Geometric continuity of plane curves in terms of Riordan matrices and an application to the F-chordal problem
aa r X i v : . [ m a t h . M G ] A p r Geometric continuity of plane curves in terms of Riordanmatrices and an application to the F-chordal problem
L. Felipe Prieto-Mart´ınez ∗ April 28, 2020
Abstract
The first goal of this article is to provide an statement of the conditions for geometric continuityof order k , referred in the bibliography as beta-constraints, in terms of Riordan matrices. Thesecond one is to see this new formulation in action to solve a theoretical cuestion about uniquenessof analytic solution for a general and classical problem in plane geometry: the F -chordal prob-lem. Keywords:
Geometric Continuity Riordan matrices F -chordal Problem F -chordal PointsEquichordal Problem For k ≥ c is of class G k (has geometric continuity of order k ) if there exists a localregular parametrization of class C k of this curve in a neighbourhood of each point of c . The case k = ∞ can also be considered and then if, in addition, we impose the local regular parametrizations to beanalytic, then we say that the curve is analytic.Geometric continuity is a concept of great importance in computer-aided geometric design, where theobjects are frequently described in terms of parametric splines. For more information see, for instance,the articles [1, 2] or the book [11].One of the main problems related to geometric continuity of plane curves can be stated as follows: Problem 1
Let c be a curve with a (countinuous) piecewise defined parametrization γ : ( − ε, ε ) → c given by γ ( t ) = γ left ( t ) = ( x left ( t ) , y left ( t )) t ∈ ( − ε, V t = 0 γ right ( t ) = ( x right ( t ) , y right ( t )) t ∈ (0 , ε ) where γ left , γ right are parametrizations of class C k . We will call the point of intersection V = γ (0) thevertex.Despite the fact of γ left , γ right being C k , a regular parametrization of c may not exist in any neigh-bourhood of V (see Figure 1). Assume that the Taylor polynomials of degree k of the functions x left ( t ) , y left ( t ) , x right ( t ) , y right ( t ) at t = 0 exist. Which compatibility conditions should satisfy the coefficientsof these Taylor polynomials if we want c to be a curve of class G k ? ∗ Department of Mathematics, Universidad Aut´onoma de Madrid (Spain), [email protected]
1s explained briefly in the abstract, this article has two main goals. The first one is to provide anew and useful formulation of the conditions on the Taylor polynomial required in Problem 1. Thistarget is reached in Section 3. The conditions for the Taylor polynomial are sometimes expressed inthe bibliography in terms of the so called connection matrices (see section 2.1 in [11]). The statementpresented here (Remark 6) is done in terms of
Riordan matrices . Finite and infinite Riordan matriceshave a well studied structure and properties (some basics are provided in Section 2) which are moreadequate for doing a certain kind of computations. For example, the inverse limit structure studied in[12], allow us to do easily proofs by induction, like the ones required in our second target. This secondgoal is to show in action this new formulation solving a theoretical problem. We show how this newstatement can be used to solve a questions about uniqueness of analytic solutions appearing in a wellknown problem in plane geometry (Problem 2).Let F be a symmetric function in two variables, defined in [0 , ∞ ) × [0 , ∞ ). Given a convex region B which boundary is c , a chord in c is any segment which endpoints belong to c . We say that P in theinterior of B is an interior F -chordal point if there exists a constant K P such that for every chord in c ,passing through P and with endpoints A, B we have F ( | A − P | , | B − P | ) = K P .Figure 1: The center of a disk is an F-chordal point for F ( a, b ) = a + b . Any point in the disk isan F -chordal point for F ( a, b ) = a · b as a consequence of Steiner’s Power of a Point Theorem (so | P − A | · | P − B | = | P − C | · | P − D | ). Klee in [10] noticed that any of the two focus of an ellipse is anF-chordal point for F = a + b (so | P − A ′ | + | P − B ′ | = | P − C ′ | + | P − D ′ | ) Problem 2 (Two points Interior F-chordal Problem)
