Applications of a New Property of Conics to Architecture: An Alternative Design Project for Rio de Janeiro Metropolitan Cathedral

Transcription

1 Jun V. Mrtín Zorrquino Frneso Grnero odrígue José uis Cno Mrtín Applitions of New Property of Conis to Arhiteture: An Alterntive Design Projet for io de Jneiro Metropolitn Cthedrl This pper desries the mthemtil disovery of new property of onis whih llows the development of numerous geometri projets for use in rhiteturl nd engineering pplitions. Illustrted is n rhiteturl pplition in the form of n lterntive projet for ío de Jneiro Metropolitn Cthedrl feturing the integrtion of irulr se nd ross in the top plne. Two lterntive designs re presented for the thedrl, sed on the hoie of either the tin Immis or Greek ross. Applitions of new property of onis in Arhiteture In 993 we set forth new property of onis [Mrtín-Zorrquino, et l. 993]. Susequent ppers presented t three interntionl onferenes (ISAMA 99, the 9th Interntionl Congress of Grphi Engineering nd Mthemtis nd Design 98) exmined some interesting exmples of the pplitions of this new property of onis in rhiteture nd in engineering [Mrtín- Zorrquino, et l. 999; Mrtín-Zorrquino, et l. 998; Mrtín-Zorrquino, et l. 997]. This pper sets forth the rhiteturl possiilities tht new mthemtil disovery llows us to hieve. First, the new property of onis hs een proved, long with their two prtiulr ses, further nlysing fmily of generted ellipses E(M) in suh wy tht y mens of simple lw of liner rrngement, it is possile to generte the three-dimensionl surfe suh tht the setions perpendiulr to the Z-xis re ellipses, proof of their ruled hrter. Next, we tke s our exmple two lterntive rrngements for io de Jneiro Metropolitn thedrl, setting forth the interpenetrtion nd prmetrition tehniques rried out in order to hieve these geometril projets. In eh rrngement irulr se is linked to ross on the ridge ove nd prllel to the plne where the se irumferene is loted. These rrngements hve the novel feture of the ross s n integrl prt of the onstrution, nd the differene etween the two possiilities depends on the style of ross hosen, tht is, whether the tin Immis or the Greek ross is preferred. The three-dimensionl development of the projet hs een hieved y the development of omputer progrms tht hve enled us to depit the two rrngements in three-spe, while working out n interesting method of prmetrition of the finl oni surfe s result of the proess of interpenetrtion of two or more surfes. Finlly, we del with spets of onstrution oth with onrete nd metlli strutures for the ttinment of the geometri projets, suh s, the originl onstrution tehniques for onrete shells used widely y Spnish rquitet Félix Cndel in ountries suh s Méxio, Colomi, the United Sttes nd Spin. Mthemtil Demonstrtion of the new property of onis prtiulried to the ellipse se In the projets set out here s lterntives to the tul io de Jneiro Metropolitn Cthedrl, we hve mde use of the new property of onis [Mrtin-Zorrquino et l, 993] dpted to the prtiulr se of the ellipse. Hene, ll the mthemtil lulus needed for the grphi design NEXUS NETWOK JOUNA VO. 3, NO., 00 43

2 of the projets re sed on the expressions tht result from of the demonstrtion of the new property of onis for the se of the ellipse, s this philosophy of lultion n e extended, s is desried elow, to the other two remining onis, nmely, the hyperol nd the prol. eginning then, with the demonsttion of the new property of onis, let us first onsider ellipse M (Figure ), defined in the plne uv y the eqution for the ellipse [see Appendix I]: u v, () where stnds for the mjor semixis of the ellipse nd the minor semixis of the ellipse. Figure A stright line norml to ellipse M t point Q hs een drwn, showing only qurter of the ellipse, s in Figure, from where the following property shll e demonstrted: The loi of the points (x,y) mirrors of points (u,v) M, through the mpping stted in this figure (k ), is in its turn nother ellipse M. The exeption point V (pex of M) together with its imge point V (pex of M ), yields to: VV k () where the vlue orresponds to the urvture rdius t point V(,0), eing represented in the figure y the distne from point C to V(,0). In the fmily E(M) of ellipses generted for 44 NEXUS NETWOK JOUNA VO. 3, NO., 00

