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A Special Conic Associated with the Reuleaux Negative Pedal Curve

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A Special Conic Associated with the Reuleaux Negative Pedal Curve INTERNATIONAL JOURNAL OF GEOMETRY Vol. 10 (2021), No. 2, 33–49 A SPECIAL CONIC ASSOCIATED WITH THE REULEAUX NEGATIVE PEDAL CURVE LILIANA GABRIELA GHEORGHE and DAN REZNIK Abstract. The Negative Pedal Curve of the Reuleaux Triangle w.r. to a pedal point M located on its boundary consists of two elliptic arcs and a point P0. Intriguingly, the conic passing through the four arc endpoints and by P0 has one focus at M. We provide a synthetic proof for this fact using Poncelet’s porism, polar duality and inversive techniques. Additional interesting properties of the Reuleaux negative pedal w.r. to pedal point M are also included. 1. Introduction Figure 1. The sides of the Reuleaux Triangle R are three circular arcs of circles centered at each Reuleaux vertex Vi; i = 1; 2; 3. of an equilateral triangle. The Reuleaux triangle R is the convex curve formed by the arcs of three circles of equal radii r centered on the vertices V1;V2;V3 of an equilateral triangle and that mutually intercepts in these vertices; see Figure1. This triangle is mostly known due to its constant width property [4]. Keywords and phrases: conic, inversion, pole, polar, dual curve, negative pedal curve. (2020)Mathematics Subject Classification: 51M04,51M15, 51A05. Received: 19.08.2020. In revised form: 26.01.2021. Accepted: 08.10.2020 34 Liliana Gabriela Gheorghe and Dan Reznik Figure 2. The negative pedal curve N of the Reuleaux Triangle R w.r. to a point M on its boundary consist on an point P0 (the antipedal of M through V3) and two elliptic arcs A1A2 and B1B2 (green and blue). A sample P is shown illustrating instantaneous tangency to N at P 0. Here, we study some properties of the negative pedal curve N of R w.r. to a pedal point M lying on one of its sides. This curve is the envelope of lines passing through points P on R and perpendicular to PM [3, Negative Pedal Curve]; see Figure2. Trivially, the negative pedal curve of arc V1V2 is a point which we call P0. We show that the negative pedal curves of the other two sides are elliptic arcs with a common focus on M and whose major axis measures 2r; see Prop. 3.1. Let V3 be the center of the circular arc where pedal point M lies, and let V1;V2 be the endpoints of said arc. Let arc A1A2 (resp. B1B2) be the negative pedal image of the Reuleaux side V1V3 (resp. V2V3) where point A1 is the image of V1, and B1 the image of V2. The endpoints of N whose preimage is V3 are respectively A2 when V3 is regarded as a point of side V1V3, and B2 when V3 is regarded as a point of side V2V3 of the Reuleaux triangle. Our main result (Theorem 2.1, Section2) is an intriguing property of the conic C∗ – called here the endpoint conic – that passes through the endpoints A1, A2, B1, B2 of the negative pedal curve N , and through P0: that one of its foci is precisely the pedal point M; see Figure3. We also give a full geometric description of its axes, directrix and vertices, and a criterion for identifying its type, according to the location of the pedal point M. A Special Conic Associated with the Reuleaux Negative Pedal Curve 35 In Section3 we prove other properties of the Reuleaux triangle and its negative pedal curve, involving tangencies, collinearities and homotheties. The proofs combine elementary techniques with inversive arguments and polar reciprocity. A review of polar reciprocity and other concepts, including the description of the negative pedal curve as a locus of points, as well as an alternative description of it as an envelope of lines is postponed to the Appendix. Figure 3. The sides of the Reuleaux R are three circular arcs centered at the vertices V1;V2;V3 of an equilateral triangle. Its negative pedal curve N w.r. to a pedal point M on its boundary consists of a point P0 (the antipode of M through V3) and of two elliptic arcs ∗ A1A2 and B1B2 (green and blue). The endpoint conic C (purple) passes through P0 and the four endpoints of the two elliptic arcs of N . It has a focus on M and its focal axis passes through the center of the Reuleaux triangle. 