20 25 1 8 1 6 14 1 2 1 0 8 6 4 2 - 2 -4 F l F g r F b A Figure 7: Three altitudes fro m a p o int to a line 6 Figure 7 shows the three coloured altitudes fr o m a p oint A to a line l , and the feet of tho se altitudes. Note that the blue and red altitudes are green per p endicular and similarly for the o ther co lours. The three triang le s formed by the four p oints are each triple right triangles , containing each a blue, red and green right vertex. Most theorems of pla nar E uclidean geometry ha ve universal versions, v a lid in each of the three geometries. This is a lar ge claim that deserves further investigation . In the red and green geometries, circles and rotations b ecome rectangular hyperb olas and Lor ent z b o osts. Ther e are no equilatera l triangles in the red a nd g reen geometr ies, so results like Nap oleo n’s theorem or Mor ley’s theorem will not have (obvious) analogs . T o see some chromogeometry in action, let’s hav e a lo ok a t conics in this more genera l framework. The ellipse as a grammola In the real num b er pla ne, o ne usua lly defines an el lipse as the locus of a po int X whose ratio of distance from a fixed p o in t (fo cus) to distance from a fixed line (directrix) is constant and less than one, and hyper bo las and pa rab olas similarly with eccentricities g reater than o ne and eq ual to one. By squaring this condition, w e can discuss the lo cus of a po int who se ratio of quadr a nc e fro m a fixed po int to a fix e d line is constant. B y quadrance from a p oint X to a line l we mean the obvious: constr uct the altitude line n fro m X to l , find its fo ot F and measur e Q ( X, F ). Let’s call such a lo cus a coni c section . Over a gener al field w e cannot distinguish ‘ellipses’ from ‘hyperb ola s’, although parab olas a re alwa ys well defined. 3 2 2 2 2 1 1 1 1 0 -1 -2 -3 3 2 1 0 -1 -2 -3 x y A B F F d d l l 3 2 2 2 1 1 1 0 -1 -2 -3 3 2 1 0 -1 -2 -3 x y A B F F d d s s Figure 8: T wo views of the ellipse 2 x 2 − 4 xy + 5 y 2 = 6 The left diagr am in Figure 8 s hows the central ellipse 2 x 2 − 4 xy + 5 y 2 = 6 7 with foci at F 1 = [2 , 1 ] a nd F 2 = [ − 2 , − 1 ], corresp onding directrices d 1 and d 2 with respec tive e q uations 2 x − y + 6 = 0 and 2 x − y − 6 = 0 , and eccen tricity e = p 5 / 6 . The familiar reflection prope r ty may b e recast a s: s pr e ads b etw een a tangent and lines to the fo ci fr o m a p oint o n the ellipse ar e equal. The right diagr am in Figure 8 illustr ates a (p erhaps?) nov el definition of an ellipse. It is mo tiv ated by the fact that a circle is the lo cus of a p oint X whose qua drances to t wo fixed per pe ndicular lines add to a constant. Define a grammola to be the lo cus of a point X suc h that the sum of the quadrances from X to tw o fix e d non-p er pendicula r in tersecting lines l 1 and l 2 is constant. This definition works for eac h of the three colo urs. It turns o ut that the lines l 1 and l 2 are unique; we call them the diagonals of the grammo la (see [2, Chapter 15]). The corners o f the grammola a re the points where the diagonal lines intersect it, and de ter mine the corner rectangle . In the blue s etting over the r eal num bers a gra mmola is alwa ys a n el lipse , while in the red and green settings a gra mmo la might b e a n ellipse, or it migh t b e a hyper bo la. The ellipse of Fig ure 8 is a blue grammola with blue diagonals  14 + 5 √ 6  x − 23 y = 0 and  14 − 5 √ 6  x − 23 y = 0 . The blue quadrance s of the sides of the co rner rectangle are 12 and 2 , whose pro duct 24 is the squared area. The right diagram in Figure 8 shows the usual fo ci a nd directrices of the grammola and its diagonals and corners. The quad- rances from a ny p oint on the conic to the t wo diagonals sum to 6 . The blue spread betw een the tw o diago nals is a n inv ar iant of the ellipse—in this case s b = 24 / 49. 3 2 2 1 1 0 -1 -2 -3 3 2 1 0 -1 -2 -3 x y A l l B 3 2 1 0 -1 -2 -3 3 2 2 1 1 0 -1 -2 -3 x y A B l l Figure 9: Ellipse as red and green gr ammola The ellips e can a lso b e describ ed as a red g rammola, as in the left diagra m of Figure 9. The r ed diago nals are  √ 22 + 2  x − 9 y = 0 and  √ 22 − 2  x + 9 y = 0 8 and the red corner rectangle has sides parallel to the red axes of the ellipse, and red qua drances 3 + √ 33 a nd 3 − √ 33 , who se pro duct is − 24. The four red corner s hav e r ather complica ted expres sions in this case. The r ed spread betw een the r ed diagonals is s r = − 8 / 3 . The sa me ellipse may a lso be v iewed a s a green g rammola, a s in the r ight diagram of Figure 9. The green diago nals a re  − 5 + √ 15  x + 5 y = 0 and  − 5 − √ 15  x + 5 y = 0 and the green corner