3 the vlues k, there re two irumferenes nd two stright lines (ellipses with null semixis). In order to prove this property of the ellipse, whih is ommon to ll three onis, s further demonstrted, let us first rrive t the slope of the stright line n norml to the genertor ellipse M t point Q(u,v). Deriving the eqution () it yields: u v (3) v Hene, the slope of the stright line norml to the tngent stright line Q(u,v) is yielded through the ondition of perpendiulrity etween the two stright lines: mm =. Thus, it results tht m = /m, nd therefore: v tg u (4) is the slope eing sought. Due to the similrities etween tringles, (ut the reder should note tht Q V, e 0 ). Agin, referring to Figure, mking lultions: yields: v e y v v y ke k v y v ( k ) v v k tg, v 0 x u x u u u x u nd sine Q V (v 0), it rings out: v u k v x (5) u k y sustituting the previous expressions of u,v into eqution (), eventully it turns out the following points loi Q (x,y): x y E M (6) k k x NEXUS NETWOK JOUNA VO. 3, NO., 00 45

4 This represents the Crtesin eqution of n ellipse for ll k, nd y depending on the prmeter k, it onstitutes the eqution of fmily of ellipses E(M), generted from the genertor ellipse M defined ording to the eqution (). Anlys is of the ellipse fmily E(M) In the ellipse fmily E(M) generted from the genertor ellipse M defined ording to eqution (), there re two irumferenes nd two stright lines, whih is further demostrted in Figure, where suh prtiulr ses hve only een represented in one qurter of the two-dimension representtion. Figure Identifying the semixis of E(M), we hve k k k k Sustituting those vlues into eqution (6) gives rise respetively to the irumferenes: x + y = ( + ) x + y = ( ) In the sme wy, y mking eh of the semixes equl to ero yields: 46 NEXUS NETWOK JOUNA VO. 3, NO., 00

5 (k ) 0 k 0 k = k= The equlity k= gives rise to symmetril nd horiontl segment whose enter mthes the point O (the origin of the oordintes), with C s one of its edges (Figures nd 3). Figure 3 With k, we hve symmetril nd vertil segment with its enter is the point O, nd C 4 s one of its edges. It n e esily proven tht / y /, re respetively the semilengths of suh segments, noting the ft tht the segment C C represents the distne etween the urvture enters in the horiontl xis; in the sme wy the segment C 3 C 4 represents the distne etween the urvture enters in the vertil xis. Figure 3 is omputer representtion of different vlues of k, two stright lines, two irumferenes nd severl ellipses, mong whih n e notied the genertor ellipse M of the fmily. Every generted point nd generted oni remins defined in their totlity y numeril prmeter k referred to s the genertor oni. This onstitutes thoroughly exlusive property in omprison to ny other known method of generting surfes with onis. In the proess of generting onis of the fmily E(M), we n find semi-stright lines or stright lines (depending on whether the genertor oni is n ellipse, hyperol or prol). NEXUS NETWOK JOUNA VO. 3, NO., 00 47

6 Thus, stright line n e delt with mthemtilly nd geometrilly s oni elonging to the fmily. In the se of the genertor ellipse, two irumferenes re otined tht n e hndled s onis of the fmily. As orollry, the following n e stted: A new onept ppers: tht of the genertor ellipse, or in generl, the genertor oni elonging to the orresponding fmily of generted onis. It is novelty to hndle the stright lines s ellipses with null semi-xis. Genertion of surfes from the fmily E(M) If it were possile to rrnge in spe, in ontinuous wy, this entire fmily of generted onis, we would otin surfe hving ross-setions tht would lso elong to the fmily. This guideline for rrnging the ellipses will e lled the Arrngement w nd shll orrespond to generting line upon whih ll trnsformed points oming from single point of the genertor ellipse shll rest. This mens tht eh k pplied to the genertor ellipse will hve its orresponding point upon the generting line in spe. Tht is to sy, eh k pplied to the genertor ellipse will hve its orresponding ellipse in spe, whih is n dditionl rrngement ondition for ll prllel plnes. Eh point elonging to the genertor oni retes lines of the sme fmily ording to the Arrngement w. So, if this Arrngement w is liner funtion of or proportionl to the generting ftor k, those lines shll e stright lines elonging to ruled surfe. It is espeilly importnt to mention here in referene to mtters of onstrution tht those points elonging to the generting lines n e defined t one, y simply linking the homologous points etween stright line segment nd irumferene. Thus whole set of ellipses re otined without the need to resort to trditionl mens for their genertion. The simpliity of this mthemtil tretment deserves note s remrkle property, in ontrst to the ft tht usully the only simple tretment of similr ruled surfes in siene nd tehnology is grphi. It is outstnding tht, thnks to the Arrngement w, n infinity of surfes, whether liner or not, n e settled up. Sustituting k for in eqution (6) ording to the liner reltion k=, for =, we hve: x y E M (7) whih is the eqution for surfe hving ross-setion urves tht re the ellipses of the fmily E(M), the -oordinte eing null in the genertor ellipse plne, where ll diretor urves or ross-setion urves re loted in prllel plnes nd perpendiulr to the vertil xis Z. Although the study of the surfe with this eqution ws first undertken with the im of representing or hrteriing the orresponding mthemtil model, it hs given rise up to the sertinment of remrkle properties nd pplitions, espeilly in onstrution, owing lrgely 48 NEXUS NETWOK JOUNA VO. 3, NO., 00

