SINGULAR POINTS OF ISOPTICS OF OPEN ROSETTES

Transcription

1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI S.N.) MATEMATICĂ, Tomul LX, 2014, f.1 DOI: /aicu SINGULAR POINTS OF ISOPTICS OF OPEN ROSETTES BY DOMINIK SZA LKOWSKI Abstract. In our previous papers we considered regular isoptics of k-th order of open rosettes. In this paper we consider open rosettes and their isoptics with singular points. We start with a definition of α-isoptic of k-th order of an open rosette and give a theorem of its parametrization z α,k t). Next, we present a corollary in which we compute z α,kt). For some t k,j we have z α,kt k,j ) = 0, so the isoptic has singular points z α,k t k,j ). Then we establish formulas for z α,kt) and z α,kt) and use them to determine the kind of singularity of the isoptic not in every case, however). We show that vectors z α,k t) and z α,k t) z α,kt) are parallel near singular points. Finally we prove the sine theorem and compute curvature of the isoptic without singular points. Mathematics Subject Classification 2010: 53A04. Key words: open rosettes, isoptics of open rosettes, parametrization of isoptics, curvature of isoptics, singular points, sine theorem. 1. Preliminaries Let C be an open rosette with the support function pt), t a, ). Recall that the parametrization of C is given by 1) zt) = pt)e it + p t)ie it. Fix t a, ) and α 0, π). By l t we denote the tangent line to C at point zt). Now consider points zt k ) = zt + 2k 1)π + α), k N, and the tangent line to C at zt k ) denote by l tk figure 1). Definition 1. The cut locus of common points of tangents l t and l tk is called an α-isoptic of k-th order of an open rosette. It is denoted by C α,k. If α = π 2 then the α-isoptic is called orthoptic.

2 86 DOMINIK SZA LKOWSKI 2 z t 3 l t z t 2 z t 1 Π Α Π Α C Α,2 Π Α l t2 C Α,3 l t3 C Α,1 l t1 C z t Figure 1: Isoptics of open rosette In the further part of this paper we consider isoptics for fixed angle α 0, π) and order k N. The symbol [, ] denotes 2) [a + bi, c + di] = ad bc. Theorem 1. Let C be an open rosette with the support function pt), t a, ), and let C α,k be its α-isoptic of k-th order. The parametrization of C α,k is given by 3) C α,k : z α,k t) = pt)e it + pt) cot α + pt ) k) ie it where t a, ). 4) We introduce the following notations q α,k t) = zt) zt k ) = pt) cos t p t) sin t pt k ) cost + α) + p t k ) sint + α) + ipt) sin t + p t) cos t pt k ) sint + α) p t k ) cost + α)),

3 3 SINGULAR POINTS OF ISOPTICS OF OPEN ROSETTES 87 5) 6) b α,k t) = [ q α,k t), e it] = p t) + pt k ) + p t k ) cos α, B α,k t) = [ q α,k t), ie it] = pt) pt k ) cos α + p t k ), so we can write 7) q α,k t) = B α,k t)e it b α,k t)ie it. We also use the functions 8) 9) λ α,k t) = b α,k t) B α,k t) cot α = p t) pt) cot α + pt k), µ α,k t) = B α,kt) = p t k ) + pt k ) cot α pt) which geometric meaning can be seen on figure 5. From 3) we get 10) z α,k t) = p t) + pt) cot α pt k) + and by using the function pt) p t) cot α + p t k ) ) e it ) ie it 11) ϱ α,k t) = B α,k t) + b α,k t) cot α = pt) p t) cot α + p t k ) we have the following corollary. Corollary 1. Let C be an open rosette with support function pt), t a, ), and let C α,k be its α-isoptic of k-th order, α 0, π). Then 12) z α,k t) = λ α,kt)e it + ϱ α,k t)ie it where t a, ) and λ α,k t), ϱ α,k t) are given by 8) and 11). 13) It is easy to verify that z α,k t) = λ 2 α,k t) + ϱ2 α,k t).

