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On the Class of Caustics by Reflection of Planar Curves 883

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On the Class of Caustics by Reflection of Planar Curves 883 Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XIV (2015), 881-906 On the class of caustics by reflection of planar curves ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE Abstract. Given any light position S P2 and any algebraic curve of P2 (with any kind of singularities), we con2sider the incident lines comingCfrom S (i.e. the lines containing S) and their reflected lines after reflection on the mirror curve . The caustic by reflection 6S( ) is the Zariski closure of the envelope of these rCeflected lines. We introduce the Cnotion of reflected polar curve and express the class of 6S( ) in terms of intersection numbers of with the reflected polar curve, thanks to Ca fundamental lemma established in [16C]. This approach enables us to state an explicit formula for the class of 6S( ) in every case in terms of intersection numbers of the initial curve . C C Mathematics Subject Classification (2010): 14H50 (primary); 14E05, 14N05, 14N10 (secondary). Introduction Let V be a three dimensional complex vector space endowed with some fixed basis. 2 We consider a light point S x0 y0 z0 P P(V) and a mirror given by an [ : : ] 22 := d irreducible algebraic curve V (F) of P , with F Sym (V_) (F corresponds C = 2 to a polynomial of degree d in C x, y, z ). We denote by d_ the class of . We consider the caustic by reflection[6 ( )]of the mirror curve with sourceCpoint S C C S. Recall that 6S( ) is the Zariski closure of the envelope of the reflected lines associated to the incCident lines coming from S after reflection off . When S is not C at infinity, Quetelet and Dandelin [9,18] proved that the caustic by reflection 6S( ) is the evolute of the S-centered homothety (with ratio 2) of the pedal of fromCS (i.e. the evolute of the orthotomic of with respect to S). This decomposCition has also been used in a modern approach bCy [2–4] to study the source genericity (in the real case). In [16] we proved formulas for the degree of the caustic by reflection of planar algebraic curves. For a presentation of the classical notions of envelope, evolute, pedal, contrapedal, orthotomic, we refer to [4,8,10,20,23]. In [7], Chasles proved that the class of 6S( ) is equal to 2d_ d for a generic ( , S). In [1], Brocard and Lemoyne gave (wCithout any proof)+a more general C Franc¸oise Pe`ne is supported by the french ANR project GEODE (ANR-10-JCJC-0108). Received April 13, 2013; accepted in revised version September 23, 2013. 882 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE formula only when S is not at infinity. The Brocard and Lemoyne formula appears to be the direct composition of formulas got by Salmon and Cayley in [19, page 137, 154] for some geometric characteristics of evolute and pedal curves. The formula given by Brocard and Lemoyne is not satisfactory for the following reasons. The results of Salmon and Cayley apply only to curves having no singularities other than ordinary nodes and cusps [19, page 92], but the pedal of such a curve is not necessarily a curve satisfying the same properties. For example, the pedal curve of the rational cubic V (y2z x3) from 4 0 1 is a quartic curve with an ordinary triple point. Therefore it−is not corre[ct:to :com] pose directly the formulas got by Salmon and Cayley as Brocard and Lemoyne apparently did (see also Section 5 for a counterexample of the Brocard and Lemoyne formula for the class of the caustic by reflection). Let us mention some works on the evolute and on its generalization in higher dimension [6, 11, 21]. In [11], Fantechi gave a necessary and sufficient condition for the birationality of the evolute of a curve and studied the number and type of the singularities of the general evolute. Let us insist on the fact that there exist irreducible algebraic curves (other than lines and circles) for which the evolute map is not birational. This study of evolute is generalized in higher dimension by Trifogli in [21] and by Catanese and Trifogli [6]. The aim of the present paper is to give a formula for the class (with multiplic- ity) of the caustic by reflection for any algebraic curve (without any restriction neither on the singularity points nor on the flex poinCts) and for any