For a given symmetric function in two vari-ables F defined in [0 , ∞ ) × [0 , ∞ ) , find a plane Jordan curve which interior region is convex, with twointerior F -chordal points. For any solution c of this problem, we call the line through P, Q the axis , and the two points where2he axis meet c the vertices . As we will see later, one of this vertices will also be a vertex in the senseof Problem 1.The F -chordal Problem is a generalization proposed in the book [4] of an older problem: the Equichordal Problem , stated in 1916-1917 independently by Fujiwara [6] and Blaschke, Rothe andWeitzenb¨ock [3]. For a convex region B with boundary c , a point P in the interior of B is an equichordalpoint if all the chords of c passing through P are of the same length (the center of a circle is an equichordalpoint, for instance). The Equichordal Problem ask wether a convex region B can have two equichordalpoints. In [6], the author already proved that no such a region can have three or more equichordalpoints. But we had to wait until 1997 when Rychlik [14] answered in a negative way the question. Inthe meantime, Wirsing showed in [19] that, if such a region exists then the curve c of the boundaryshould be analytic. This is one of the reasons why analytic solutions for Problem 2 are of interest. Otherproblems (some of which remain open) with an interesting history can be considered as particular casesof the F -chordal Problem too. More is said about this in Section 6.In Section 4, G solutions of Problem 2 are considered. In Section 5 we prove the following theorem,which answer the question of uniqueness of analytic solutions of Problem 2 for most of the cases studiedin the bibliography. Theorem 3
Let four different collinear points V , P, Q, V , where P, Q are between V , V . Let F :[0 , ∞ ) → R be a symmetric function in two variables, satisfying the following conditions:(i) it is C ∞ ,(ii) ∂F∂b | ( k P − V k , k P − V k ) , ∂F∂b | ( k Q − V k , k Q − V k ) = 0 ,(iii) ∂F∂a | ( k P − V k , k P − V k ) , ∂F∂a | ( k Q − V k , k Q − V k ) = 0 .(iv) ∀ n ∈ N , ∂F∂a (cid:12)(cid:12) ( k Q − V k , k Q − V k ) · ∂F∂b (cid:12)(cid:12) ( k P − V k , k P − V k ) ∂F∂a (cid:12)(cid:12) ( k P − V k , k P − V k ) · ∂F∂b (cid:12)(cid:12) ( k Q − V k , k Q − V k ) ! n = k V − Q kk V − P k k V − P kk V − Q k Then if there exists an analytic solution for the interior F -chordal Problem with interior F -chordalpoints P, Q and vertices V , V , this solution is unique. In this case, k P = F ( k V − P k , k V − P k ) and k Q = F ( k V − Q k , k V − Q k ).As a consequence of this theorem, we will provide an alternative proof of part of the results involvedin the articles by Rychklik [14, 15] for the Equichordal Problem. We also comment on some of the moststudied cases appearing in the bibliography, and we give a generalization of Theorem 3 for a differentdefinition of F -chordal point that does not require the curve c to be a Jordan cuve. This is done inSection 6.Finally, in Section 7, we propose some problems of uniqueness of analytic solutions in plane geometry,that are suitable to be studied using the techniques in this article . Riordan matrices first appeared in [16], although the original definition was slightly different to theone in current use. The classical survey [17] contains more information with a similar notation to theone used here. Riordan matrices and generalized Riordan matrices are special types of infinite lowertriangular matrices: ( a ij ) ∞ i,j =0 = a a a a a a ... ... ... . . . efinition 4 An infinite matrix ( a ij ) ∞ i,j =0 is a generalized Riordan matrix over the reals if and onlyif there exists two formal power series: (2.1) d ∈ R [[ t ]] , h ∈ t R [[ t ]] such that, for every ≤ i, j , a ij = [ t i ]( d · h i ) , where [ t i ] f denotes the i − th coefficient of the formal powerseries f . In other words, the generating function of the i -th column a i , a i , a i , . . . is d · h i . In this case,we write: ( a ij ) ∞ i,j =0 = R ( d, h ) . If we replace the condition (2.1) by the stronger one d ∈ R [[ t ]] \ t R [[ t ]] , h ∈ t R [[ t ]] \ t R [[ t ]] , we would have an (ordinary) Riordan matrix instead. The condition h ∈ t R [[ t ]] ensures that gereneralized Riordan matrices are always lower triangular.Generalized Riordan matrices have an important property known as the First Fundamental Theoremof Riordan Matrices (1FTRM). If we multiply any generalized Riordan matrix by an infinite columnvector, we obtain a new infinite column vector: R ( d, h ) F F F ... = G G G ... And if F is the generating function of the sequence F , F , F , . . . , then the generating function of G , G , G , . . . is d · ( F ◦ h ). As a consequence of this theorem, for every two generalized Riordanmatrices, we have that(2.2) R ( d, h ) · R ( f, g ) = R ( d · f ◦ h, g ◦ h )It can be proved straightforward that the set of (ordinary) Riordan matrices is a group, which adescription of the inverse in terms of the corresponding formal power series too. But this is not necessaryfor this article.We will need something else, concerning the inverse limit structure of the Riordan group. Thisstructure will allow us to do proofs by induction. Define a generalized partial Riordan matrix R n ( d, h ) tobe the principal submatrix of size ( n +1) × ( n +1) of a generalized Riordan matrix R ( d, h ) = ( a ij ) ≤ i,j< ∞ R ( d, h ) = . . .R n ( d, h ) ...0 . . .a n +1 , . . . a n +1 ,n a n +1 ,n +1 ... ... ... . . . Remark 5
As a consequence of matrix block multiplication for triangular matrices, we have that: R ( d, h ) F F F ... = G G G ... ⇐⇒ ∀ n ∈ N , R n ( d, h ) F F ... F n = G G ... G n Moreover, see that in the matrix R n ( d, h ) depends only on the coefficients of T aylor n ( d ) , T aylor n ( h ) ( T aylor n ( f ) denotes the Taylor polynomial of degree n at t = 0 of f ). In particular, R n ( d, h ) = R n ( e d, e h ) ⇐⇒ ( T aylor n ( d ) = T aylor n ( e d ) T aylor n ( h ) = T aylor n ( e h )4 ee also that if (2.3) R n ( d, h ) F F ... F n = G G ... G n holds for n = k , then it holds for every m ≤ k .Aditionally, if for n = k we already have a partial Riordan matrix R k ( d, h ) satisfying (2.3) then wecan search for a matrix R k +1 ( d, h ) satisfying (2.3) for n = k + 1 just looking at the last entry in thecolumn vector obtained in each side of the equality, and we have only two new parameters in R k +1 ( d, h ) to achieve this. We will call this proccess extending the matrix. Much more can be said about generalized partial Riordan matrices. For example an intrinsic defi-nition (not depending on the definition of a “bigger” generalized Riordan matrix) is also possible. Werecommend [12] for more information about this finite dimensional matrices.
Now that we have introduced Riordan matrices, we will go back to Problem 1. In the context of thisproblem, c is G k if there exists a C k regular reparametrization u of γ left (or equivalently of γ right ) suchthat u (0) = 0 and e γ : ( − δ, δ ) −→ c e γ ( t ) = γ left ( u ( t )) t ∈ ( − δ, V t = 0 γ right ( t ) t ∈ (0 , δ )For each k , let the corresponding Taylor polynomials at t = 0 T aylor n ( x left ( t )) = a + a t + . . . + a k t k , T aylor n ( y left ( t )) = b + b t + . . . + b k t k T aylor n ( x right ( t )) = c + c t + . . . + c k t k , T aylor n ( y right ( t )) = d + d t + . . . + d k t k The conditions that this parameters a i , b i , c i , d i for i = 0 , . . . , k must satisfy for the curve c are a setof linear equations. In the bibliography, the linear relations (equivalent to those proposed here) areknown as beta-constraints , and is frequently stated in terms of the so called connection matrices (see[11]). But we suggest here to express these conditions in terms of Riordan matrices, which provide usa powerful tool for doing computations. Remark 6