rec tangle ha s sides pa r allel to the gre e n axes of the ellipse, and green quadrances 4 + 2 √ 10 and 4 − 2 √ 10 , whose pro duct is aga in − 24. Except for a sign, the three squared area s of the blue, red and gr e en corner rectangles are the sa me. The green spr ead betw een the gr een diago nals is s g = − 3 / 2 . The relatio nship b etw een the blue, red and green sprea ds of an ellipse is 1 s b + 1 s r + 1 s g = 1 . The ellipse as a quadr ola Another w ell known definition of an ellipse is as the lo cus of a point X whose sum of distances from t wo fix e d po in ts F 1 and F 2 is a constant k . T o determine a universal analo g of this, we co nsider the lo c us of a p oint X s uch that the quadrances Q 1 = Q ( F 1 , X ) a nd Q 2 = Q ( F 2 , X ), to gether with a num ber K, satisfy Arc himedes formula A ( Q 1 , Q 2 , K ) = 0 . This is the quadra tic analog to the equation d 1 + d 2 = k , just as the T r iple quad formula is the a nalog to a linear relatio n b etw een three distances. Such a loc us we c a ll a quadr ola . This a lgebraic for m ulation applies to the relativistic geometries, and a lso extends to general fields. The notion captures bo th that of ellipse and hyperb ola in the Euclidean setting, and while it is in general a different concept than a gra mmo la, it is p ossible for a co nic to b e b oth, as is the case o f a n ellipse in Euclidea n (blue) geo metry . The left diagram in Figure 10 shows that in the red geo metr y , a new phe- nomenon occur s: our same ellipse as a qua drola has t wo p airs o f foci { F 1 , F 2 } and { G 1 , G 2 } . Ea ch of these p oints is also a fo cus in the context of a conic sec- tion, and there are tw o pairs of corresp o nding dir ectrices { d 1 , d 2 } and { h 1 , h 2 } . Directrices are parallel or red p erp endicular, and int ers ect a t points on the el- lipse, and tange n ts to these directrix p oints pass throug h tw o fo ci, forming a parallelog ram which is both a blue and a gr e en r e ctangle . It turns out that the red spre a ds b etw een a tangent and lines to a pair o f red foc i ar e equal, as shown at po int s A and B . The right diagra m in Figur e 10 shows the same ellipse as a green quadrola , with ag ain tw o pa irs of green fo ci, tw o pair s of corresp onding g reen dire c trices (whic h are para llel or green p erp endicular), and the tangents at directrix p o ints forming a blue a nd r e d r ectangle. 9 3 2 2 2 2 2 1 1 1 1 1 0 -1 -2 -3 3 2 1 0 -1 -2 -3 x y A F F G G B C d h d h s s r r 3 1 1 1 1 1 0 -1 -2 -3 3 2 2 2 2 2 2 1 0 -1 -2 -3 x y A C B F F G G d d h h s s r r Figure 10: Red and green fo ci a nd dir ectrices The red and g reen directrix p oints are easy to find: they are the limits of the ellipse in the null and the co ordinate directions. So the red and gr een directrices and fo ci are also then simple to lo cate geo metr ically . This is no t the cas e for the usual (blue) fo ci and directrices , and s ug gests that co nsidering ellipses from the re la tivistic pers pe c tives can be practically useful. I n algebr aic ge o metry the ‘other’ pair of blue fo ci are not unknown; they require complex ific a tion and a pro jective view (see for example [1, Chapter 1 2]). 3 1 0 -1 -2 -3 3 2 2 0 -2 -3 1 x y -1 Figure 11: Three sets of fo ci a nd dir ectrices When we put a ll three colour ed pictures to gether, as shown in Figur e 11, another curio us phenomenon app ea r s—there are three pairs of co loured fo ci that app ear to b e close to the in terse c tions o f directrice s o f the oppo site colour. The r eason for this will become clearer la ter when we co nsider pa rab olas. 10 Hyp erb olas Over the re al num b ers some of what we saw with ellipses extends also to hyper - bo las, although there a re differences . The central h yp erb ola shown in Fig ure 12 with equation 7 x 2 + 6 xy − 17 y 2 = 12 8 is a r ed quadro la with re d fo ci F 1 = [3 , 1] and F 2 = [ − 3 , − 1], mea ning that it is the lo cus of a p oint X = [ x, y ] such that A  ( x − 3) 2 − ( y − 1) 2 , ( x + 3) 2 − ( y + 1) 2 , 64  = 0 . 