New Geometric Constructions to Determine the Radius of Curvature of Conics at any Point

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1 roceedings of the th WSEAS Interntionl Conference on ALIED MATHEMATICS, Dlls, Texs, USA, Mrch 22-24, New Geometric Constructions to Determine the Rdius of Curvture of Conics t ny oint FELIE JIMÉNEZ, FRANCISC GRANER Deprtment of Applied Mthemtics University of the Bsque Country (UV/EHU) Escuel Técnic Superior de Ingenierí; Ald. Urquijo s/n; 480 Bilo SAIN Astrct: - Compss-nd-strightedge constructions hve long trdition nd still re of very genuine mthemticl interest []. Some recent results in the geometry of conics with remrkle constructive nd engineering pplictions [2] lso suggest originl constructions to otin the rdius of curvture of conics t ny given point using compss nd strightedge lone. Three versions of the method re presented with figures, descriptions of constructive steps, nd justifictions of their exctitude to solve the prolem stted. Key-Words: - Compss-nd-Strightedge Constructions; Geometry of Conics; Ellipse; Hyperol; rol; Rdius of Curvture. Introduction Strightedge nd compss geometric constructions re clssic, dting ck to the ncient Greeks; their interest is certinly historicl, ut lso geometricl nd mthemticl in very sic sense []. As is well-known, the opertions llowed in these constructions re only two: to drw stright line joining two given points, nd to drw circle whose center nd rdius re known. Ech point must then e determined s the intersection of two stright lines, or line nd circle, or two circles. Conic sections re mong the most intensively pplied curves in Computtionl Geometry, Computer Grphics nd other fields relted with Geometry nd Geometric Design, nd the rdius of curvture is crucil differentil property of ny curve in the ove-mentioned fields. This, together with the relevnce of strightedge-nd-compss constructions, mkes it worthy to study ny new construction finding the rdius of curvture of conic t ny given point y mens of strightedge nd compss lone. Biliogrphy presenting solution to such n ovious prolem of clssic type is not s common s one might expect. Clssics in the field like [] or [4] offer no direct ccount of it (lthough it my e possile to infer solution sed on the principles discussed therein). A method to solve the prolem for n ellipse ws pulished in []; it hs the slight disdvntge tht it does not work with its vertices. In [6] the sme method is shown, ut two lterntive constructions re provided to solve the vertices cse. Such lterntives, in contrst, re not necessry with the constructions presented in this pper. The pproch proposed here is different nd originl. It is sed on recently found property of conic sections, nmely, tht the cue root of the rdius of curvture of conic t ny of its points is directly proportionl to the length of the segment perpendiculr to the conic from tht point to ny one of its min xes. It my e surprising tht such simple new insights on conics cn still come to light nowdys. roof nd discussion of this property, long with consequences for the genertion of fmilies of surfces of conic sections with mny interesting pplictions, cn e found in [2]. More specificlly, prticulr cse of these fmilies of surfces mkes them ruled, which gretly fcilittes their constructive pplictions (even if they do not turn out to e develople, they cn e relized y mens of stright ems, tie rods, etc., nd indeed mny pplictions hve lredy een crried out in Civil Engineering, Art nd Architecture, Geometric Design, etc.) In this pper, tht sme property of conics inspires some different, ut relted, geometric constructions. For ech one of them, the figure nd description of the successive steps required to drw it re followed y mthemticl justifiction of its exctitude to solve the strightedge-nd-compss prolem. For the ellipticl cse, the dt re hlf-xes, of ellipse E nd its point, in which the rdius of curvture R is sought, s cn e seen in Fig.. It hs een drwn with > (specificlly, / = /2), lthough the cse > cn e drwn nlogously, presenting essentilly no difference. The only prt of the figure tht cnnot e ctully drwn with

2 roceedings of the th WSEAS Interntionl Conference on ALIED MATHEMATICS, Dlls, Texs, USA, Mrch 22-24, strightedge nd compss is precisely the ellipse E; nevertheless, E hs lso een represented to help visuliztion, nd it is possile nd esy to otin ny numer of ritrry points of E with strightedge nd compss. Fig. elow shows wellknown wy to do it: fter drwing the min circle c nd the minor circle c of E (only qurter of which re represented in the figure), drw ny stright line from the origin nd determine the respective points of intersection nd 2. The horizontl line y 2 nd the verticl one y intersect in, which is point of the ellipse, s is proved y its prmetric equtions {x = cos t, y = sin t}, where the prmeter t is the ngle of the stright line y, 2 nd with the xis of scisss (incidentlly, t = rd in Fig.). 2 Justified Constructions In this section, ll constructions re fully explined, including figures, with justified constructive steps. A first construction for the ellipticl cse will e followed y solution to the hyperolic one. The prolic cse is similr to them; insted of showing its solution explicitly, more interesting, optimized version for the cse of n ellipse will e presented. 2. First Construction for the Ellipticl Cse Fig. shows the first originl construction we present to solve the prolem for the ellipticl cse. 2.. Description of the Constructive Steps First, the positions of foci F nd F of E re determined ( construction tht hs not een represented, ut is trivil tking into ccount tht F = 2 ). Both foci re joined with nd the isecting line is drwn, whose segment etween nd the xis we cll e x. Next, e x is trnslted horizontlly until its ottom end coincides with the origin of coordintes (the trnslted segment hs lso een leled e x in the figure, to indicte its length). This, like mny other elementry constructs, is trivilly chievle with strightedge nd compss. We cll its upper end nd r the stright line y nd (we only drw prt of it in the first qudrnt). With center t, is rotted onto xis +, clling 2 the point otined. Let us cll,,, the points of coordintes (,0), (,0), (0,), (0,), respectively. By 2 prllel to is drwn until intersecting r t, which is in turn rotted upon, clling 4 the 6 2 / r 2 4 E c c e x e x F F Fig.. Rdius of curvture of n ellipse with strightedge nd compss: First Construction.