light posi- tion S (including the case when S is at infinity or when S is on the curve ). C In Section 1, we define the reflected lines m at a generic m and the 2 2 R 2 C (rational) “reflected map” R ,S P 99K P mapping a generic m to the equation of . C : 2 C Rm In Section 2, we define the caustic by reflection 6S( ), we give conditions ensuring that 6 ( ) is an irreducible curve and we prove thaCt its class is the degree S C of the image of by R ,S. In Section 3C, we gCive formulas for the class of caustics by reflection valid for any ( , S). These formulas describe precisely how the class of the caustic depends on geCometric invariants of and also on the relative positions of S and of the two cyclic points I, J with respCect to . As a consequence of this result, we obtain the C following formula for the class of 6S( ) valid for any of degree d 2 and for a generic source position S: C C ≥ class 6S( ) 2d_ d , ` µI ( ) µJ ( ), C = + − C 1 − C − C where ( , ` ) is the contact number of with the line at infinity ` and where C 1 C 1 µI ( ) and µJ ( ) denote the multiplicity numbers of respectively I and J on . CIn SectionC4, our formulas are illustrated on two examples of curves (theClem- niscate of Bernoulli and the quintic considered in [16]). In Section 5, we compare our formula with the one given by Brocard and Lemoyne for a light position not at infinity. We also give an explicit counter- example to their formula. ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 883 In Section 6, we prove our main theorem. First we give a formula for the class of the caustic in terms of intersection numbers of with a generic “reflected polar” C at the base points of R ,S. We then compute these intersection numbers in terms of C the degree d and of the class d_ of but also in terms of intersection numbers of with each line of the triangle I J S. C C In appendix A, we prove a useful formula expressing the classical intersection number in terms of probranches. ACKNOWLEDGEMENTS. The authors thank Jean Marot for stimulating discussions and for having indicated them the formula of Brocard and Lemoyne. 1. Reflected lines m and rational map R ,S R C 2 Recall that we consider a light position S x0 y0 z0 P and an irreducible [ : : ] 2 algebraic (mirror) curve V (F) of P2 given by a homogeneous polynomial F Symd (V) with d C2.=We write Sing( ) for the set of singular points of . 2 ≥ C C For any non singular point m, we write m for the tangent line to at m. We set T C 2 C S(x0, y0, z0) V 0 . For any m x y z P , we write m(x, y, z) V 0 . 2 \ { } [2 : : ] 2 2 \ { } We write as usual ` V (z) P for the line at infinity. For any P(x1, y1, z1) V 0 , we define 1 = ⇢ 2 \ { } d 1 1 F x F y F z F Sym − V_ . P := 1 x + 1 y + 1 z 2 2 Recall that V (1P F) is the polar curve of with respect to P x1 y1 z1 P . C 2 [ : : ] 2 Since the initial problem is euclidean, we endow P with an angular structure for which I 1 i 0 P2 and J 1 i 0 P2 play a particular role. To this end, let[us:reca:ll t]he2definition o[ f t:he−cro:ss-]ra2tio β of 4 points of ` . Given 1 four points (Pi ai bi 0 )i 1,...,4 such that each point appears at most 2 times, we [ : : ] = define the cross-ratio β(P1, P2, P3, P4) of these four points as follows: b3a1 b1a3 b4a2 b2a4 β P1, P2, P3, P4 − − , = b3a2 b2a3b4a1 b1a4 − − 1 with convention 0 . For any distinct lines 1 and 2 not equal to ` , con- taining neither I nor=J1, we define the oriented angAular meAasure between 1and A1 A2 by ✓ (modulo ⇡Z) such that 2i✓ a1 ib1 a2 ib2 e− β P1, P2, I, J + − = = a1 ib1a2 ib2 − + 2 (where Pi ai bi 0 is the point at infinity of i ). Let Q Sym (V_) be defined by[ Q(:x, y, z:) ] x2 y2. It will be worthAnoting that Q2( F) F2 := + r = x + 884 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE 2 Fy 1I F1J F. For every non singular point m of ` , we recall that tm Fy = 2 C \ 1 [ : Fx 0 P is the point at infinity of m and so tm I, J is equivalent to m− V: (Q] (2 F)). T C 62 { } 62 r Now, for any m (` Q( F)) and any incident line ` containing m, we de- 2C\ 1[ r fine as follows the associated reflected line Rm(`) (for the reflexion on at m with respect to the Snell-Descartes reflection law Angle(`, ) Angle( C, R (`))). TmC = TmC m Definition 1.1. For every m (` V (Q( F))), we define rm ` ` 2 C \ 1 [ r : 1 ! 1 mapping P ` to the unique rm(P) such that β(P, tm, I, J) β(tm, rm(P),I,J).
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