Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the rnhes of the hperol, nd whih is direted towrd one fol point of the hperol will e refleted towrd the other fol point when it strikes the rnh of the hperol towrd whih it ws direted. Gerr Del Fio Mth Center Metropolitn Stte Universit St. Pul, Minnesot Jul 6, 9 Definition of the Hperol The reder of this pper is presumed to e fmilir with the definition of hperol nd with the derivtion of the eqution nd orresponding grph of the hperol. The following is psule summr of this definition nd its interprettion. A hperol is the lous of points for whih the solute vlue of the differene of the respetive distnes from n point on the hperol to two fied fol points is onstnt. The grph of suh lous of points, tht is, the grph of hperol, hs the following pperne: Refletion Propert of Hperol Pge
P Fol Point Fol Point If P is n ritrr point on either rnh of the hperol, the solute vlue of the distne from one fol point to P minus the distne from the other fol point to P is onstnt, regrdless of the point P tht is seleted on either rnh of the hperol. In the emple tht we re depiting in this pper, we ssume without loss of generlit tht the eqution of the hperol is: where > nd > This ssumption results in the fol points -, ) nd, ) eing loted on the -is where: tht is Furthermore, the onstnt solute vlue of the differene of the respetive distnes etween n point P on either rnh of the hperol nd the two respetive fol points is equl to. The following is the grph of the hperol with its smptotes nd with the oordintes of the -interepts nd the fol points identified: Refletion Propert of Hperol Pge
, ) -, ), ) -, ), ) Fol Point, -) Fol Point Definition of the Refletion Propert of the Hperol The following digrm depits the definition of the refletion propert of the hperol: λ Tngent Line λ P Fol Point Fol Point Refletion Propert of Hperol Pge 3
3 If P is n ritrr point on n rnh of the hperol, n ngle λ is formed the intersetion of the tngent line t P nd the line from one fol point to the point P. And, n ngle λ of equl mesure is formed the intersetion of the tngent line t P nd the line from the other fol point through the point P. Proof of the Refletion Propert of the Hperol In order to provide n nltil proof of the refletion propert of the hperol, we will mke use of the following digrm: β, ) 4 θ θ -, ), ) β In order to prove the refletion propert, we need to demonstrte tht 3 4. But, we know tht 3 euse these re vertil ngles. And, we know tht 4 euse these re vertil ngles. Therefore, our ojetive in the following proof will e to demonstrte tht. We will omplish this proving tht ) ). We egin otining the slope of the tngent line t the point, ) performing impliit differentition of the eqution of the hperol: Refletion Propert of Hperol Pge 4
d d d d Therefore, the slope of the tngent line t the point, ) is: β ) We lso will need β ) whih is determined s follows: β ) β ) Net, we otin the slope of the line from the point -, ) to the point, ) : θ ) Net, we otin the slope of the line from the point, ) to the point, ) : θ ) Sine we will need θ ), we mke the following djustment: θ ) θ ) Now, we proeed to otin ) s follows: β θ Refletion Propert of Hperol Pge 5
Refletion Propert of Hperol Pge 6 Therefore: θ β θ β θ β This epression for n e simplified lgerill mking use of the ft tht: Therefore: And, we lso will mke use of the ft tht: Proeeding with the lgeri simplifition, we hve the following:
Refletion Propert of Hperol Pge 7 In summr: Net, we proeed to otin s follows: θ β Therefore: θ β θ β θ β In summr: Thus, we hve demonstrted tht: nd Tht is, we hve proven tht:
Therefore,. This ompletes the nltil proof of the refletion propert of the hperol. Geometri Eplntion of the Refletion Propert of the Hperol In the preeding pper titled Refletion Propert of n Ellipse, the refletion propert of the ellipse ws eplined in geometri terms using the following digrm: P F M D R E There is one ellipse whih hs the points D nd E s its fol points nd whih psses through the point P. Correspondingl, there is one hperol whih hs the points D nd E s its fol points nd whih psses through the point P. If we eliminte the ellipse from the ove digrm nd reple it with the orresponding hperol, we hve the following digrm: Refletion Propert of Hperol Pge 8
