Retroreflecting Curves in Nonstandard Analysis

Retroreflecting Curves in Nonstandard Analysis

We present a direct construction of retroreflecting curves by means of Nonstandard Analysis. We construct non self-intersecting curves which are of class C¹, except for a hyper-finite set of values, such that the probability of a particle being reflected from the curve with the velocity opposite to...

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Datum:2009
Hauptverfasser: Almeida, R., Neves, V., Plakhov, A.
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Sprache:English
Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2009
Schriftenreihe:Журнал математической физики, анализа, геометрии
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spelling irk-123456789-1065302016-10-01T03:01:44Z Retroreflecting Curves in Nonstandard Analysis Almeida, R. Neves, V. Plakhov, A. We present a direct construction of retroreflecting curves by means of Nonstandard Analysis. We construct non self-intersecting curves which are of class C¹, except for a hyper-finite set of values, such that the probability of a particle being reflected from the curve with the velocity opposite to the velocity of incidence, is infinitely close to 1. The constructed curves are of two kinds: a curve infinitely close to a straight line and a curve infinitely close to the boundary of a bounded convex set. We shall see that the latter curve is a solution of the problem: find the curve of maximum resistance in nitely close to a given curve. 2009 Article Retroreflecting Curves in Nonstandard Analysis / R. Almeida, V. Neves, A. Plakhov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 12-24. — Бібліогр.: 7 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106530 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description We present a direct construction of retroreflecting curves by means of Nonstandard Analysis. We construct non self-intersecting curves which are of class C¹, except for a hyper-finite set of values, such that the probability of a particle being reflected from the curve with the velocity opposite to the velocity of incidence, is infinitely close to 1. The constructed curves are of two kinds: a curve infinitely close to a straight line and a curve infinitely close to the boundary of a bounded convex set. We shall see that the latter curve is a solution of the problem: find the curve of maximum resistance in nitely close to a given curve.
format Article
author Almeida, R.
Neves, V.
Plakhov, A.
spellingShingle Almeida, R.
Neves, V.
Plakhov, A.
Retroreflecting Curves in Nonstandard Analysis
Журнал математической физики, анализа, геометрии
author_facet Almeida, R.
Neves, V.
Plakhov, A.
author_sort Almeida, R.
title Retroreflecting Curves in Nonstandard Analysis
title_short Retroreflecting Curves in Nonstandard Analysis
title_full Retroreflecting Curves in Nonstandard Analysis
title_fullStr Retroreflecting Curves in Nonstandard Analysis
title_full_unstemmed Retroreflecting Curves in Nonstandard Analysis
title_sort retroreflecting curves in nonstandard analysis
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2009
url http://dspace.nbuv.gov.ua/handle/123456789/106530
citation_txt Retroreflecting Curves in Nonstandard Analysis / R. Almeida, V. Neves, A. Plakhov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 12-24. — Бібліогр.: 7 назв. — англ.
series Журнал математической физики, анализа, геометрии
work_keys_str_mv AT almeidar retroreflectingcurvesinnonstandardanalysis
AT nevesv retroreflectingcurvesinnonstandardanalysis
AT plakhova retroreflectingcurvesinnonstandardanalysis
first_indexed 2025-07-07T18:36:31Z
last_indexed 2025-07-07T18:36:31Z
