Constructing cubic curves with involutions

In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles’ Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter’s construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions. As an application of Schroeter’s construction we provide a new parametrisation of elliptic curves with torsion group Z/2Z×Z/8Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/8\mathbb {Z}$$\end{document} and give some configurations with all their points on a cubic curve.


Introduction
Heinrich Schroeter gave in [2] a surprisingly simple ruler construction to generate points on a cubic curve. Since he did not provide a formal proof for the construction, we would like to present this here. Schroeter's construction can be interpreted as an iterated construction of line involutions. Thus, we first define the notion of a line involution with cross-ratios, and then we show how one can construct line involutions with ruler only.
For the sake of simplicity, we introduce the following terminology: For two distinct points P and Q in the plane, P Q denotes the line through P and Q, P Q denotes the distance between P and Q, and for two distinct lines l 1 and l 2 , l 1 ∧ l 2 denotes the intersection point of l 1 and l 2 . We tacitly assume that the plane is the real projective plane, and therefore, l 1 ∧ l 2 is defined for any distinct lines l 1 and l 2 . For the cross-ratio of four lines a, b, x, y of a pencil we use the notation cr(a, b, x, y).
Line involution. Given a pencil. A line involution Λ is a mapping which maps each line l of the pencil to a so-called conjugate linel of the pencil, such that the following conditions are satisfied: • Λ is an involution, i.e., Λ • Λ is the identity, in particular we have Λ(l) = l.
Notice that any line involution is defined by two different pairs of conjugate lines. We shall use the following construction for line involutions (for the correctness of the construction see Chasles [1, Note X, §34, (28), p. 317]): Given two pairs a,ā and b,b of conjugate lines which meet in P . Suppose, we want to find the conjugate lined of a line d from the same pencil. Choose a point D = P on d and two lines through D which meet a and b in the points A and B, andā andb in the pointsĀ andB, respectively (see Fig. 1). LetD = AB ∧ĀB. Then the conjugate lined of d with respect to the line involution defined by a,ā, b,b is the line PD. • Given three di↵erent pairs of conjugate lines a,ā, b,b, c,c, and let l 1 , l 2 , l 3 , l 4 be four lines among a,ā, b,b, c,c from three di↵erent pairs of conjugate lines, then cr (l 1 , l 2 , l 3 , l 4 ) = cr l 1 ,l 2 ,l 3 ,l 4 .

Vice-versa, let
Notice that any line involution is defined by two di↵erent pairs of conjugate lines. We shall use the following construction for line involutions (for the correctness of the construction see Chasles [1, Note X, §34, (28), p. 317]): Given two pairs a,ā and b,b of conjugate lines which meet in P . Suppose, we want to find the conjugate lined of a line d from the same pencil. Choose a point D 6 = P on d and two lines through D which meet a and b in the points A and B, andā andb in the pointsĀ andB, respectively (see Fig. 1). LetD = AB^ĀB. Then the conjugate lined of d with respect to the line involution defined by a,ā, b,b is the line PD. Notice that this construction can be carried out using only a ruler.

Schroeter's Construction for Cubic Curves
Based on line involutions, Schroeter provided in [2] a simple ruler construction for cubic curves. Notice that this construction can be carried out using only a ruler.

Schroeter's Construction for Cubic Curves
Based on line involutions, Schroeter provided in [2] a simple ruler construction for cubic curves.
Schroeter's Construction. Let A,Ā, B,B, C,C be six pairwise distinct points in a plane such that no four points are collinear and the three pairs of points A,Ā, B,B, C,C are not the pairs of opposite vertices of the same complete quadrilateral. Now, for any two pairs of points P,P and Q,Q, we define a new pair S,S of points by stipulating Then all the points constructed in this way lie on a cubic curve. At first glance, it is somewhat surprising that all the points we construct lie on the same cubic curve, which is defined by three pairs of points (recall that a cubic curve is defined by 9 points). The reason is that we have three pairs of points and not just 6 points. In fact, if we start with the same 6 points but pairing them differently, we obtain a different cubic curve. It is also not clear whether the construction generates infinitely many points of the curve. Schroeter claims in [2] that this is the case, but, as we will see in the next section, it may happen that the construction gives only a finite number of points.