The conditions for having geometric continuity of order n at V in the notation above areequivalent to the existence of a partial (ordinary) Riordan matrix R k (1 , u ) such that: (3.1) R n (1 , u ) a ... a n = c ... c n R n (1 , u ) b ... b n = d ... d n Recall that u ∈ t R [[ t ]] \ t R [[ t ]] . connection matrices . But it has twoadvantages: (1) If, for instance, ( x left , y left ) are fixed and known functions, we hace a bridge betweenfunctional equations and the linear restrictions for the corresponding Taylor polynomials (the sameoccurs for different types of restrictions between ( x left , y left ) and ( x right , y right )). (2) For the G ∞ curves case, we can easily do proofs by induction (as we will do to show uniqueness of analytic solutionsfor Problem 2).In this second case, we need to find, ∀ n ∈ N , matrices R n (1 , u ) satisfying (3.1). The stategy (which isthe one used in Theorem 3) is the following. First we find a solution for the case n = 2. Then we extend (as explained in Remark 5) this solution to a solution for the case n = 3 and so on. if we have a matrix R k (1 , u ) satisfying (3.1) for n = k , we then prove that there is a unique matrix R k +1 (1 , u ) satisfying thissame equation for n = k + 1. The existence of this matrices often implies certain restrictions betweenthe coefficients a i , b i , c i , d i . G solutions for Problem 2 First of all, we will discuss the necessity of the conditions imposed for the function F in Theorem 3.Usually, for most of the particular cases appearing in the bibliography of the F -chordal Problem, in theequation F ( a, b ) = k P we can write b explicitely as a function ϕ P ( a ). For example: • For the
Equichordal Problem , F ( a, b ) = a + b = k P , so we can consider that b = k P − a . • For the
Equiproduct Problem (except in the trivial case k P = 0), F ( a, b ) = a · b = k P , and againwe can take b = k P a .But, what can we do in the rest of the cases? Remark 7
For an arbitrary function in two variables F ( a, b ) , the Implicit Function Theorem, togetherwith condition (ii) implies that in a neighbourhood of ( k P − V k , k P − V k ) , ( k Q − V k , k Q − V k ) wecan find two real functions ϕ P , ϕ Q such that F ( | A − P k , k B − P k ) = k P ⇐⇒ k B − P k = ϕ P ( k A − P k ) F ( | A − Q k , k B − Q k ) = k Q ⇐⇒ k B − Q k = ϕ Q ( k A − Q k ) Moreover, condition (i) ensures that ϕ P , ϕ Q are C ∞ functions. We will denote the corresponding Taylorseries as: ϕ P ( a ) = ϕ P + ϕ P ( a − k P − V k ) + ϕ P ( a − k P − V k ) + . . .ϕ Q ( a ) = ϕ Q + ϕ Q ( a − k Q − V k ) + ϕ Q ( a − k P − V k ) + . . . Now we can re-state conditions ( iii ) , ( iv ) in terms of these functions ϕ P , ϕ Q . See that: ϕ P = − ∂F∂a | ( k P − V k , k P − V k ) ∂F∂b | ( k P − V k , k P − V k ) , ϕ Q = − ∂F∂a | ( k Q − V k , k Q − V k ) ∂F∂b | ( k Q − V k , k Q − V k ) , so condition ( iii ) implies ϕ P , ϕ Q = 0 , and condition ( iv ) implies that (4.1) ∀ n ∈ N , (cid:18) ϕ Q ϕ P (cid:19) n = k V − Q kk V − P k k V − P kk V − Q k = ϕ Q ϕ P x − x + 1The relation between Problem 2 and Problem 1 arises from the fact that each F -chordal point inducesan involutive correspondence between points in c . The image of a point A through this correspondenceis the other point lying in the intersection between c and the line through the F -chordal point and A .And thank to this, we have that: 6 emark 8 Let c be a curve with two F -chordal points P, Q . Any parametrization γ : ( − ε, ε ) → c with γ ( t ) = ( x ( t ) , y ( t )) of c in a neighbourhood of one of the vertices V induces two (one for each F -chordalpoint) parametrizations of c in a neighbourhood of V . γ P ( t ) = P + ϕ P ( | P − ( x ( t ) , y ( t )) k ) P − ( x ( t ) , y ( t )) k P − ( x ( t ) , y ( t )) k γ Q ( t ) = Q + ϕ Q ( | Q − ( x ( t ) , y ( t )) k ) Q − ( x ( t ) , y ( t )) k Q − ( x ( t ) , y ( t )) k Figure 2: This picture may help to understand this double parametrization and the correspondencebetween points in c . So if we want a solution of Problem 2 to be G n near V , it should exist a C n regular reparametrization u ( t ) , such that u (0) = 0 and such that satisfies (4.2) γ P ( u ( t )) = γ Q ( t )Equation (4.2) is a restrictions for the coefficients of the Taylor polynomial of degree n of x ( t ) , y ( t ).And we can state (4.2) as a functional equation, suitable to be expressed in terms of Riordan matrices,as done in Remark 6. This leads to the following: Remark 9