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fulltext Journal of Mathematical Physics, Analysis, Geometry 2009, vol. 5, No. 1, pp. 12�24 Retrore�ecting Curves in Nonstandard Analysis R. Almeida1, V. Neves1, and A. Plakhov1;2 1Department of Mathematics, University of Aveiro Campus Universit�ario de Santiago, 3810-193 Aveiro, Portugal E-mail:ricardo.almeida@ua.pt, vneves@ua.pt and a.plakhov@ua.pt 2presently visiting Institute of Mathematical and Physical Sciences University of Aberystwyth, Aberystwyth SY23 3BZ, Ceredigion, UK E-mail:axp@aber.ac.uk Received March 29, 2008 We present a direct construction of retrore�ecting curves by means of Nonstandard Analysis. We construct non self-intersecting curves which are of class C1, except for a hyper-�nite set of values, such that the probability of a particle being re�ected from the curve with the velocity opposite to the velocity of incidence, is in�nitely close to 1. The constructed curves are of two kinds: a curve in�nitely close to a straight line and a curve in�nitely close to the boundary of a bounded convex set. We shall see that the latter curve is a solution of the problem: �nd the curve of maximum resistance in�nitely close to a given curve. Key words: Nonstandard Analysis, retrore�ectors, maximum resistance problems, re�ection, billiards. Mathematics Subject Classi�cation 2000: 26E35, 49K30, 49Q10. 1. Introduction A. A retrore�ector is an optical device that sends incident beams of light back to their origin. If the retrore�ector is much smaller than the size of the source of light, it actually reverses the direction of light. We proceed to de�ne a mathematical retrore�ector. Consider a set with piecewise smooth boundary, and the billiard in the comple- ment of this set. The set is called mathematical retrore�ector (or just retrore�ec- tor), if almost all incident particles are re�ected in such a way that the velocity of re�ection is opposite to the velocity of incidence. In this paper we shall construct two-dimensional retrore�ectors by means of Nonstandard Analysis. c R. Almeida, V. Neves, and A. Plakhov, 2009 Retrore�ecting Curves in Nonstandard Analysis As far as we know, it is the �rst time that nonstandard analysis techniques are used within the framework of mathematical retrore�ectors theory. In [6], an asymptotically retrore�ecting sequence of sets was constructed. More precisely, the sets in the sequence presented in [6] are contained in one �xed bounded convex set and contain another one. �Asymptotically retrore�ecting� means that the sum of the incidence velocity and the re�ection velocity converges in measure to zero, with both the velocities being considered as functions on the (measurable) set of all incident particles. In [5], an asymptotically retrore�ecting sequence of unbounded sets was constructed, each of them containing a �xed half-plane and contained in another one. One can easily construct a partial retrore�ector; from Fig. 1, one can see that only a part of the incident particles is reversed. Fig. 1: A partial retrore�ector. B. Let us formulate the main results of the paper. First, consider a set with piecewise smooth boundary, contained in the lower half-plane, � f(x; y) j y � 0g � R 2 and de�ne the mapping (�; �) 7! � + (�; �) as follows (see Fig. 2). �Ω ξ �� � � � Fig. 2: Angle of re�ection. Consider the billiard in R 2 n . Tag billiard particles incident on by their point of the �rst intersection with the straight line y = 0 and by the velocity Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 13 R. Almeida, V. Neves, and A. Plakhov at the moment of intersection. That is, let a particle intersect the line at the point (�; 0) and let the velocity at this point be v = �(cos �; sin �); then tag this particle by (�; �) 2 R� [0; �]. The particle makes several re�ections from @ and �nally intersects the line y = 0 again and moves freely afterwards. Denote the �nal velocity by v + = (cos �+ (�; �); sin � + (�; �)). The mapping (�; �) 7! � + (�; �) is de�ned on a subset of R � [0; �]. Theorem 1. There exists such that its boundary @ is a nonsel�ntersecting curve in�nitely close to the line y = 0 and invariant with respect to the shift (x; y) 7! (x+1; y). Moreover, for all (�; �) 2 [0; 1]� [0; �], �+ (�; �)�� � 0 holds, except for a set of measure � 0. Theorem 1 means that nearly all incident particles almost reverse direction, and the re�ecting set is obtained from the half-plane by an in�nitely small modi- �cation near its boundary. C. Now �x a convex bounded set B � R 2 with nonempty interior and consider a set � � B with piecewise smooth boundary @�. De�ne the mapping (�; �) 7! � + � (�; �) in a similar way. Namely, consider the billiard in R2 n�. Let an incident particle intersect @B for the �rst time at the point � and let the velocity at this point form the angle � with the tangent to @B at �. The particle makes several re�ections from �, then intersects @B again and �nally moves freely, the �nal velocity making the angle �+� (�; �) with the tangent. The mapping � + � is de�ned on a subset of @B � [0; �]. Theorem 2. There exists a set �� such that the boundary @�� is a closed nonsel�ntersecting curve in�nitely close to @B and such that for all (�; �) 2 @B� [0; �], �+��(�; �)� � � 0 holds, except for a set of measure � 0. D. There is an application of these results in Newtonian aerodynamics. Sup- pose that a body � moves forward through a highly rare�ed medium, and at the same time slowly rotates. Due to elastic collisions between the body and the medium particles, a braking force acting on the body in the direction opposite to its motion is created. This force is called the force of aerodynamic resistance, or just resistance. The mean value of resistance is given by the formula R(�) = 3 8 Z @B �Z 0 � 1 + cos(�+� (�; �)� �) � sin � d� d�; (1) the factor 3=8 is chosen in such a way that substituting � = B one gets R(B) = j@Bj, that is, resistance of the convex set B is just its perimeter. 14 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 Retrore�ecting Curves in Nonstandard Analysis Consider the problem: maximize R(�) over all sets � � B such that @� is near @B. The solution is given, say dynamically, by the sets �� determined in Th. 2, for which R(��) � 1:5. The paper is organized as follows. Theorems 1 and 2 are proved in Sects. 3 and 4, respectively. In Section 5 the maximization problem is examined in more detail. 2. Self-Intersecting Mirrors We present a rather elementary direct approach to this problem by means of (nonstandard) In�nitesimal Calculus. As in [5], we use the basic re�ection property of the ellipse (Fig. 3): rays which hit between the foci are also re�ected between the foci. F F1 2F F1 2 � Fig. 3: Re�ection in an ellipse. In particular, if the ellipse has foci F1(�c; 0); F2(c; 0); equation x 2 a2 + y 2 b2 = 1; and eccentricity c=a � 0; then the angle of re�ection � is in�nitesimal, i.e., re�ec- tion is almost opposite to incidence. Assume light rays may have any direction whatsoever from above a line seg- ment of length 1 and �x internal sequences Mi; Ni 2 � N1 for i 2 � N (where �N1 denotes the set of in�nite hypernatural numbers). Divide the segment [0; 1] in N1 equal parts and in each of them de�ne an ellipse with the major axis on the initial segment, as shown in Fig. 4, where F2i�1;1 and F2i;1 denote the foci of the i� th ellipse, i = 1; : : : ; N1. Each of the N1 ellipses veri�es the following conditions for exactness of sub- division: Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 15 R. Almeida, V. Neves, and A. Plakhov 0 0 F F1,1 2,1F F1,1 2,1 F F3,1 4,1F F3,1 4,1 F F5,1 6,1F F5,1 6,1 F7,1F7,1 F2N -2,1 1 F2N -2,1 1 F F2N -1,1 2N ,1 1 1 F F2N -1,1 2N ,1 1 1 1(N -1)/N1 1(N -1)/N1 11/N11/N1 2/N12/N1 3/N13/N1 F F1,1 2,1F F1,1 2,1 1/k11/k1 1/k11/k12/(k M )1 12/(k M )1 1 1/N11/N1 Fig. 4: First step. k1 = 2N1 � 1 + 1 M1 � ; a1 = 1 k1 + 1 k1M1 ; b1 = 1 k1 r 1 + 2 M1 ; c1 = 1 k1M1 : Therefore the eccentricity e1 � 0 as required; but the probability P1 that a light ray falls out of the foci window is given by P1 = N1 2 k1 = M1 M1 + 1 � 1: Next de�ne new ellipses in each of the segments [(j � 1)=N1; F2j�1;1] and [F2j;1; j=N1] for j = 1; : : : ; N1. Note that both segments have the length 1=k1 and divide each of them into N2 equal parts wherein ellipses are de�ned again with foci F2i�1;2 and F2i;2, i = 1; : : : ; N2, according to the following conditions: k2 = 22N1N2 � 1 + 1 M1 �� 1 + 1 M2 � ; a2 = 1 k2 + 1 k2M2 ; b2 = 1 k2 r 1 + 2 M2 ; c2 = 1 k2M2 : The probability P2 that a light ray falls out of the foci windows is given by P2 = 2N1N2 2 k2 = � M1 M1 + 1 �� M2 M2 + 1 � : 16 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 Retrore�ecting Curves in Nonstandard Analysis Iteration of this procedure