A Proof of Schroeter's Construction
It is very likely that Schroeter discovered his construction based on his earlier work on cubics (see [3,4]). However, he did not give a rigorous proof of his construction, and the fact that he claimed wrongly that the construction generates always infinitely many points of the curve might indicate that he overlooked something. Below we give a simple proof of Schroeter's construction using Chasles' Theorem (see Chasles [1, Chapitre IV, §8, p. 150]) and the terminology of elliptic curves.
Theorem 1 (Chasles' Theorem). If a hexagon ABCĀBC is inscribed in a cubic curve Γ and the points AB ∧ĀB and BC ∧BC are on Γ, then also CĀ ∧CA is on Γ (see Fig. 2).
With Chasles' Theorem we can prove the following q.e.d.
As an immediate consequence of Proposition 2 we get Corollary 3. The unique cubic curve passing through the 9 points A,Ā, B,B, C,C, D, E, F contains also the 3 pointsD,Ē,F .
In order to show that all the points constructed by Schroeter's construction lie on the same cubic curve, we interpret the construction in the setting of elliptic curves. For this, let be a cubic curve and let O be a point of inflection of -recall that every cubic curve in the real projective plane has at least one point of inflection. For two points P and Q on let P # Q be the third intersection point (counting multiplicities) of P Q with , where for P = Q, P Q is the tangent on with contact point P . Furthermore, for each point P on , let P := O # P . As usual, we define the binary operation + on the points of by stipulating P + Q := (P # Q) . q.e.d.
As an immediate consequence of Proposition 2 we get Corollary 3. The unique cubic curve Γ passing through the 9 points A,Ā, B,B, C,C, D, E, F contains also the 3 pointsD,Ē,F .
In order to show that all the points constructed by Schroeter's construction lie on the same cubic curve, we interpret the construction in the setting of elliptic curves. For this, let Γ be a cubic curve and let O be a point of inflection of Γ -recall that every cubic curve in the real projective plane has at least one point of inflection. For two points P and Q on Γ let P # Q be the third intersection point (counting multiplicities) of P Q with Γ, where for P = Q, P Q is the tangent on Γ with contact point P . Furthermore, for each point P on Γ, let −P := O # P . As usual, we define the binary operation + on the points of Γ by stipulating P + Q := − (P # Q) .
(b) Vice versa, if P := P #P =P #P ∈ Γ for two points P,P ∈ Γ, then we have for all Q ∈ Γ the following: Notice that P + P = O and, since O is a point of inflection, we have O = O. It is well known that the operation + is associative and the structure ( , O, +) is an abelian group with neutral element O, which is called an elliptic curve. and Adding (1) and (2)and subtracting Q+Q yields P +P =P +P and hence P #P =P #P .
Exchanging left and right hand in (1) and adding (2) gives, upon subtracting P +P , Q + Q =Q +Q and hence Q # Q =Q #Q. S # S =S #S follows by exchanging the pair Q,Q by the pair S,S. Proof. (a) By assumption we have P # Q =P #Q = S and P #Q =P # Q =S. With a point O ∈ Γ of inflection, we get and Adding (1) and (2)and subtracting Q+Q yields P +P =P +P and hence P #P =P #P .
Exchanging left and right hand in (1) and adding (2) gives, upon subtracting P +P , Q + Q =Q +Q and hence Q # Q =Q #Q. S # S =S #S follows by exchanging the pair Q,Q by the pair S,S.
(b) For the second part, we proceed as follows: By assumption, we have P #P =P #P and therefore P +P = O#(P #P ) = O#(P #P ) =P +P . We add S and subtract P +P to get S + P −P = S +P − P or (O # (S # P )) # (O #P ) = (O # (S #P )) # (O # P ). It follows that (S # P ) #P = (S #P ) # P , i.e., Q #P =Q # P =S. Finally, Q # Q =Q #Q = Q follows from the first part. q.e.d.
For the sake of simplicity we write 2 * P for P + P . Let A,Ā be a pair of points with A # A =Ā #Ā on a cubic curve Γ, and with respect to a given point of inflection O, let T A :=Ā − A. Then A + T A =Ā, which implies that Now, by assumption we have 2 * A = 2 * Ā and therefore we get that 2 * T A = O. In other words, T A is a point of order 2. Now we are ready to prove the following It follows that all points we obtain by Schroeter's construction belong to the same curve Γ. q.e.d.
The above proof shows that the Schroeter points have the following additional properties • If P,P is a pair of Schroeter points on Γ, then the tangents in P andP meet on Γ.
• With respect to a chosen point O of inflection, we have thatP − P = T is a point of order 2 on Γ which is the same for all Schroeter pairs P,P .
The following result shows that we can construct the tangent to Γ in each Schroeter point by a line involution (hence with ruler alone).
Proposition 6. Let Γ be the cubic from Proposition 2. Assume that S,S, P,P , Q,Q are three of the pairs A,Ā, B,B, C,C, D,D, E,Ē, F,F or of the pairs which are constructed by Schroeter's construction, such that SP , SQ, SP , SQ are four distinct lines. Let s = SS ands its conjugate line with respect to the involution given by the lines SP , SQ, SP , SQ. Thens is tangent to Γ in S (see Fig. 4).
Before we can prove Proposition 6, we have to recall a few facts about cubic curves. It is well-known that every cubic curve can be transformed into Weierstrass Normal Recall that A # B := (A + B). In particular, if C = A # A, then the line through C and A is tangent to a,b with contact point A.
The following result gives a connection between conjugate points and tangents. q.e.d.
In homogeneous coordinates, the curve y 2 = x 3 + ax 2 + bx becomes : Y 2 Z = X 3 + aX 2 Z + bXZ 2 . Recall that A # B := −(A + B). In particular, if C = A # A, then the line through C and A is tangent to Γ a,b with contact point A.
The following result gives a connection between conjugate points and tangents. q.e.d.