From now on, to study (4.2) , we will take P = (1 , , Q = ( − , , V = ( x , with x > .In the notation of the previous remark, to find a G n solution for Problem 2, we need to solve ∀ k ≤ n the following system of two matricial equations: (4.3) ... + R n (1 , u ) R n (1 − x, p (1 − x ) + y − ( x − ↑ Coefficients of ϕP (( x − t ) x − t ↓ == R n ( − − x, p ( x + 1) + y − ( x + 1)) ↑ Coefficients of ϕQ (( x t ) x t ↓ R n (1 , u ) R n ( y, p (1 − x ) + y − ( x − ↑ Coefficients of ϕP (( x − t ) x − t ↓ == R n ( y, p ( x + 1) + y − ( x + 1)) ↑ Coefficients of ϕQ (( x t ) x t ↓ Proposition 10
Let c be G solution for Problem 2, with vertices V , V , interior F -chordal points P, Q , and such that F satisfies conditions (i)-(iv) from Theorem 3. Then the tangent vector of c at V is either parallel, either perpendicular to the axis of c .Proof: Suppose that we have a parametrization γ ( t ) = ( x ( t ) , y ( t )) such that γ (0) = V = ( x , t = 0 are T aylor ( x )( t ) = x + x t T aylor ( y )( t ) = y t Then the case n = 1 of (4.3) is " + " u − x − x (1 − x ) x ϕ P x − − ϕ P ( x − + ϕ P ( x − == " − − x − x ( − − x ) x ϕ Q x +1 − ϕ Q ( x +1) + ϕ Q ( x +1) u y ϕ P x − − ϕ P ( x − + ϕ P ( x − = " y ϕ Q x +1 − ϕ Q ( x +1) + ϕ Q ( x +1) where T aylor n ( u )( t ) = u t . This system leads to a system of 4 equations, each of them correspondingto one of the entries of the column vectors of length two obtained in each side of each matricial equation: − ϕ P = − ϕ Q − u ϕ P x = − ϕ Q x u y ϕ P x − = y ϕ Q x +1 The first and third equations are trivial, so we only need to matter the other two. Taking into accountthat γ ( t ) is a regular parametrization and so γ ′ ( t ) = ( x , y ) = (0 , • Case 1: x , y = 0, which is not possible since implies a contradiction with condition ( iv ) (seeEquation (4.1), which recall that is a version of condition (iv) in terms of ϕ P , ϕ Q ) ( u ϕ P = ϕ Q u ϕ P x − = ϕ Q x +1 ⇒ ϕ Q ϕ P = ϕ Q ϕ P x − x + 1 • Case 2: x = 0 , y = 0 ( u ϕ P = ϕ Q ⇒ u = ϕ Q ϕ P • Case 3: x = 0 , y = 0 ( u ϕ P x − = ϕ Q x +1 ⇒ u = ϕ Q ϕ P x − x + 1 ✷ Proof of Theorem 3
First of all, we need to check that Case 2 in the end of the previous proof does not correspond to anyanalytic solution of Problem (2).
Proposition 11
Let F satisfying conditions (i)-(iv) of Theorem 3. An analytic curve c that satisfies (4.3) for every n ∈ N is a line segment contained in the line through P, Q . So it does not correspond toany analytic solution of Problem 2, which should be a Jordan curve with
P, Q in its interior region.Proof:
We are going to do the proof by induction. This argument is similar but more simple thatthe one of Theorem 3. The base case has already been considered in Proposition 10. Let x + x t + . . . + x k +1 t k +1 y t + . . . + y k +1 t k +1 be the Taylor polynomials at t = 0 of x ( t ) , y ( t ) respectively. Assume that, for some k ≥
1, if y , . . . , y k =0 and we have some x , . . . , x k , u , . . . , u k that are a solution of (4.3) for n = k . Then the secondequation of the system (4.3) for n = k + 1 is of the type: u ... ... . . .0 u k +1 . . . u k +11 . . . y k +1 . . . ↑ Coef f icients of ϕ P (( x − t ) x − t ↓ == . . . y k +1 . . . ↑ Coef f icients of ϕ Q (( x +1)+ t ) x +1+ t ↓ and we are going to see that it implies that y k +1 = 0.The result in each side of the matricial equation is a column vector. The equality between the twolast entries in each is: u n +11 y n +1 ϕ P x − y n +1 ϕ Q x + 1 ⇒ ( u n +11 ϕ P x − − ϕ Q x + 1 ) y n +1 = 0The only solution of the above equation is y n +1 = 0, since u = ϕ Q ϕ P and so: u n +11 = ϕ Q ϕ P x − x + 1This shows that the case x = 0 , y = 0 implies the Taylor series of y ( t ) at 0 equals 0. And so, ina neighbourhood of V , c is a segment contained in the line through P, Q . Since c is analytic, by the Principle of Analytic Continuation c must be a line segment, which cannot be a solution for Problem2. ✷ Now that we have discard Case 2, we now that any analytic solution for Problem 2 has its tangentvector at any of its vertices perpendicular to its axis, and we can complete the proof of Theorem 3.