follows the pattern ki = 2i iY j=1 Nj iY j=1 � 1 + 1 Mj � ; ai = 1 ki + 1 kiMi ; bi = 1 ki r 1 + 2 Mi ; ci = 1 kiMi : Interestingly enough, whatever the sequence Ni might be Pi = iY j=1 Mj Mj + 1 : In particular, if for some �xed N 2 � N1 all the Mj = N , then PN2 = � 1� 1 N + 1 �N2 � e � N2 N+1 � 0: (2) Assume from now on that for some �xed N 2 � N1; Mj � N so that (2) holds. The possibility that a ray entering a foci window hits one of the smaller ellipses and is not re�ected conveniently must also be considered. The following discusses this situation. Consider Fig. 5, where one ellipse is centered at the origin of coordinates for simplicity. Let the light ray r pass through the window [F1;i�1F2;i�1] with inclination �. 2/(k N)i-12/(k N)i-1 F1,i-1F1,i-1 F2,i-1F2,i-1-ai-ai -bi-bi � r E Fig. 5: Avoiding inconvenient hits. As a matter of notational simpli�cation, de�ne A := 2Ki�1Ni and B := Ki�1N 2 : Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 17 R. Almeida, V. Neves, and A. Plakhov The centered ellipse is given by x 2 a2i + y 2 b2i = 1 with ai = 1 A and bi = N A(N + 1) r 1 + 2 N : An equation of the light ray is yt = tan � � x� 1 A � t B � for some t 2]0; 1[: The light ray intersects the ellipse at a point (x; yt) when � = arctan p 1� x2A2 B +At�ABx �B � p N(N + 2) N + 1 ! necessarily with 0 < x < 1=A; but then 0 < � < arctan B At � p N(N + 2) N + 1 ! = arctan N Nit � p N(N + 2) 4(N + 1) ! therefore � � 0 as long as N Nit � 0 and this happens whenever t � 1 N and Ni = N 3, thus the probability that the entering light rays hit a smaller ellipse is approximately N2 �1X j=1 2j+1 N2kj jY i=1 Ni = 2 N2 N2 �1X j=1 � N N + 1 �j = 2 N 1� � N N + 1 �N2 �1 ! � 2 N � 1� e � N2 �1 N+1 � hence in�nitesimal. Summarizing: As long as all the Mi = N and Ni = N 3 , for some N 2 � N1 , the N 2 -th step of the foregoing procedure entails a sel�ntersecting "mirror" which re�ects light rays along lines in�nitely near the incidence lines with probability in�nitely near 1. Although sel�ntersecting, our curve is �� continuous and in�nitely resistant. 18 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 Retrore�ecting Curves in Nonstandard Analysis 3. Simple Mirrors From now on we will take all the Ni = N 3. We eliminate self-intersections �indirectly� as illustrated in Fig. 6: extend the mirror in�nitesimally towards the center of each ellipse [�ci;�P ] [ [P; ci], and connect with the ellipse itself by means of two straight line segments r and r of adequate inclination �. -ci-ci � r ciciP r Fig. 6: Eliminating self-intersections. The angle � must of course be in�nitesimal, but also such that the line r, and its symmetric r, do not intersect any of the inner ellipses. Finally, having thus created more �re�ective� regions, their total length must be in�nitesimal. We now sketch calculations ci = 1 kiN ; ai+1 = 1 2kiN3 ; bi+1 = 1 2(N + 1)kiN2 r 1 + 2 N : For some positive � to be determined, the center C of the �rst inner ellipse and the end point P verify C = ci + ai+1 = 2N2 + 1 2kiN3 ; P = ci 1 + � : Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 19 R. Almeida, V. Neves, and A. Plakhov The line r and inner ellipse E satisfy r � y = tan� (x� P ) ; E � (x�C)2 a2i+1 + y 2 b2i+1 = 1: The angle � for which r is tangent to E is given by � = arctan � bi+1 ai+1 q a2i+1 � (x� C)2 x� P ; ci < x < C: Now, � � 0 whenever q a2i+1 � (x� C)2 x� P � 0; but, 0 � q a2i+1 � (x� C)2 x� P � ai+1 x� P � ai+1 ci 1 + � � � 1 N2� and � � 0 when � = 1 N . Any in�nitesimal angle � > � may be used to eliminate the self-intersection. Moreover, as ci � P = 1 kiN(N + 1) < 1 N 2 kiN ; the probability of a ray being inadequately re�ected by this procedure is in�nite- simal. Summarizing, the probability of a ray being re�ected with opposite direction of incidence is given by dPN2 � 1� � e � N2 N+1 + 2 N � 1� e � N2 �1 N+1 �� � 1: 4. Convex Mirrors As a matter of making terminology more precise, let � : �[0; 1] ! � R 2 be the curve thus de�ned in Sect. 3. When one wants to take into account the size and the position of the mirror, an a�ne transformation is in order: given distinct points P and Q in R2 , let (v1; v2) := Q� P; M := � v1 �v2 v2 v1 � ; �PQ(t) := P +M�(t); t 2 �[0; 1]; 20 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 Retrore�ecting Curves in Nonstandard Analysis �PQ describes the (simple plane) mirror positioned along �! v , which we may re-parametrize in I := [a; b]; a < b, by � I PQ(t) := �PQ � t� a b� a � ; t 2 I: (3) Suppose now that � : [0; `] � R ! R 2 is a C 1 regular curve parameterized by arc length?. Let the �re�ective side� of � be its convex side as illustrated in Fig. 7. �´´�´´ Fig. 7: Convex mirror. A mirror of almost maximum resistance adjusted to the curve may be described in the following way 1. Pick an in�nite N 2 � N1 and de�ne for 0 � j � 2N : aj := ( j=2 N ; if j is even; (j+1)=2 N � 1 N2 ; if j is odd; bj := `aj ; so that [0; `] = 2N[ j=1 [bj�1; bj ]; bj � bj�1 = ( ` N2 ; j is even; ` N � ` N2 ; j is odd; 1 � j � 2N: ?Actually it su�ces that � is recti�able so that the following general procedure may be adapted. Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 21 R. Almeida, V. Neves, and A. Plakhov 2. De�ne Pj := �(bj); 0 � j � 2N; Ij := [bj ; bj+1]; 0 � j � 2N � 1: and consider the polygon [P0; P1; : : : ; P2N ]. Also de�ne, for j 2 f0; : : : ; 2N � 1g (vide (3) above) �j(t) := 8>>><>>>: � Ij PjPj+1 (t); if t 2 Ij & j is even; Pj + N 2 ` (t� bj)(Pj+1 � Pj); if t 2 Ij & j is odd: Finally, �0 + � � � + �2N�1 is a mirror of almost maximum resistance whose standard part is �. Under in�nite magni�cation, the geometry between Pj and Pj+2 with j even is exempli�ed in Fig. 8 below. PjPj Pj+1Pj+1 Pj+2Pj+2 �/N 2 /N 2 �/N Fig. 8: Curve under in�nitesimal microscope. 5. Calculus of the Resistance We will now evaluate the resistance of the curve obtained in Sect. 3 by mini- mizing R. To do so, we must maximize the angle �+��. We assume that the light ray hits one inner ellipse between the foci, so that the direction of the re�ected ray is almost inverted (elsewhere the probability is approximately zero). Therefore the angle of re�ection � + � � is less than the angle of re�ection when a ray light hits one of the foci (and consequently the ray is re�ected to the second foci). 22 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 Retrore�ecting Curves in Nonstandard Analysis F 1 F 2 � Fig. 9: Maximizing the angle of re�ection. Let us consider the general case (the i-step) and let � be a half of the maximum angle of re�ection, as exempli�ed in Fig. 9. Therefore tan� = ci bi = 1p N(N + 2) and so cos(�+ � �) � cos 2 arctan 1p N(N + 2) ! = 1� 2 (N + 1)2 and R >� 3 8 � 2� 2 (N + 1)2 � 1Z 0 �Z 0 sin � d� d� = 3 4 � 2� 2 (N + 1)2 � � 1:5: We also remark that the maximum resistance of any curve in�nitely close to the segment [0; 1] � f0g is 1:5. Acknowledgments. The work was supported by Centre for Research on Optimization and Control (CEOC) from the �Funda�c�ao para a Ci�encia e a Tec- nologia� FCT, co�nanced by the European Community Fund FEDER/POCTI. References [1] F. Bagarello and S. Valenti, Nonstandard Analysis in Classical Physics and Quan- tum Formal Scattering. � Intern. J. Theoret. Phys. 27 (1988), 557�566. [2] F. Bagarello, Nonstandard Variational Calculus with Applications to Classical Me- chanics. 1. An Existence Criterion. � Intern. J. Theoret. Phys. 38 (1999), 1569� 1592. [3] F. Bagarello, Nonstandard Variational Calculus with Applications to Classical Me- chanics. 2. The Inverse Problem and More. � Intern. J. Theoret. Phys. 38 (1999), 1593�1615. Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 23 R. Almeida, V. Neves, and A. Plakhov [4] A.E. Hurd and P.A. Loeb, An Introduction to Nonstandard Real Analysis. Pure and Appl. Math. 118. Acad. Press, Inc., Orlando etc., 1995. [5] A.Y. Plakhov, Billiards Inverting the Direction of Particles' Motion. � Russian Math. Surveys 61 (2006), 179�180. [6] A.Y. Plakhov and P.F. Gouveia, Problems of Maximal Mean Resistance on the Plane. � Nonlinearity 20 (2007), 2271�2287. [7] K.D. Stroyan and W.A.J. Luxemburg, Introduction to the Theory of In�nitesimals. Pure and Appl. Math. 72. Acad. Press, New York, London, 1976. 24 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1

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