The next result gives a connection between line involutions and conjugate points.
Lemma 8. Let A = (x 0 , y 0 ) be an arbitrary but fixed point on Γ α,β,γ . For every point P on Γ α,β,γ which is different from A andĀ, let g := AP andḡ := AP . Then the mapping I A : g →ḡ is a line involution.
Proof. It is enough to show that there exists a point ζ 0 (called the center of the involution) on the line h : x = 0, such that the product of the distances between ζ 0 and the intersections of g andḡ with h is constant.
SinceT = T + T = O, with respect to T we have g : y = y 0 andḡ : x = x 0 , which implies that ζ 0 = (0, y 0 ). Now, let P = (x 1 , y 1 ) be a point on Γ α,β,γ which is different from A,Ā, T, O, and let g := AP andḡ := AP . SinceP = α γx 1 , −y 1 , the slopes λ P and λP of g andḡ, respectively, are Thus, the distances s P and sP between ζ 0 and the intersections of g andḡ with h, respectively, are and using the fact that for i ∈ {0, 1}, y 2 i = α x i + β + γx i , we obtain which is independent of the particular point P = (x 1 , y 1 ). q.e.d.
Since line involutions are invariant under projective transformations, as a consequence of Lemma 8 we obtain the following Fact 9. Let be the cubic from Proposition 2 with two pairs of Schroeter points P,P = T + P , Q,Q = T + Q, and let R be a point on such that RP, RP , RQ, RQ are four di↵erent lines. Let S be a further point on andS = T + S. Then the lines s = RS ands = RS are conjugate lines with respect to the line involution given by the lines RP, RP , RQ, RQ (see Fig. 6). Now we are ready to prove Proposition 6.
Proof of Proposition 6. First notice that S andS are distinct, since otherwise,S = T + S = S, which implies that T = S S = O.
Assume that the line s intersects in a point U which is di↵erent from S andS. ThenŪ := T + U belongs tos. If the lines intersects in a point V which is di↵erent fromŪ , then, with respect to the involution given by the linesŪS,ŪP ,ŪS,ŪP , the pointV belongs to s. Hence,V =S, which shows thats is tangent to in S. which is independent of the particular point P = (x 1 , y 1 ). q.e.d.
Since line involutions are invariant under projective transformations, as a consequence of Lemma 8 we obtain the following Fact 9. Let Γ be the cubic from Proposition 2 with two pairs of Schroeter points P,P = T + P , Q,Q = T + Q, and let R be a point on Γ such that RP, RP , RQ, RQ are four different lines. Let S be a further point on Γ andS = T + S. Then the lines s = RS ands = RS are conjugate lines with respect to the line involution given by the lines RP, RP , RQ, RQ (see Fig. 6). Now we are ready to prove Proposition 6.
Proof of Proposition 6. First notice that S andS are distinct, since otherwise,S = T + S = S, which implies that T = S − S = O.
Assume that the line s intersects Γ in a point U which is different from S andS. ThenŪ := T + U belongs tos. If the lines intersects Γ in a point V which is different fromŪ , then, with respect to the involution given by the linesŪ S,Ū P ,ŪS,ŪP , the pointV belongs to s. Hence,V =S, which shows thats is tangent to Γ in S. Now, assume that the line s intersects just in S andS. Then, the line s is tangent to either in S or inS. We just consider the former case, the latter case is handled similarly. Let P n (for n 2 N) be a sequence of points on which are di↵erent from S and which converges to S, i.e., lim n!1 P n = S. Since for each n 2 N we havē P n = T + P n (whereP n := T + P ), by continuity of addition we have lim n!1Pn =S. For each n 2 N let t n := P n S. Then, for each n 2 N,t n =P n S. Since s is tangent to in S, by continuity, on the one hand we have lim n!1 t n = s, and on the other hand we have lim n!1tn = s, which implies thats = s and shows thats is tangent to in S. q.e.d.
As a corollary of Proposition 6 and Lemma 4(a) we obtain the following: Corollary 10. Let be the cubic from Proposition 2. Then we have: (a) In each Schroeter point it is possible to construct the tangent by a line involution, i.e., with a ruler construction.