7 to the ft tht these surfes re ruled ones, s shll e further demonstrted. In Figure 4, k = / hs een hosen ording to the liner Arrngement w. Confirmtion o f the ondition of ruled surfes In order to prove this importnt hrteristi of our surfes defined ording to eqution (7), its Crtesin eqution shll hve to e expressed through its prmetri form: x m( ) s( ) y n( ) q( ) s demonstrted in Appendix III. Estlishing the reltion etween the eqution: (8) x y ( ) nd the fundmentl trigonometri form: sin os it is evident tht: x sin y os = (+ ) x sin y = os + sin + os (9) nd, identifying the equtions (8) nd (9), it n e proved tht the prmenter n e esily removed to led to eqution (7), quod ert demos rndum. t Exmple of using the Arrngement w As hs een pointed out in the preeding setion, depending on the reltionship estlished etween the prmeter k tht defines the set of ross-setion urves elonging to the fmily of ellipses nd the vrile, n infinity of surfes will result. A speil se of these surfes ours when the reltionship etween nd k is liner, resulting in ruled surfe, s ws previously demonstrted. This prolem ould e resolved y finding the eqution of the ruled surfe nd its generting stright lines in order to develop over etween stright ridge nd n ellipti se, given: Semimjor xis A nd semiminor xis of the ellipti se. NEXUS NETWOK JOUNA VO. 3, NO., 00 49

8 50 NEXUS NETWOK JOUNA VO. 3, NO., 00 Figure 4

9 Height H of the stright ridge. ength of the stright ridge (ligned with A). The generl eqution of ll generted ellipses y the genertor ellipse with semixes nd will e s lredy stted, ording to eqution (6): x y k ( ) k In the prtiulr se for the se: (0) k A () ( k) ko = () For the stright ridge, the orresponding ellipse would hve semixis / nd 0, for k= (3) tking the enter of the genertor ellipse s the origin of the oordintes, nd onsidering it loted etween the stright ridge nd the se t distne elow the stright ridge. The liner reltion etween k nd n e set forth when it is known tht in the stright ridge = for k=, nd in the genertor ellipse =0 for k=0: k whih for the oordinte H H k (4) From () nd (3), A k (5) From () nd (5), ( / ) (6) A / NEXUS NETWOK JOUNA VO. 3, NO., 00 5

10 From (3) nd (6), (A / ) (7) From (4) nd (5), H (8) A / With vlues, nd the resulting surfe eqution f(x,y,)= is: x y Another interesting vlue ould e the distne I etween the two stright lines (ellipses with null semixis): I (0) The two irumferenes nswer to the ondition tht their semixes A nd re equl (A =), for vlues of k suh tht k= / nd k= /. All of wht hs een lulted is shown in Figure 4. In order to otin the stright lines (generting lines ording to the onept of ruled surfes), it is suffiient to tke severl points of the genertor ellipse nd pply to them the vlues k= (stright ridge) nd k = ko (se irumferene). First, we shll hve to otin the eqution of the norml stright line t the seleted point of the genertor ellipse. Seond, we shll drw the intersetion etween the norml stright line previously tred nd the trnsformed ellipses (k= nd k = ko), otining in this wy the oordintes x,y of the end of the stright line. The oordinte stnds for the distne with respet to the genertor ellipse. Geometril nd Mthemtil Developmen t of the Projet The projet of the lternte version of the io de Jneiro Cthedrl in rsil hs een developed s n exmple of diret pplition of the new property of onis to rhiteture, with the prtiulr requirement tht the design shll inlude ross t its stright ridge s n integrl prt of suh onstrution. Two ptterns of rosses hve een set out in this pper, the tin Immis ross nd the Greek ross. The Greek ross hs two rms of equl length tht ross eh other t right ngles, the point of intersetion eing t the enter points of the rms. The tin Immis ross hs rms of unequl length tht ross eh other t right ngles; the point of intersetion, s in the Greek ross, is t the enter points of the rms. This is different from the more ommon tin ross, in whih the point of intersetion does not oinide with the midpoint of the long rm. (9) 5 NEXUS NETWOK JOUNA VO. 3, NO., 00