3 roceedings of the th WSEAS Interntionl Conference on ALIED MATHEMATICS, Dlls, Texs, USA, Mrch 22-24, point otined. Agin the prllel to y 4 is drwn to meet r t. Besides, the prllel to y is drwn to meet + t 6. And finlly, the prllel to is drwn y 6 until intersecting r t. The rdius of curvture of E t is exctly distnce Mthemticl Justifiction of the First Construction It is well-known tht the two isecting lines of the stright lines joining with oth foci of E re the tngent nd the norml to E y (only the norml e x is drwn in Fig.; its perpendiculr y, not drwn is tngent to E). Therefore, y virtue of the ove-stted property tken from [2], the length of e x is proportionl to the cue root the rdius of curvture R of E t. More precisely: the rtio etween e x nd the cue root of R (serched) is equl to the rtio etween hlf-xis nd the cue root of the rdius of curvture t. This is well-known to e 2 /, so it cn e esily otined, y the Theorem of Thles, y mens of the two dotted prllels in Fig. (i.e, the prllel to y so s to otin 6, of ordinte 2 /). Therefore, to solve the prolem, it is enough to rise the rtio e x / to the third power (which is chieved y mens of the two dshed rottions upon + from nd 2 to 4 nd, respectively followed y the corresponding dshed prllels until r 2 nd 4, which re drwn prllel to ) nd then to multiply the resulting rtio y the rdius of curvture t (which is chieved y the two dsh-dotted prllels in Fig., i.e., nd 6 ). 2.2 Construction for the Hyperolic Cse Fig.2 shows the construction to solve the prolem for the cse of hyperol. As will e pprent, the ide ehind it is very similr to the previous one. It hs een drwn for = 4, =. oint, of coordintes ( cosh t, sinh t), hs een drwn for t = Description of the Constructive Steps Agin, the positions of foci F nd F of H re determined in the first plce (which hs not een represented, ut is trivil since F 2 = ). Both foci re joined with nd the isecting line is drwn, the length of whose segment etween nd the xis t we cll e y. Using this distnce, point 2 is drwn on the xis with ordinte e y. With center t, 2 is rotted onto xis +, clling the point otined. By prllel to 2 is drwn until intersecting the xis t 4, which is in turn rotted upon +, clling the point otined. A second prllel y is drwn to meet the xis t ( 2 = e y ) 2 H R e y F 2 / F Fig.2. Rdius of curvture of hyperol t ny point with strightedge nd compss.

4 roceedings of the th WSEAS Interntionl Conference on ALIED MATHEMATICS, Dlls, Texs, USA, Mrch 22-24, Besides, is rotted onto the xis, from where prllel to is drwn to meet the xis. When this lst point is rotted gin onto the xis, its sciss is 2 /. Drwing from this point prllel to previously otined 6 meets the xis t point whose ordinte is exctly the serched rdius of curvture R of the hyperol t its point Mthemticl Justifiction of the Construction for the Hyperolic Cse It is well-known tht the two isecting lines of the stright lines joining with oth foci of H re the tngent nd the norml to H y ; therefore, y virtue of the property stted in Section, the length e y is proportionl to the cue root the rdius of curvture R of H t. More precisely: the rtio etween e y nd the cue root of R (serched) is equl to the rtio etween hlf-xis nd the cue root of the rdius of curvture t (,0). This is well-known to e 2 /, so it cn e esily otined, y the Theorem of Thles, y mens of the dotted lines in Fig.2. Therefore, to solve the prolem, it is enough to rise the rtio e y / to the third power (which is chieved y mens of the two dshed rottions upon + followed y the corresponding dshed prllels until ) nd then to multiply the resulting rtio y the lredy otined rdius of curvture t (,0). A very similr construction sed on the sme principles my e derived for the prolic cse. However, we think it is more interesting to show how the first construction presented, for exmple, my e optimized in the two senses explined elow. 2. Second, ptimized Construction for the Ellipticl Cse Fig. shows the optimized construction proposed. The min dvntge of the first constructions is pedgogic one, nd it lies in the esiness with which it cn e reproduced y simple resoning just knowing tht the cue root of the rdius of curvture t is proportionl to the length of norml until ny xis of the conic. This lst construction is vrition tht requires less pper re nd less precision in the determintion of intersections. It lso results in more hrmonic (uniform) distriution of lines on the drwing. It requires less pper re ecuse the points used re nerer origin of coordintes, eing ll in the sme qudrnt s (first one, in our cse). Besides, it is not necessry to drw the lrgest rdius of curvture of E, which is t if >. To illustrte this difference, it my e noted tht Fig. ws drwn with =., = (nd t = rd), while Fig. is drwn with = 2, = (nd eqully t = rd), so the lrgest rdius of curvture hs effectively chnged from 2 / =. 2 / = 2.2 to 2 2 / = 4. This mens tht Fig., if drwn with the dt of Fig., would hve needed considerle reduction to fit in the pge, while Fig. cn e seen to fit quite comfortly even in just one text column. t 2 r 8 c Fig.. Rdius of curvture of n ellipse with strightedge nd compss: Second Construction. This second ellipticl construction lso requires less precision in the determintion of intersections ecuse none tkes plce under smll ngles, nd ll the segments to which prllels hve to e drwn re quite longer thn nd, especilly, in the First Construction. In other words, the conditioning of the prolem is etter with this method. 2.. Description of the Constructive Steps As with the previous ones, we will first descrie the steps to crry out the construction, nd then we will go on to justify its mthemticl exctitude. The dt re the sme s efore: hlf-xes, of ellipse E, nd its point in which its rdius of curvture must e clculted. Agin, the ellipse itself is lso represented for illustrtive resons, even if only discrete numer of its points my e drwn y the sole use of strightedge nd compss. We strt y drwing the min circle c nd the minor circle c of E (only their portions in the first E 6 4 c