4 88 DOMINIK SZA LKOWSKI 4 2. Singular points Now we prove that z α,k t) vanishes when there exists such t that 14) zt) = zt k ), which means that the rosette intersects itself at points used to construct its α-isoptic. The set of all such angles we denote by 15) T k = {t k,j : zt k,j ) = zt k,j + 2k 1)π + α), j = 1, 2,...}. Thus for any t k,j T k we have 16) 17) 18) 19) 20) q α,k t k,j ) = 0, λ α,k t k,j ) = 0, µ α,k t k,j ) = 0, ϱ α,k t k,j ) = 0, z α,k t k,j) = 0. Definition 2. Let C be an open rosette and let C α,k be its α-isoptic of k-th order. Point z α,k t k,j ) C α,k, t k,j T k, is said to be a singular point of α-isoptic of open rosette of k-th order. Since for any α 0, π) and any t a, ) we have 21) zt) zt + α), the α-isoptic of the first order does not have any singular points. We can write the parametrization of C α,k as 22) z α,k t) = zt) + λ α,k t)ie it. From equations 22) and 17) for t k,j T k we have 23) z α,k t k,j ) = zt k,j ), so the singular point lies on the rosette. Now we describe behaviour of the unit tangent vector to α-isoptic near the singular point.

5 5 SINGULAR POINTS OF ISOPTICS OF OPEN ROSETTES 89 C Α,k z t 2 k 1 Π Α Μ Α,k t z Α,k t z Α,k t k,j Λ Α,k t C C z t Figure 2: t < t k,j The α-isoptic in neighbourhood of singular point, case when Theorem 2. Let C be an open rosette with support function pt), t a, ), and let C α,k be its α-isoptic of k-th order, α 0, π). Then 24) lim z α,k t) z α,k = lim t) z α,k t) z α,k t) for every t k,j T k, where z α,k t) is given by 12). Proof. On figures 2 and 3 we have parts of α-isoptic of k-th order in neighbourhood of singular point z α,k t k,j ). The functions λ α,k t) and µ α,k t) change their signs at t = t k,j and we have 25) 26) λ α,k t) > 0, µ α,k t) > 0 for t < t k,j, λ α,k t) < 0, µ α,k t) < 0 for t > t k,j.

6 90 DOMINIK SZA LKOWSKI 6 z t C Α,k Λ Α,k t z Α,k t z Α,k t k,j Μ Α,k t C C z t 2 k 1 Π Α Figure 3: t > t k,j The α-isoptic in neighbourhood of singular point, case when Let 27) Rt) = pt) + p t) denotes curvature radius of rosette C at point zt). First we obtain 28) 29) and then λ α,k t) = ϱ α,kt) Rt), ϱ α,k t) = λ α,kt) Rt) cot α + Rt k), ϱ α,k t) 30) lim t t k,j λ α,k t) = Rt k,j) cos α Rt k,j + 2k 1)π + α). Rt k,j ) Now 31) lim z α,k t) z α,k t) = Rt k,j) M α,k t k,j ) e it k,j + Rt k,j) cos α Rt k,j + 2k 1)π + α) ie it k,j, M α,k t k,j )

7 7 SINGULAR POINTS OF ISOPTICS OF OPEN ROSETTES 91 where 32) M α,k t) = R 2 t) 2Rt)Rt k ) cos α + R 2 t k ). Taking into account the opposite sign of λ α,k t) we obtain right-handed limit as 33) lim z α,k t) z α,k t) = Rt k,j) M α,k t k,j ) eit k,j Rt k,j) cos α Rt k,j + 2k 1)π + α) ie it k,j. M α,k t k,j ) From equations 12), 28) and 29) we have 34) 35) z α,k t) = 2ϱ α,kt) + Rt)) e it + z α,k t) = + 2λ α,k t) Rt) cot α + Rt k) ) ie it, 4λ α,k t) + 3Rt) cot α 3Rt ) k) + R t) 4ϱ α,k t) + 3Rt) R t) cot α + R t k ) e it ) ie it, and hence we get the following corollary for singular points. Corollary 2. Let C be an open rosette with support function pt), t a, ), and let C α,k be its α-isoptic of k-th order, α 0, π). Then z α,k t k,j) = Rt k,j )e it k,j + Rt k,j ) cot α + Rt ) k,j + 2k 1)π + α) ie it k,j 36), z α,k t k,j) = 3Rt k,j ) cot α 3Rt ) k,j+2k 1)π+α) +R t k,j ) e it k,j ) + 3Rt k,j ) R t k,j ) cot α + R t k,j + 2k 1)π + α) ie it k,j 37), where t k,j T k. In particular z α,k t k,j) 0.