2. Main Result: The Endpoint Conic Referring to Figure3. ∗ Theorem 2.1. The conic C which contains P0 and the endpoints A1, A2, B1, B2 of the negative pedal curve of the Reuleaux triangle R with respect to a pedal point M located on a side of R, has one focus on M and its axes pass through the circumcenter of 4V1V2V3. The proof will require some additional steps which steadily use an inversive approach and polar duality. 36 Liliana Gabriela Gheorghe and Dan Reznik The main idea is to use polar reciprocity: in order to prove that the focus of the C∗ is M, we show that there exists a circle C to which the five polars of the endpoints A0;A1;B0;B1 and P0 are tangent. Recall that whenever we perform a polar transform (or a polar duality) of a conic, w.r. to an inversion circle (see Figure4), points on the conic transform into their polars w.r. to the inversion circle, and those polars become tangents to the dual curve of the (initial) conic [2, Article 306]. This indirect and inversive approach is appropriate as the polars of A1, A2, B1, B2, and P0 can be readily analyzed. Classic facts about polar reciprocity guarantee that the dual (curve) of a conic is a circle iff the center of the inversion circle (which, in our case, is M) is the focus of the endpoint conic; see Proposition A.2. The reader not familiar with the topic may find useful the details in Appendix and the references therein. Referring to Figure4. Lemma 2.1. Let M be a point on the side V1V2 of the Reuleaux triangle and 0 0 0 I the inversion circle. Let V1;V2;V3 be the inverses of points V1;V2;V3 and 0 0 0 0 let arcs V1V3 and V2V3 be, respectively, the inverses of arcs V1V3 and V2V3 of the Reuleaux triangle. Then: 0 0 (1) The polars of A1 and B1 are the tangents at V1 and V2 to circular 0 0 0 0 0 0 arcs V1V3 and V2V3. The poles of the tangents at V1 and V2 to arcs 0 0 0 0 V1V3 and V2V3, are the points A1, B1. 0 0 0 (2) The polars of points A2, B2 are the tangents in V3 to arcs V1V3 and 0 0 0 0 0 V2V3, respectively. The poles of the tangents at V3 to arcs V1V3 and 0 0 V2V3, are the points A2, B2. 0 0 0 0 (3) The inverse of arc V1V2 is the line V1V2 excluding segment [V1 V2] 0 0 and the polar of P0 is the line V1V2: Proof. Inversion w.r. to a circle maps circles to either circles or lines: thus, 0 0 0 0 the inverses of arcs V1V3 and V2V3 are the two circular arcs V1V3 and V2V3, respectively. On the other hand, since arc V1V2 passes through the inversion center, its image is the union of two half-lines. All other statements derive from the description of the negative pedal curve as the dual of its inverse, as shown in Proposition A.3. Lemma 2.2. Using the notation in Lemma 2.1: 0 0 0 0 0 0 0 0 0 (1) the angles at V1, V2, and V3 between arcs V1V2, V1V3, and V1V2 respectively (the inverses of the sides of the Reuleaux triangle), are 120◦. 0 0 0 0 0 (2) 4V1O V2 determined by the tangents at V1, V2 to said arcs and the 0 0 line V1V2 is equilateral. Proof. These statements derive from the fact that inversion is preserves angles between curves. Given the above results, Theorem 2.1 is equivalent to the following Lemma; see Figure4: Lemma 2.3. The five polars of endpoints A1, A2, B1, B2, and P0 are tangent 0 0 0 to circle c, the exinscribed circle in 4V1O V2, which is externally-tangent to 0 0 side V1V2. A Special Conic Associated with the Reuleaux Negative Pedal Curve 37 0 0 0 0 Figure 4. The Reuleaux triangle R and its inverse: (1) arcs V1 V3 (blue) and V2 V3 (green) 0 0 0 0 are the inverses of sides V1V3 and V2V3 of R, while line V1 V2 except for segment [V1 V2 ] 0 itself (violet) is the image of arc V1V2; (2) the polars of A1 and B1 are the tangents in V1 0 0 0 0 0 0 and V2 to arcs V1 V3 and V2 V3 ; (3) the polars of A2 and B2 are the tangents in V3 to arcs 0 0 0 0 0 0 V1 V3 and V2 V3 ; (1)v) line V1 V2 is the polar of P0. (v) all five polars above are tangent to 0 0 0 the exinscribed circle c of 4V1 O V2 (purple). The angle between circles c0 (green) and C0 0 ◦ 0 0 (blue) in V3 is 120 iff V3 is on the circle C concentric with c and passing through V1 and 0 0 V2 (not shown); in this case, the tangents at V3 (dashed green and blue) to circles c0 and C0, are also tangent to the exinscribed circle c.
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