10 5 0 -5 -10 10 5 0 -5 -10 1 1 1 1 2 2 2 2 x y F F G G d d h h A B Figure 12: A hyperb ola as re d qua drola and gr een gr ammola As a conic section the corresp onding directrices hav e equations 3 x − y − 16 = 0 and 3 x − y + 16 = 0 . This hyperb ola also ha s a nother pair of re d fo ci G 1 = [1 , 3] and G 2 = [ − 1 , − 3], with a sso ciated directr ic es x − 3 y + 8 = 0 a nd x − 3 y − 8 = 0 . As in the c a se of the ellips e w e considered earlier, in eac h case the focus is the p ole of the co rresp onding directrix, meaning that it is the int ers e ction of the tangents to the hyper bo la a t the directrix p oints. These tangents pass through t wo foci at a time, and are parallel to the red null dire c tio ns. The para llelogra m formed by the four fo ci is a blue and g r een rectangle. So we could have found the red fo c i and directrices purely geometr ically , by finding those po ints on the hype r b ola where the ta ngents are parallel to the red nu ll directions, and then forming in tersec tions b et ween these po in ts. This is again quite different from finding the usual blue fo ci and dir ectrices. Note that if w e try to find gr e e n fo ci, vertical tangents are easy to find, but there are no horizontal tangents, thus the s ituation will necessa rily b e somewhat different. Is the hyper bo la als o a gr ammola? It cannot be a blue g rammola, since these are all ellipses, and it turns out not to be a red g rammola either. But it is a 11 green grammola with equa tio n  119 + 8 √ 238  x + 51 y  2 2  119 + 8 √ 238  51 +  119 − 8 √ 238  x + 51 y  2 2  119 − 8 √ 238  51 = 128 3 . The g reen diago na ls ar e shown in Figure 1 2. The parab ola F rom the v iewp o int of universal (a ffine) g eometry , the most interesting co nic is the p ar ab o la . Giv en a p oint F and a generic line l not passing throug h F , the lo cus of a p oint X suc h that Q ( X , F ) = Q ( X , l ) is what we usually c a ll a par ab ola, independent of which geometry we ar e consider ing. The generic parab ola ha s a distinguished blue, red and gr e e n fo cus, and also a blue, red a nd green directrix. 10 20 30 -10 0 10 20 30 0 F F F r g g r b b l l l Figure 13: Three fo ci and dir ectrices of a para bo la Figure 13 shows a par ab ola in the Cartesia n plane and all three fo ci and directrices. A remar k able phenomenon app ear s: F b is the intersection of l r and l g , F r is the in terse c tion of l b and l g , and F g is the in terse c tion of l b and l r . F urthermore l r and l g are blue per pendicula r, l b and l g are red perp endicular , and l b and l r are gre e n per pendicula r—in other w ords w e g et a triple right triangle of fo ci. This means that once we know one o f the fo cus/ dir ectrix pairs, the other tw o can b e found simply b y construc ting the a ppropriate altitudes from the fo cus to the directrix together with their feet. Although the v arious dir ectrices ar e in different dire c tions, the axis direc - tion, defined as b eing p erp endicular to the dir ectrix, is common to all. Figure 14 shows the familiar reflection prop erty of the para bo la, where a pa rticle P approaching the parab ola a long the axis direction and reflecting off the tan- 12 10 20 30 -10 0 10 20 30 0 F F F r g g r b b l l l Z P x y Figure 14: Reflection pr op erties of a pa rab ola gent (in either a blue, red or green fashio n) a lwa ys then passes through the corres p o nding fo cus. The following figure shows some interesting co llinearities asso ciated to a parab ola, involving coloured v ertices V (in tersections o f axes with the pa rab ola), bases X (intersections o f ax es with dir e ctrices) and points Y formed by tange nts to vertices. F F F V V X Y Y Y X X V r g g b r r b b r r g g g g r b b b l l l a a a x y 10 20 30 -10 0 10 20 30 Figure 15: Collinearities for a chromatic pa rab ola Finally we show the three parab olas which hav e a given fo cus F and a given directrix l , bo th in black, each in terpre ted in o ne of the three geometr ies. Each 13 of the thr ee parab olas that share this focus and directrix hav e a fo ca l triang le consisting of F and t w o of the feet o f the altitudes from F to l , la belle d F b , F r and F g . The dotted line passe s through the intersections of the red a nd green parab olas . V ar ious vertices and axes ar e shown, and w e leav e the reader to notice int ere s ting collinearities, a nd to try to pr ove them. 10 20 30 -10 0 10 20 30 x F F F F g r b l Figure 16: Three parab olas with a co mmon fo c us and dir ectrix In conclusion, ther e may very well b e other useful metrical definitions o f conics; there are certainly s till many rich discov eries to b e made ab out these fascinating and most imp or tant g eometric ob jects. Chromog eometry extends to many other asp ects o f planar geometry , for example to tria ng le g eometry in [4]. References [1] C. G. Gibson, Elementary Ge ometry of Algebr ai c Curves , Cambridge Uni- versit y P ress, Cambridge, 19 98. [2] N. J. Wildberg e r , Divine Pr op ortions: Ra tional T rigonometry to Universal Ge ometry , Wild Egg Bo oks, Sydney , 2 0 05, h ttp://wildeg g.com. [3] N. J. Wildberg er, Affine and Pr oje ctive R ational T rigonometry , prepr int , 2006. [4] N. J. Wildb erger , Chr omo ge ometry , preprint 2006 . 14