8 92 DOMINIK SZA LKOWSKI 8 The kind of singularity depends on consecutive derivatives of parametrization [3]). We know that z α,k t k,j) 0 but, in general, we don t know anything about z α,k t k,j) and the next derivatives. However, if we have a particular explicit support function then we can describe the kind of singularity as in the following example. Example 1. Let C be an open rosette with the support function 38) pt) = 2 + 5π t sin t, t R. Orthoptic of second order of this rosette is given by ) 2 + 5π 2 + 5π 39) z π 2,2 t) = t sin t e it t + 5π ) ) cos t ie it 2 where t R. We have 40) 41) 42) The set 43) λ π 2,2 t) = 2 + 5π π cos t sin t, 2 ϱ π 2,2 t) = 2 + 5π 2 2 5π sin t + cos t, 2 z π,2t) = λ π 2 2,2 t)e it + ϱ π 2,2 t)ie it. { z π 2,2 t k,j ): t k,j = 3π 4 + 2πj, j Z } contains all singular points of orthoptic of second order, so we have 44) 45) 46) At singular points we obtain λ π 2,2 t k,j ) = 0, ϱ π 2,2 t k,j ) = 0, z π,2t k,j) = ) 48) lim lim z π z π z π 2,2t) 2,2t) = 1 + 0i,,2t) 2 z = 1 + 0i. π,2t) 2

9 9 SINGULAR POINTS OF ISOPTICS OF OPEN ROSETTES 93 Similarly, for the derivatives we have 49) 50) z π,2t k,j) = 1 5π i, z π,2t k,j) = 1 15π ) i. 2 2 so both of them are nonzero and linearly independent. This means that at singular point the orthoptic has a cusp [3]). Considered curves are presented on figure 4. C CΠ,2 2 Figure 4: Open rosette and its orthoptic of second order from example 1 From equations 31), 33), 36) we see that vector z α,k t) z α,k t) in its limit position when t t k,j ) is parallel to the vector z α,k t k,j). It also follows from the Taylor expansion of zt) eq. 1)).

10 94 DOMINIK SZA LKOWSKI 10 Corollary 3. Let C be an open rosette and let C α,k be its α-isoptic of k-th order, α 0, π). Then [ z α,k lim t) ] 51) t t z z α,k k,j α,k t), t k,j) = 0, [ ] 52) lim z α,k t) z α,k z t), α,k t k,j) = 0, where z α,k t), z α,k t k,j) are given by 12) and 36). Example 2. Vectors from example 1 given by 47), 48) are parallel to the vector given by 49). Now we give a counterpart of the sine theorem for isoptics of ovals [1], [2]). Theorem 3. Let C be an open rosette with support function pt), t a, ), and let C α,k be its α-isoptic of k-th order, α 0, π). By ξ k we denote the angle between vectors z α,k t) and ieit and by η k we denote the angle between vectors z α,k t) and ieit k. Then we have 53) q α,k t) = λ α,kt) sin ξ k = µ α,kt) sin η k, for t a, ), t / T k, and 54) 55) q α,k t) q α,k t) = lim λ α,k t) µ α,k t) = lim = 0, t t sin ξ k,j k t t sin η k,j k = lim λ α,k t) µ α,k t) = lim = 0, t t + sin ξ k,j k t t + sin η k,j k for t k,j T k, where q α,k t), λ α,k t), ϱ α,k t) are given by 4), 8) and 11). Proof. Consider a regular point on the isoptic and a case when λ α,k t) > 0 and µ α,k t) < 0 as in figure 5. Using equations 8), 11) and 13) we obtain 56) z α,k t) b 2 α,k t) + B2 α,k t) =.

11 11 SINGULAR POINTS OF ISOPTICS OF OPEN ROSETTES 95 z t 2 k 1 Π Α Μ Α,k t z Α,k t q Α,k t Π Η k Π Α Ξ k ie it ie i t 2 k 1 Π Α Λ Α,k t C Α,k C z t Figure 5: Sine theorem for isoptics of open rosette, case when λ α,k t) > 0, µ α,k t) < 0 It follows from 7) that 57) q α,k t) = b 2 α,k t) + B2 α,k t). The sine of the angle between vectors z α,k t) and ieit is equal to [ ] z α,k t), ieit 58) sin ξ k = ie it z α,k. t) Using the formula 12) we obtain 59) [ z α,k t), ieit] = λα,k t), and then by 56), 57)) we have 60) sin ξ k = λ α,kt). q α,k t)