11 Figure 5 NEXUS NETWOK JOUNA VO. 3, NO., 00 53

12 The ft tht eh ross is omposed of two rms perpendiulr to eh other, irrespetive of their lengths, mde us think out the intersetion of two surfes hving stright ridges tht were rotted y 90. On the other hnd, sine the se to e linked with the stright ridge onsists of irumferene (in this prtiulr se, of n ellipse), the only possiility we hve, judging y ll tht hs een delt with in the preeding setions, is the employment of the fmily of ellipses E(M) expressed y y eqution (6), whih is repeted here: x y E M () k k Endowing the former expression with liner Arrngement w of the form k = /, where represents the height etween the ridge plne nd the plne of the genertor ellipse, we otin the following expression oth in Crtesin nd prmetri forms: x y x sin t y ost where nd t re the two prmeters of the previous expression, nd the prmeter stnds for the height of the ross-setion urves with respet to the plne of height ero, or plne where the genertor ellipse is loted. The prmeter t stnds for the ngle formed y the vertil plne holding the generting line for onstnt vlue of t with the vertil plne x = 0. A etter exposition for the prmetri representtion of surfe, is detiled in Appendix II. () Figure 6 As ws mentioned efore, surfe tht links the stright ridge to the irulr se will hve to e otined, nd it n esily e seen tht suh surfe omes out s the result of extrting it from the generl one (see the enirled detil of Figure 5), thus otining the elementl surfe tht will e used for the development of the thedrl. To use our own terminology, strting with 54 NEXUS NETWOK JOUNA VO. 3, NO., 00

13 the generl surfe shown in Figure 5, lled elementl surfe, nd rotting it y 90 we otin elementl surfe, shown in Figure 6. The positioning of these figures is suh tht elementl surfe will orrespond to the surfe hving oth the mjor xis of the genertor ellipse nd its stright stright ridge re prllel to the x- xis. Therefore, oth the mjor xis of the genertor ellipse nd the stright ridge of elementl surfe re prllel to the y-xis. From wht hs een disussed up to this point, the projet will onsist of the interpenetrtion etween the two surfes rotted 90 with respet to eh other, or in othe words, ll the rosssetion or diretor ellipses of the two elementl surfes utting eh other perpendiulrly. The following dt re known: dius of the se irle; Height H etween the ridge plne nd the se one; engths nd of the stright ridges of elementl surfes nd respetively. The eqution from whih we egin the further mthemtil development of the projet is the given y eqution () in its Crtesin form, where the liner reltionship is given y k= /, with eing the distne etween the genertor ellipse loted in the plne = 0 (k = 0), nd the stright ridge (k = ). The next step onsists in lulting the intersetion etween eh pir of perpendiulr diretor ellipses tht re loted in the sme horiontl plne for every vlue of the prmeter, yielding the four intersetion points s solution of the eqution system orresponding to the two elementl surfes, in suh wy tht y hieving the sme proess for the entire set of the hosen vlues for the prmeter, nd further joining the homologous intersetion points, we rrive t the four intersetion urves. The intent of the preeding onsidertions is to uild three point rrys, for one to store the oordintes (x, y, ) of the set of points elonging to the finl surfe, the rows of suh rrys hrteried y the t prmeter nd the olumns hrteried y the prmeter. The quest for the vlues of the semixes nd of the genertor ellipse, nd of the prmeter, in terms of the known dt (se irle rdius), H (height etween the ridge plne nd the se one) nd (length of the stright ridge), led us to the results tht were otined in the previous setion where we disussed how to del with the Arrngement w. In expressions (3), the vlues A nd refer to the vlues of the semi-mjor nd semi-minor xes of the ellipse (in this se to the rdius of the se irle) loted with respet to the plne of the stright ridge t distne H, eing the length of the stright ridge. NEXUS NETWOK JOUNA VO. 3, NO., 00 55

14 A H A A (3) The setions tht follow onsist of prtiulriing the forementioned vlues of, nd for eh of the two thedrl types in terms of whether the tin Immis ross or the Greek ross is hosen for the ridge pln, long with the desription of the interpenetrtion proess etween the two elementl surfes nd in order to generte the eventul resultnt surfes for the designs. The projet for eh type of ross is illustrted. Projet of thedrl sed on the tin Immis ross Defining the tin Inmis ross s tht whose lims re unequl nd entered t point loted upon the vertil -xis, we will designte s the length of the minor rm of the ross prllel to the x- xis nd elonging to elementl surfe. We will designte s the length of the mjor rm of the ross prllel to the y-xis nd elonging to elementl surfe, suh tht the vlues of the semixes nd the reltive lotions of every one of the elementl surfes will e determined ording to the expressions (3). The expression for elementl surfe is dedued from eqution () nd given y: t os t os y sin sin t x t A A (4) The expression for elementl surfe is dedued from the eqution () y rottion of 90, nd is given y: 56 NEXUS NETWOK JOUNA VO. 3, NO., 00