5 roceedings of the th WSEAS Interntionl Conference on ALIED MATHEMATICS, Dlls, Texs, USA, Mrch 22-24, qudrnt re drwn). We cll nd their respective intersecting points with +. By verticl line is drwn until c t, nd horizontl until c t 2 ; horizontl y nd verticl y 2 meet t, nd we cll r the stright line y coordinte origin nd. With center t, is rotted onto + t 4 ; y 4 prllel to until r t, which is gin rotted onto 6, nd y 6 nother prllel to until. Finlly, y prllel to is drwn until r t 8, eing distnce 8 the exct rdius of curvture of E t Mthemticl Justifiction of the Second Construction The coordintes of point of the ellipse re, s cn e seen in Fig.2, ( cos t, sin t). Note tht prmeter t is not the ngle defined y the xis nd the position vector of itself, ut of uxiliry points nd 2, which re ligned with the origin of coordintes. By the Theorem of ythgors, numertor of well-known eqution: ( 2 sin 2 t + 2 cos 2 t) / 2 is the R = () s e.g. in [2], p. 96, so it will e enough to rise to the cue nd divide y to solve the prolem. Assuming, to simplify the explntion, =, nd y the Theorem of Thles, oth rottions upon + followed y prllels until r do rise to the third power, while the dsh-dotted prllels chieve the division y =. Finlly note tht ssuming = implies no loss of generlity, ecuse the unit of length cn lwys e chosen to e equl to the length of segment (i.e., ), nd if the rdius of the osculting circle is the 8 segment drwn, this will hold whtever length unit is used. Finlly, note tht due to the lso well-known equtions / cos = cos t ( sin t + cos t) (2) / 2 sin = sin t ( sin t + cos t) s e.g. in [2], p. 96, the ngle shown s on Fig.2 is indeed the one of the norml to E t, which would sve the representtion of F, perhps too fr on the left, in the First Construction s well. Conclusions nd Further Work New geometric constructions to find the rdius of curvture of conics t ny one of their points y mens of strightedge nd compss hve een presented. These constructions re consequences of newly discovered property of conic sections with mny engineering nd constructive pplictions [2]. Deeper comprison with lredy existing methods [], [6], nd study from the projective viewpoint [] my constitute the oject of further investigtion. References: [] J. Delttre nd R. Bkouche, Why ruler nd compss?, in History of Mthemtics: History of rolems, pp. 89-, ris, 99. [2] F. Grnero Rodríguez, F. Jiménez Hernández nd J. J. Dori Irirte, Constructing fmily of conics y curvture-dependent offsetting from given conic, Computer Aided Geometric Design, No. 6, 999, pp [] J. etersen, Methods nd Theories for the Solution of rolems of Geometricl Construction Applied to 40 rolems, Stechert, New ork, 92. [4]. uig Adm, Curso de Geometrí Métric; Tomo II - Complementos, Biliotec Mtemátic, Mdrid, 92. [] T. H. Egles, Constructive geometry of plne curves. With numerous exmples, p. 46, Mcmilln & Co., London, 88. [6] V. Lkshminrynn nd R. S. Vishwnr, A Text Book of rcticl Geometry or Geometricl Drwing, p. 29, Ed. Jin Brothers, ilni- Jodhpur-Udipur, 962. [] M. A. enn nd R. R. tterson, rojective Geometry nd its Applictions to Computer Grphics, rentice Hll, New Jersey, 986.

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in