12 96 DOMINIK SZA LKOWSKI 12 Now consider the vectors z α,k t) and ieitk) and compute the sine of the angle between them [ z k)] α,k t), ieit 61) sin η k = z α,k t) ie it k). Hence we obtain 62) [ ] z α,k t), ieit k) = µα,k t), so 63) sin η k = µ α,kt). q α,k t) Combining 60) and 63) we obtain the formula 53). At other regular points signs of λ α,k t) and ϱ α,k t) might be different) the reasoning is similar. At the singular point z α,k t k,j ) of α-isoptic we have to calculate appropriate one-handed limits. There are two cases: for t < t k,j we have λ α,k t) > 0 and µ α,k t) > 0, thus 64) 65) lim sin ξ k = lim lim sin η k = lim λ α,k t) z α,k = t) Rt k,j), M α,k t k,j ) µ α,k t) z α,k = Rt k,j + 2k 1)π + α). t) M α,k t k,j ) for t > t k,j we have λ α,k t) < 0, µ α,k t) < 0 and 66) 67) lim sin ξ k = lim lim sin η k = lim λ α,k t) z α,k = t) Rt k,j), M α,k t k,j ) µ α,k t) z α,k = Rt k,j + 2k 1)π + α). t) M α,k t k,j ) The function M α,k t) is given by 32). Notice that in both cases the sines of ξ k and η k are nonvanishing.

13 13 SINGULAR POINTS OF ISOPTICS OF OPEN ROSETTES 97 There is an interesting geometric interpretation of the above theorem. The normal lines of curve C at points zt) and zt k ) intersect at a point which is located on the normal line of C α,k at z α,k t). These normal lines are presented of figure 5 as dashed lines. We can also point out that the theorem is a consequence from basic kinematics and from the fact that the two tangent lines l t and l tk with the constant angle α between them represent a moving rigid body. Now we compute the curvature of α-isoptic. Theorem 4. Let C be an open rosette with the support function pt), t a, ), and let C α,k be its α-isoptic of k-th order, α 0, π). If t / T k then the curvature of C α,k is given by 68) κ α,k t) = λ2 α,k t) + ϱ2 α,k t) + λ α,k t)ϱ α,kt) λ α,k t)ϱ α,k t) λ 2 α,k t) + ϱ2 α,k t))3/2 where t a, ) and functions λ α,k t), ϱ α,k t) are given by 8), 11). 69) 70) Proof. Using 12) we obtain z α,k t) = λ α,k t) ϱ α,kt))e it + ϱ α,k t) λ α,kt))ie it and [ z α,k t), z α,k t)] = λ 2 α,k t) + ϱ2 α,k t) + ϱ α,kt)λ α,k t) λ α,kt)ϱ α,k t). The curvature is derived from the formula [ 71) κ α,k t) = z α,k ] t), z α,k t) z α,k t) 3. In general we are not able to describe curvature near a singular point. The following example suggests that always 72) lim κ α,k t) = lim κ α,k t) =, where t k,j T k. Example 3. Consider orthoptic C α,2 from example 1. Its curvature at regular points) is equal to λ 2 π 73) κ π 2,2 2 t) =,2t) + ϱ2 π,2t) + λ π,2t)ϱ π 2 2 2,2 t) λ π 2,2 t)ϱ π,2t) 2 ) 3/2, λ 2 π,2t) + ϱ2 π,2t) 2 2

14 98 DOMINIK SZA LKOWSKI 14 so 74) κ π 2,2 t) = π4 + 5π) π)6 + 5π)cos t sin t) 40π sin 2t π2 + 5π) π) 2 cos t sin t) 20π sin 2t ) 3/2 and 75) 76) at singular points. lim κ π t t 2,2 t) =, k,j lim κ π t t + 2,2 t) =. k,j 3. Future work It would be interesting to consider similar problems for isoptics of a pair of open rosettes and for isoptics of a pair of quasiperiodic open rosettes. REFERENCES 1. Benko, K.; Cieślak, W.; Góźdź, S.; Mozgawa, W. On isoptic curves, An. Ştiinţ. Univ. Al.I. Cuza Iaşi, Mat., ), Cieślak, W.; Miernowski, A.; Mozgawa, W. Isoptics of closed strictly convex curve, Global differential geometry and global analysis Berlin, 1990), 28 35, Lecture Notes in Math., 1481, Springer, Berlin, Rutter, J. Geometry of Curves, Chapman & Hall/CRC Mathematics, Chapman & Hall/CRC, Boca Raton, FL, Sza lkowski, D. Isoptics of open rosettes, Ann. Univ. Mariae Curie-Sk lodowska Sect. A ), Sza lkowski, D. Isoptics of open rosettes II, An. Ştiinţ. Univ. Al.I. Cuza Iaşi, Mat. N.S.), ), Received: 9.XI.2011 Institute of Mathematics, Revised: 6.II.2012 Maria Curie-Sk lodowska University, Accepted: 7.II.2012 Pl. M. Curie-Sk lodowskiej 1, Lublin, POLAND dominik.szalkowski@umcs.lublin.pl