15 sin sin y t t x A (5) Additionlly, we wnt to tke into ount the following reltionships: 0 A A (6) where the vlues,, nd,,, for elementl surfes nd respetively, re given y the following expressions otined from (3) y simply mking A== nd identifying y nd, depending on the elementl surfe to whih it refers ) H ( H ; ) ( ; ) H ( H ; ) ( ; (7) It is interesting to mention s well, s n e proved y mking the lultions, tht: H, whih justifies the ft tht yields to = = 0. Eventully it n e verified tht the sum of the semixes nd, s well s nd from the genertor ellipses orrespond to the vlues of the rdius of the se irumferene, whih is given dtum. We wnt to emphsie the ft tht the genertor ellipses of the elementl surfes does not lie in the sme plne (Figure 7), hene the mthemtil expression for one of the two surfes will NEXUS NETWOK JOUNA VO. 3, NO., 00 57

16 hve to e referred to in terms of the other. Therefore the vlues of the prmeter of the seond surfe with respet to the plne =0 of the genertor ellipse of the first surfe, ording to = ( ), reltionship tht ppers expliity in the vlue A. When the system mde up of equtions (4) nd (5) is resolved, the result yields the loi of the spe points ommon to the two surfes from whih the eventul thedrl is formed, loi tht will lso e the outome of the interpenetrtion proess. In onsequene of wht hs een disussed ove, the equtions tht define the nonplnr urves tht result from the proess of interpenetrtion for the se of tin Inmis ross re given thus: x y A 0 A 0 0 A 4 0 A A 0 A 4 0 A A (8) Figure 7 The proess of interpenetrtion onsists, then, of first otining the four possile points of intersetion generted y eh of the ellipses elonging to elementl surfes nd situted on 58 NEXUS NETWOK JOUNA VO. 3, NO., 00

17 Figure 8 the sme plne defined y prmeter. The next step is to eliminte the ones or strethes of the diretion urves tht re oneled (Figures 7 nd 8). This gives the utilile enirling surfe of the finl resultnt surfe. The meeting of ll homologous points of intersetion will define the four interseting urves etween the two elementl surfes (Figures 7, 8, 9). For the next phse, the strting point will e the previous lultion proess, whih onsisted of seleting determined numer of diretion urves tht re ommon to the two surfes of the thedrl, lulting their respetive dimensions with referene to the plne ontining the generting urve nd whih will e determined y prmeter. This will lso supply the height of eh diretion plne with respet to the plne of the se irumferene, the vlue of whih is given y h = H +. Then suitle nd suffiient numer of generting stright lines is determined using prmeter t, defined s the ngle formed y the vertil plne tht ontins the diretion stright line with the plne x = 0. Vlues of t in the diretion of the Y xis towrds the X xis re tken s positive, tht is to sy, lokwise in Crtesin referene system. NEXUS NETWOK JOUNA VO. 3, NO., 00 59

18 Figure 9 The next step is to determine ll the points t whih eh previously seleted generting stright line uts eh nd every one of the diretion urves lso seleted in ordne with wht ws disussed in the prgrph ove. Hving otined ll the points of intersetion, point mtrix is reted for eh of the x, y, oordintes defined s [X], [Y] nd [Z]. The olumns for ll the previous mtries will reflet the order numer orresponding to the diretion plne, eginning with the ridge plne nd moving s fr s the se plne. The rows, on the other hnd, represent the order numer of the generting stright lines eginning with the vlue t = 0 nd finishing with t = in lokwise diretion. Hene eh ple within the orresponding mtrix defined y its row nd olumn will orrespond to the vlue of the x oordinte for the [X] mtrix, y for [Y] nd for the orresponding [Z] mtrix. From ll the points otined ove, those whih re inside the resultnt surfe will hve to e disregrded using the proess of interpenetrtion. It is dvisle for the previously seleted genertors to inlude those tht pss through the points of intersetion of eh pir of diretion ellipses situted in the plnes whose prmeter vlues were lso seleted. From the point of view of onstrution, this thedrl projet ould e omplished in two wys. One would involve the tehnique of reinfored onrete shells [Cndel 985; Fer 970] for spns tht re not too lrge, while the other would involve the use of metl strutures for lrge spns. In either of these methods, one importnt feture is the sene of intermedite pillrs, 60 NEXUS NETWOK JOUNA VO. 3, NO., 00