P F M D R E In the first of these two digrms ove, the line through points P nd M is tngent to the ellipse nd the line through points R nd P is perpendiulr to the tngent line. In the seond of these two digrms, the respetive roles of these two lines re reversed. Tht is, in the seond of these two digrms ove, the line through points R nd P is tngent to the hperol nd the line through points P nd M is perpendiulr to the tngent line. Refer to Appendi A for supplementr detil.) In either of these two digrms ove, the tringle EPF is n isoseles tringle, nd the line segment from point P to point M is the isetor of ngle EPF. We mde use of this disover in the geometri eplntion of the refletion propert of the ellipse nd in the orollr to the refletion propert of the ellipse. Refletion Propert of Hperol Pge 9
In the orollr to the refletion propert of the ellipse, we determined tht the following ngles re equl: φ φ P Ф Ф Ф 3 F M D R E Furthermore, in this digrm ove, ngles φ nd φ 3 re equl euse the re vertil ngles. Therefore: φ φ 3 This ompletes the geometri eplntion of the refletion propert of the hperol. Applition of the Refletion Propert of the Hperol The refletion propert of the hperol n e used in vrious pplitions, suh s the design of high-preision optil equipment. The following is n emple of the design of high-preision telesope whih mkes use of oth hperoli surfe nd proli surfe within the interior of the telesope. This devie is known s Cssegrin refleting telesope whose design ws ttriuted to Lurent Cssegrin, 7th entur siene teher in Frne. Refletion Propert of Hperol Pge
The following digrm is two-dimensionl ross-setion of Cssegrin telesope, whih oviousl is three-dimensionl ojet. In the design of this telesope, onve proli mirror forms the k of the telesope. An inoming light r strikes the onve surfe of the proli mirror nd is refleted towrd the onve surfe of hperoli mirror: Inoming Light R Proli Mirror Fol Point of Prol nd Hperol Hperoli Mirror Other Fol Point of Hperol The left-most fol point of the hperol is the sme point s the fol point of the prol. When the light r strikes the onve surfe of the hperol, it is redireted towrd the other fol point of the hperol whih resides t the eepiee of the telesope. Referenes http://www.mthdem.om/pr/prime/rtiles/onis/inde.sp Refletion Propert of Hperol Pge
Refletion Propert of Hperol Pge Appendi A This ppendi provides supplementr detil for the geometri eplntion of the refletion propert of the hperol. Refer to the ssoited digrms on pges 8 nd 9 of this pper.) The geometri eplntion mde use of n ellipse nd hperol whih shred the sme fol points nd whose respetive tngent lines t ommon point of intersetion were perpendiulr to eh other. Assume tht the eqution of the ellipse is s follows: > > > nd where nd where Assume tht this ellipse psses through the point ),. Tht is: Assume tht the fol points of this ellipse re -, ) nd, ). Tht is: The hperol whih psses through the point ), nd whih hs its fol points t the points -, ) nd, ) hs n eqution of the following form: B nd where where A B A To prove this ssertion, we first demonstrte tht the point ), lies on the hperol whih hs this given eqution ove: B A
Seondl, we demonstrte tht A B. This will verif tht the hperol whih hs the given eqution ove hs the points -, ) nd, ) s its fol points: A B Finll, we will demonstrte tht the line whih is tngent to the ellipse t the point, ) is perpendiulr to the line whih is tngent to the hperol t the point, ). We will omplish this demonstrting tht the negtive reiprol of the slope of the line whih is tngent to the ellipse t the point, ) equls the slope of the line whih is tngent to the hperol t the point, ). Using impliit differentition of the eqution of the ellipse, the slope of the line whih is tngent to the ellipse t the point, ) ws previousl determined to e: The negtive reiprol of this slope is: Using impliit differentition of the eqution of the hperol, the slope of the line whih is tngent to the hperol t the point, ) ws previousl determined to e: B A Refletion Propert of Hperol Pge 3
Refletion Propert of Hperol Pge 4 But, we re given the following: B nd A Therefore: A B A B This ompletes the demonstrtion tht the line whih is tngent to the ellipse t the point ), is perpendiulr to the line whih is tngent to the hperol t the point ),.