(b) In addition to the Schroeter points on one can construct for each Schroeter pair P,P the point P # P =P #P 2 by ruler alone: These are the intersection points of the tangents in P and inP . Now, assume that the line s intersects Γ just in S andS. Then, the line s is tangent to Γ either in S or inS. We just consider the former case, the latter case is handled similarly. Let P n (for n ∈ N) be a sequence of points on Γ which are different from S and which converges to S, i.e., lim n→∞ P n = S. Since for each n ∈ N we havē P n = T + P n (whereP n := T + P ), by continuity of addition we have lim n→∞Pn =S. For each n ∈ N let t n := P n S. Then, for each n ∈ N,t n =P n S. Since s is tangent to Γ in S, by continuity, on the one hand we have lim n→∞ t n = s, and on the other hand we have lim n→∞tn = s, which implies thats = s and shows thats is tangent to Γ in S. q.e.d.
As a corollary of Proposition 6 and Lemma 4(a) we obtain the following: Corollary 10. Let Γ be the cubic from Proposition 2. Then we have: (a) In each Schroeter point it is possible to construct the tangent by a line involution, i.e., with a ruler construction.
(b) In addition to the Schroeter points on Γ one can construct for each Schroeter pair P,P the point P # P =P #P ∈ Γ by ruler alone: These are the intersection points of the tangents in P and inP .  A priori it might be possible that Schroeter's construction does not yield all cubic curves. However, the next theorem says that in fact all cubic curves carry Schroeter's construction.
Theorem 11. Let Γ be a non-singular cubic curve. Let A, B, C be three different arbitrary points on Γ. Then, there are pointsĀ,B,C on Γ such that D = AB ∧ĀB, E = BC ∧BC, F = CA ∧CĀ are points on Γ and so do all the points given by Schroeter's construction.
Proof. ChooseĀ such that A # A =Ā #Ā andB :=Ā # (A # B). In particular, we have A # B =Ā #B, and, by Lemma 4, A #B =Ā # B and B # B =B #B. LetC :=B # (B # C). In particular, we have B # C =B #C, and, by Lemma 4, B #C =B # C and C # C =C #C. It follows from Chasles' Theorem 1 that A #C =Ā # C. From the above, we obtain by applying Proposition 2 with C andC exchanged, that A # C =Ā #C. Hence all points constructed from these points by Schroeter's construction lie on Γ. q.e.d.
Remarks. Let Γ 0 be the cubic curve passing through A,Ā, B,B, C,C, D, E, F , let O be a point of inflection of Γ 0 , and let E 0 = (Γ 0 , O, +) be the corresponding elliptic curve.
(1) If C n is a cyclic group of order n, then there is a point on Γ 0 of order n (with respect to E 0 ). This implies that if we choose the six starting points in a finite subgroup of E 0 , then Schroeter's construction "closes" after finitely many steps and we end up with just finitely many points. However, if our 6 starting points are all rational and we obtain more than 16 points with Schroeter's construction, then, by Mazur's Theorem, we obtain infinitely many rational points on the cubic curve Γ 0 .
(2) If the elliptic curve E 0 has three points of order 2, then one of them, say T , has the property that for any point P on Γ 0 we haveP = P + T . In particular, we haveT = T + T = O. Furthermore, for the other two points of order 2, say S 1 and S 2 , we have S 1 = S 2 + T and S 2 = S 1 + T , i.e., S 1 =S 2 .
(3) If we choose another point of inflection O on the cubic curve Γ 0 , we obtain a different elliptic curve E 0 . In particular, we obtain different inverses of the constructed points, even though the constructed points are exactly the same (see Fig. 8).
respect to E 0 ). This implies that if we choose the six starting points in a finite subgroup of E 0 , then Schroeter's construction "closes" after finitely many steps and we end up with just finitely many points. However, if our 6 starting points are all rational and we obtain more than 16 points with Schroeter's construction, then, by Mazur's Theorem, we obtain infinitely many rational points on the cubic curve 0 .
(2) If the elliptic curve E 0 has three points of order 2, then one of them, say T , has the property that for any point P on 0 we haveP = P + T . In particular, we haveT = T + T = O. Furthermore, for the other two points of order 2, say S 1 and S 2 , we have S 1 = S 2 + T and S 2 = S 1 + T , i.e., S 1 =S 2 .
(3) If we choose another point of inflection O 0 on the cubic curve 0 , we obtain a di↵erent elliptic curve E 0 0 . In particular, we obtain di↵erent inverses of the constructed points, even though the constructed points are exactly the same (see Fig. 8).