Proof of Theorem 3:
Let the Taylor series of x ( t ) , y ( t ) (recall that γ ( t ) = ( x ( t ) , y ( t )) is a parametriza-tion of an analytic solution c in a neighbourhood of V ) be respectively x ( t ) = x + x t + . . . y ( t ) = y t + y t + . . .
9e know (Proposition 10) that, since the tangent vector of c at V is perpendicular to the axis, then(4.3) for n = 1 implies that u = ϕ Q ϕ P x − x + 1 .We are going to prove by induction in k the following statement: for each choice of x > V ), y = 0, and y , . . . , y n , there exists a unique choice of x , . . . , x k , u , . . . , u k such that (4.3) holds. This determines univocally γ ( t ) up to reparametrization (this is thereason of the freedom of the parameters y , . . . , y n , . . . ) and thus c , according to the Principle of AnalyticContinuation .Although we have already studied the case k = 1, for a better understanding of this proof, we includethe case k = 2 which is the first significant one. In this case (4.3) is: + u u u − x − x [( x − x − y ] 0 ϕ P x − a a == − − x − x | [( x + 1) x + y ] 0 ϕ Q x +1 b b u u u y y ϕ P x − a a = y y ϕ Q x +1 b b where ϕ P (( x − t ) x − t = ϕ P x − + a x + a x + . . . and ϕ P (( x +1)+ t ) x +1+ t = ϕ Q x +1 + b x + b x + . . . Each matricial equation lead to a system of 3 linear equations in the indeterminates x , x , u , u :(5.1) − ϕ P = − ϕ Q − u ϕ P x = − ϕ Q x − u ϕ P x − x + [ u a (( x − x − y )] == − ϕ Q x +1 x + [ b (( x + 1) x + y )](5.2) u y ϕ P x − = y ϕ Q x +1 ϕ P x − y u + [ u ϕ P x − y ] = [ ϕ Q x +1 y ]We have already discussed in Proposition 10 the values of u that make the two first equations in eachsystem to hold. The last equation in (5.2) does not depend on x , and has nontrivial coefficient of theindeterminate u (the coefficient is y ϕ P x − u ) for the indeterminate x (the coefficient is − ϕ P x − u + ϕ Q x + 1 ) andso it has a unique solution in the indeterminate x too.Now assume that the statement is true for k − ≥
2. We want to solve (4.3), for n = k . The secondmatricial equation in (4.3) is of the type: u ... ... . . .0 u n . . . u k y y k . . . . . . 0 ϕ P x − a ... a k = y y k . . . . . . 0 ϕ Q x +1 b ... b k u k of the form: y ϕ P x − u n + [ C ] = [ C ]where nothing in the brackets depend on u k , x k (they do on y k ). So we have a unique solution on theindeterminate u k that makes this equation hold. On the other hand, the first matricial equation in(4.3) is of the type: + u ... ... . . .0 u k . . . u n − x − x k . . . . . . 0 ϕ P x − a ... a n = − − x − x k . . . . . . 0 ϕ Q x +1 b ... b k The equation corresponding to the last entry in the column vector is: − u k ϕ P x − x k + [ C ] = − ϕ Q x + 1 x k + [ C ]And nothing in the brackets depends on x k (they do on y k , u k ). The number ϕ Q x +1 − u k ϕ P x − is not zeroaccording to the hypothesis of the theorem. So this equation has a unique solution in the indeterminate x k . ✷ First of all, we want to point out that in the proof of Theorem 3 we have not used the fact that c mustbe the boundary of a convex region, neither a Jordan curve. This theorem still holds for a more generaldefinition of interior F -chordal point: Definition 12
We say that P is an interior F -chordal point of a curve c if there exists k P such thatfor every line through P either (1) l does not intersect c or (2) l meets c at two points A, B satisfying F ( k A − P k , k B − P k ) = k P and such that P is in the interior of the segment AB . For example, for F = a − b , the center of symmetry P of any hyperbola is an interior F -chordal pointin this sense, with k P = 0. With this definition, if we have two point P, Q the line through them canalso be considered to be a solution of Problem 2 (modifying the statement of Proposition 11).Secondly, we want to remark that, with almost the same proof, we can obtain this more generalversion of Theorem 3 that will be required in this section:
Theorem 13