19 Figure 0 NEXUS NETWOK JOUNA VO. 3, NO., 00 6

20 Figure llowing ompletely open nd diphnous spes. In those strutures where the use of lminr reinfored onrete strutures involves high uilding osts due in prt to n exessive inrese in thikness of the onrete lyers, nd where memrne stress lultions re no longer relile for this type of struture, it is preferle to resort to metl strutures involving the use of stronger, lighter nd more resistnt steels mking it possile to hieve spns of up to 300 meters without intermedite supports. Hving otined the point mtries, in the onstrution proess using the tehnique of reinfored onrete lminr strutures, the phse of ereting the wooden formwork would involve rising wooden pillrs of length h for eh point Q(x,y,) projeted on the se irumferene plne. The oordintes re stored in the orresponding mtries mentioned ove, llowing the ssemly of wooden strips or ords in pillr form. The ends of the pillrs re susequently joined, giving rise to the generting stright lines suh tht in ertin setions of the thedrl they will go from the stright ridge to the se irumferene, nd in others from the stright ridge to the orresponding point on the urve of intersetion, etween the two surfes, therey forming the thedrl surfe. After the formwork, the neessry reinforing rs re positioned nd the onrete poured in to form the reinfored onrete. One the onrete hs set, the formwork is removed, reveling urved onrete shell of ertin thikness ple of providing dequte rigidity nd strength to withstnd nd sor the stresses generted. The finl surfe 6 NEXUS NETWOK JOUNA VO. 3, NO., 00

21 Figure Figure 3 NEXUS NETWOK JOUNA VO. 3, NO., 00 63

22 Figure 4 will rete totlly open interior spe with no internl olumns to tke wy light nd spe, one one of the merits of this type of onstrution, devised y Spnish rhitet Félix Cndel [Cndel 985; Fer 970]. Designed with re, these surfes would llow the retion of openings to ommodte rose windows nd other lrge pertures to let light enter. Gling these openings with suitly olored glss, in onjuntion with design of the top prt of the spe s trnsprent ross would projet n extrordinry hlo of light onto the ltr nd inside the thedrl. The result of the thedrl projet with the tin Immis ross is shown in Figure 0 nd Figure, whih tkes ount of the dt = 53 metres, H = 80 metres (whih orresponds to the min dimensions of the tul ío de Jneiro thedrl), s well s the rms of the ross with vlues of = 40 meters nd = 50 meters ( golden proportions). Projet for the thedrl sed on the Greek ross The Greek ross is prtiulr se of the tin Immis ross where the rms hve the sme length equl to. The orresponding expressions given in (7) now eome those given in (9). It n e esily verified, s with the tin Immis ross, tht the sum of the semixis nd of the genertor ellipse orresponds to the rdius of the se irumferene whih is given. 64 NEXUS NETWOK JOUNA VO. 3, NO., 00

23 4 ) H ( H k H 4 ) ( 4 (9) In the disussion up to this point, we hve worked out the vlues of,, ording to the equtions of the previous se, sine the Greek ross is prtiulr se of the tin Immis ross. The orresponding equtions will e s follows: For elementl surfe, the stright ridge of whih lies prllel to the X xis, we hve: os os sin sin t t y t A t x A (30) For elementl surfe, the stright ridge of whih lies prllel to the Y xis, nd turned 90 with respet to elementl surfe, we hve: os os sin sin t A t y t t x A A (3) In the previous expressions it n e oserved tht A =A =A 0 nd = = 0, these oeffiients dependent on the prmeter. Thus, the system formed with the four previous equtions will led to the loi of points elonging to the plne urves in ordne with the solution (3) shown elow, suh tht the intersetion of suh plne urves y whtever plne perpendiulr to the Z xis gives rise to the four ommon points of the two ellipses loted in the sme utting plne. NEXUS NETWOK JOUNA VO. 3, NO., 00 65

24 Figure 5 Figure 6 66 NEXUS NETWOK JOUNA VO. 3, NO., 00

25 x y A0 0 A0 0 x y A0 0 A0 0 (3) In Figures, 3 nd 4, the proess of interpenetrtion disussed in the previous setions is illustrted shemtilly for the Greek ross, showing the prllel plnes of the se nd of the stright ridge, etween whih the surfe is formed, s well s the plnes elonging to elementl surfes nd. In this se the two genertor ellipses lie in the sme plne, sine the rms of the ross re of the sme length s well s the heights of nd from eh respetive rm to the genertor ellipse plne ( = 0). These figures lso illustrte the four intersetion points generted y the two genertor ellipses, through whih the four intersetion urves pss, s result of the proess of interpenetrtion etween the two surfes. The finl result of the Greek ross projet is illustrted in Figures 5 nd 6, where the given prmeters were gin = 53 meters nd H = 80, with the length of the rms of the ross = =30 metres. The rhiteturl onlusion In ddition to the high level of originlity of these design projets from struturl nd mthemtil point of view, demonstrted in the ft tht the light generted in the ridge plne is n integrl prt while the stright ridge itself forms the roof nd support, the symoli religious hrter suggested y these surfes is unquestionle nd worthy of note. This religious hrter is expressed y the wy the upper level in the form of ross seems to emre the whole spe enlosed y the surfe of the thedrl developed here, thus spiring to join together the upper, divine, level, with the lower, terrestril, one. Appendix I. Definition of the oni ellipse The ellipse s oni setion n e onsidered s generted y the utting of one y plne tht uts ll its generting lines, nd it is therefore defined s the loi of points of the plne whose sum of distnes etween two points is onstnt. Figure I. NEXUS NETWOK JOUNA VO. 3, NO., 00 67