Let four different collinear points V , P, Q, V , where P, Q are between V , V . Let F :[0 , ∞ ) → R be a symmetric function in two variables, satisfying the following conditions:(i) it is C ∞ ,(ii) ∂F∂b | ( k P − V k , k P − V k ) , ∂F∂b | ( k Q − V k , k Q − V k ) = 0 ,(iii) ∂F∂a | ( k P − V k , k P − V k ) , ∂F∂a | ( k Q − V k , k Q − V k ) = 0 .(iv*) ∀ n ∈ N , n ≥ , ∂F∂a (cid:12)(cid:12) ( k Q − V k , k Q − V k ) · ∂F∂b (cid:12)(cid:12) ( k P − V k , k P − V k ) ∂F∂a (cid:12)(cid:12) ( k P − V k , k P − V k ) · ∂F∂b (cid:12)(cid:12) ( k Q − V k , k Q − V k ) ! n = k V − Q kk V − P k k V − P kk V − Q k hen if we fix the tangent direction at V , if there exists an analytic solution for the interior F -chordalProblem with interior F -chordal points P, Q , vertices V , V and this given tangent direction at V , thissolution must be unique.Proof: In Proposition 10 the case x , y = 0 cannot be excluded, but anyway u = ϕ Q ϕ P x − x +1 . Laterin the proof of Theorem 3, one of the matrices in the system (4.3) is different but it does not affect theargument. ✷ Finally we are going to collect some consequences of theorems 3, 13. Several particular cases of F -chordal points have been studied in the bibliografphy. We have already discussed about equichordalpoints ( F ( a, b ) = a + b ). In Figure 1 we can see on the left a disk, for which any point in the interior isan equiproduct point ( F ( a, b ) = a · b , see [4, 5, 7, 20] for more information) and on the right an ellipse, forwhich any of its two focus is an equireciprocal point ( F ( a, b ) = a + b , see [10]). In general, the familyof F -chordal points for F ( a, b ) = a α + b α has also be considered, for α ∈ R (see [4]).The following three results are a direct consequence of Theorem 3: Theorem 14 (concerning the Equichordal Problem)
For every four collinear points V , P , Q , V , if it exists an analytic curve with equichordal points P, Q and vertices V , V , this curve is unique. Theorem 15 (concerning the Equireciprocal Problem)
For every four collinear points V , P , Q , V , if it exists an analytic curve with two equireciprocal points, this curve is unique.If k V − Q k = k V − P k , the ellipse with foci P, Q and major axis the segment with endpoints V , V is this unique curve (see [5, 10] to see that such an ellipse has these properties). Theorem 16 (concerning the F -chordal Problem for F ( a, b ) = a α + b α ) For every four collinearpoints V left , P, Q, V right , for F ( a, b ) = a α + b α , α = 0 , if it exists an analytic curve with two F -chordalpoints, this curve is unique. And this last one is a consequence of Theorem 13:
Theorem 17 (conerning the Equiproduct Problem)
For every four collinear points V , P , Q , V , the circles that pass through V , V are the unique analytic curves with F -chordal points P, Q andvertices V , V . In relation to Theorem 14, the fact that it does not contradict the Theorem by Rychlik in [14]deserves a little explanation. The author already discussed in [15] that the Equichordal Problem hada local analytic regular solution, pointing out the Helfenstein was wrong in his article [9]. From thelocal point of view, the family of all the interior F -chordal problems studied here behaves in a similarway: we have a unique candidate (up to reparametrization) for the power series of a parametrizationnear a vertex. To study wether those local solutions can be extended or not to solutions of Problem2 needs other type of global techniques. For example, for the Equichordal Problem , Rychlik provedthat extension cannot be done becouse of the hiperbolicity of the problem. But this is not the case ofthe Equireciprocal Problem, for instance. The techniques used in this article are not suitable for thisglobal analysis. Anyway, maybe the desciption of the coefficients of the local solution (specially the firstterms) appearing in the proof of Theorem 14 could be use in the search for a more simple proof of theresult by Rychlik, which remains open for the classical statement of the problem (where the solutionsmust be the boundary of a convex region). 12igure 3: Any parametrization γ ( t ) of a neighbourhood of the vertex V (any of the two points wherethe line from P is tangent to c ) induces another parametrization γ P ( t ) in the same neighbourhood Finally we will state two other classical problems in plane geometry. The techniques appearing in thisarticle seem to be suitable to prove uniqueness of analytic solutions for them, but an improved argumentmay be required. Those questions are open, up to our knowledge.There exists another version of the F -chordal Problem, that we could call the Exterior F -chordalProblem which could be stated as follows. For a convex region B with boundary c , we say that a point P in the exterior region of c is an exterior F -chordal point if there exists a constant k P such that forevery chord with endpoints A, B in c , F ( | A − P | , | B − P k ) = k P . Problem 18 (one point and two points exterior F -chordal Problems) For a given symmetricfunction in two variables F defined in [0 , ∞ ) × [0 , ∞ ) :(a) find a plane Jordan curve which interior region is convex with one exterior F-chordal point,(b) find a plane Jordan curve which interior region is convex with two exterior F -chordal points. Some results are known concerning particular cases of this problem, see for example [20] for the