26 Tking into ount the ellipse illustrted in the Figure I., the following points re defined. The fixed points F, F re lled the foi of the ellipse; The distne FF is lled fol length; The distnes F P nd FP re lled rdius vetors; The sum PF PF' is onstnt vlue, s it n e demonstrted ording to the definition of the ellipse itself ; The differene will e lled nd will e positive, sine ; The line segment distne AA' will e lled the mjor xis of the ellipse; The line segment distne ' will e lled the minor xis of the ellipse. Tking line segment middle of the line segment ove, it omes to: FF ' s the horiontl xis nd the perpendiulr to the point in the FF ' s the vertil xis, nd from the definition of the ellipse given x - y x y PF PF' y operting upon tht eqution nd tking into ount the reltion following expression is otined for the nonil eqution of the ellipse:, the x y Appendix II. Surfes nd urves on surfe [Grnero odrigue 985] In the ffine three-spe, the term urve pplies to the entire olletion C 3 of three oordintes (x (t ), y (t ), (t )), suh tht x (t ), y (t ), (t ) re ontinuous funtions t ertin intervl I. Hene, the equtions: x x( t) y y( t) ( t) (I.) (II.) re lled the prmetri equtions of the urve C (tridimensionl line). Considering the existing isomorphism etween the free vetors of spe nd the points of E 3, it is often more suitle to define the urve C y free vetor v ( t ) of omponents (x (t ), y (t ), (t )). In this se it is ler tht y hnging the prmeter t, the vetor end will drw the urve C (Figure II.). 68 NEXUS NETWOK JOUNA VO. 3, NO., 00

27 Figure II. The eqution v( t ) x( t )ux y( t )u ( t ) u is lled the vetoril eqution of C. y When the urve is loted in plne is lled plne urve. Otherwise it is lled nonplne urve. In the ffine three-spe the term surfe pplies to the entire olletion of points S 3 of oordintes (x(,), y(,), (,)), suh tht x(,), y(,), (,) re ontinuous funtions in ertin domin D. The equtions x x(, ) y y(, ) (, ) (II.) re lled prmetri equtions of the surfe S. In the sme wy, sed on the isomorphism etween free vetors nd points, we would e le to define the surfe S y free vetor or position vetor v (, ) of omponents (x(,), y(,), (,)). In these onditions, y vrying the prmeters nd, the position vetor end will trvel through the surfe S (Figure II.). Figure II. The eqution v (, ) x(, ) ux y(, ) uy (, ) u NEXUS NETWOK JOUNA VO. 3, NO., 00 69

28 will e lled the vetoril eqution of suh surfe. Often it is fesile to remove the two prmeters nd mong the three equtions (II.), giving rise to n eqution of the form F(x,y,) = 0, or = f(x,y), whih is lled the Crtesin eqution of the surfe. eiprolly if the surfe is defined y = f(x,y), mking x =, y = yields the following prmetri equtions x y f (, ) (II.3) ikewise, let us notie tht if whtever reltionship, s for instne = f(), etween oth prmeters is estlished, the position vetor v (, ) v f ( ), will depend solely on one of those prmeters, nd onsequentely, urve upon the surfe will e drwn. Agin, from wht hs preeded, the result will e tht when one of the two prmeters is fixed, or when these re ( t) under the reltionship = f(), or, we will hve urve on the surfe. ( t) Exmple: Given the surfe S defined in prmetris s: x S y y isolting nd in terms of x nd y etween the two first equtions, whih re liner, y sustituting in the third we will hve = f(x,y). However in this se we n very esily otin = f(x,y) y just notiing tht: where x y x y Therefore, 6 x y or x y 6 0 represents the Crtesin eqution of the surfe. It is very simple to disply the preeding surfe, sine ll its intersetion through plnes = k prllel to the horiontl re irumferenes with rdii tht eome uniformly shorter until eoming null t point (0,0,6). ikewise, let us notie tht: x S y 6, 70 NEXUS NETWOK JOUNA VO. 3, NO., 00

29 re less diffiult prmetri equtions of this surfe thn the preeding ones. Appendix III. uled surfes [Grnero odrigue 985]. It is well known from nlyti geometry tht every stright line in spe n e expressed in the form: x = m + (III.) y = n + where m,n,,. It is ler tht when the oeffiients m,n,, re fixed, the system (III.) will represent sole stright line in spe; in other words, to every qudruple set of (m,n,,) 4 will orrespond sole stright line of suh spe. If we mke these four oeffiients e in terms of prmeter, then for every vlue of we will hve well-defined stright line suh tht y vrying this prmeter the result will e loi of stright lines, tht is, in surfe mde up of stright lines. A ruled surfe is tht whih is generted y stright line (generting line) whih moves ording to given mthemtil lw. Figure III. et e (III.) vrile stright line whose oeffiients m,n,, depend on prmeter. In these onditions we will hve the equtions: x m( ) ( ) (III.) y n( ) ( ) representing the prmetri equtions of surfe (two equtions with sole prmeter whose isoltion will give rise to the Crtesin form). Sine the vetoril eqution of the surfe is defined y mens of the position vetor of given point P(x,y,) (Figure III.), we hve: v( ) x i + y j + k where: v( ) [m( ) + ( ) ] i + [n( ) + ( )] j + k will e its vetoril eqution. Hene we will e le to sy: NEXUS NETWOK JOUNA VO. 3, NO., 00 7

30 x m( ) ( ) y n( ) ( ) = (III.3) re s well the prmetri equtions of tht surfe. et us tke notie tht lst eqution of the preeding system n e verified y every point of spe nd in onsequene = will e ll 3. It is ler tht prmetri equtions (III.) nd (III.3) represent the sme surfe, sine the system (III.3) n e onsidered s intersetion of the surfe defined s the system (III,) with =, tht is, with 3. Furthermore, tking into ount tht Appendix II demonstrted the prmetri expression of the surfes through three equtions with two prmeters nd, in this pper we hve hoosen the system (III.3) to hieve it, nd y onveniene t nd hve een seleted s prmeters. First pulished in the NNJ online Spring 00 eferenes CANDEA, Félix En defens del formlismo y otros esritos. ilo: Ediiones Xrit. FAE, Colin s estruturs de Cndel. Mexio: CECSA. GANEO-ODÍGUEZ, Frniso Álger y geometrí nlíti. Mdrid nd Mexio City: MGrw- Hill. MATÍN-ZOAQUINO, Jun V., Frniso Grnero-odrígue, José uis Cno-Mrtín, J.J. Dori-Irirte, J.J A Novel Version of the Cthedrl Inspired in the Alredy uilt one in io de Jneiro. Pp in ISAMA 99. Nthniel Friedmn nd Jvier rllo, eds. Proeedings of the Interntionl Soiety of the Arts, Mthemtis nd Arhiteture (ISAMA) 99 onferene, 7- June 999, Sn Sestián. MATÍN-ZOAQUINO, J.V.. Frniso Grnero-odrígue, J.J.Dori-Irirte, J.J "Otrs Aportiones y esultdos sore l Nuev Fmili de Figurs Geométris Elementles on Cónis". 9º Interntionl Congress of Grphi Engineering. ilo - S. Sestián: UPV-EHU. MATÍN-ZOAQUINO, J.V., Frniso Grnero-odrígue, J.J. Dori-Irirte "New Properties of Coni Setions nd the uled Surfes Deriving from them. Seleted Applitions in Arhiteture nd Engineering". Pp in Mthemtis & Design 98. J. rrllo, ed. Proeedings of the Seond Interntionl Conferene of Mthemtis & Design 98. Sestián: UPV-EHU. MATÍN-ZOAQUINO, J.V., Frniso Grnero-odrígue nd J.J. Dori-Irirte "Superfiies Tridimensionles de Seiones Cónis." Ptent numer P93066/993. The Authors Prof. Dr. Mrtín Zorrquino is n industril engineer nd Professor in Thermi Engineering t the High Tehnil Shool of Engineering in ilo (Spin) holding his professor s hir sine 966. He hs een diretor of the Het Engines Deprtment t the sme university sine 987. During the lst yers, he hs run ten reserh projets finned y the sque Country University nd the sque Government itself. He hs written severl reserh rtiles Dr. Grnero odrigue is n industril engineer nd eturer in Applied Mthemtis t the High Tehnil Shool of Engineering in ilo. He hs written severl reserh workings nd severl ooks on mthemtis, five of them pulished y the pulishing house MGrw-Hill. Dr. Cno Mrtín is n industril engineer. He works s n engineer for lrge firm loted in the sque Country. 7 NEXUS NETWOK JOUNA VO. 3, NO., 00