Equiprod-uct Problem case. For these problems, the double parametrization, analogous to the one for the Interior F -chordal Problem that appears in Remark 8, can be obtained with only one exterior F -chordal point(we omit the details, but we offer a picture, see Figure 3). So it makes sense to study using our tech-niques the One Point case. Again, a new definition is possible for exterior F -chordal point , not requiring c to be a Jordan curve.On the other hand, we have a problem related to Geometric Tomography, which is a field thatfocuses on problems of reconstructing plane regions from tomographic data. The term was introducedby R. J. Gardner in the book [7]. We could state one of the main and most simple problems in thisfield as follows. Problem 19 (One Point, Two Points, One Line or Two Lines Tomographic Reconstruc-tion Problem) A tomographic image from a point P in the exterior of a convex region B is a real function f P suchthat f ( θ ) is the length of l θ ∩ B , where l θ is the line passing through P and with angle θ with respectthe axis OX . Given one or two tomographic images from a point, find a Jordan curve c such that theinterior region of c is convex and has this or these tomographic images.Equivalently, the tomographic image from the OY axis r , is a real function g r such that g r ( t ) is thelength of the segment l t ∩ B , where l t the horizontal line which y coordinate equals t . The analoguecan be defined for any line with the corresponding modifications. Given one or two tomographic imagesfrom a line, find a Jordan curve c such that the interior region of c is convex and has this or thesetomographic images. B , assuming that it is G k for some k . References [1] Barsky, B. A. and DeRose T. D.: Geometric Continuity of Parametric Curves: Three Equivalent Characterizations,IEEE Computer Graphics and Applications 9.6, 60-69 (1989).[2] Barsky, B. A. and DeRose, T. D.: Deriving the beta-constraints for geometric continuity of parametric curves, Rendi-conti del Seminario Matematico e Fisico di Milnano, 63.1 (1993).[3] Blaschke, W., Rothe, W., and Weitzenb¨ock., R. Aufgabe 552. Arch. Math. Phys., 27-82 (1917).[4] Croft, H., Falconer, K. J., and Guy, R. K.: Unsolved Problems in Mathematics, Vol. II, Unsolved Problems in Geometry.Springer-Verlag, New York (1991).[5] K. J. Falconer, K. J.: On the equireciprocal point problem, Geom. Dedicata 14,113126 (1983).[6] Fujiwara, M.: ¨Uber die Mittelkurve zweier geschlossenen konvexen Curven in Bezug auf einen Punkt, Tohoku MathJ. 10, 99103 (1916).[7] Gardner R. J.: Geometric Tomography. Cambridge University Press, New York (1995).[8] Gardner, R. J., Kiderlen, M.:
A solution to Hammer’s X-ray reconstruction problem , Advances in Mathematics 214(2007).[9] Helfenstein, H. J.:
Ovals with equichordal points , J. London Math. Soc. 31, 5457, (1956).[10] Klee, V.: Can a plane convex body have two equireciprocal points? American Mathematical Monthly 76, 5455 (1969).[11] Kiciak, P.: Geometric Continuity of Curves and Surfaces, Synthesis Lectures on Visual Computing: Computer Graph-ics, Animation, Computational Photography, and Imaging 8.3 (2016).[12] Luz´on, A., Merlini, D., Mor´on, M. A., Prieto-Mart´ınez, L. F., and Sprugnoli, R.: Some inverse limit approaches tothe Riordan group, Linear Algebra and its Applications 491, 239-262 (2016).[13] Rosenbaum, J., Amer. Math. Monthly 53, 36 (1946)[14] Rychklik, M. R.: A complete solution to the equichordal point problem of Fujiwara, Blaschke and Weizenb¨ock, Inv.Math 129, 141-212 (1997).[15] Rychklik, M. R.: Why is Helfenstin’s claim about equichordal points false?, New York J. Math 18, 499-521 (2012).[16] Shapiro, L., Getu, S., Woan, W. J. and Woodson, L. C.: The Riordan Group, Discrete Applied Mathematics 34,229-239 (1991).[17] Shapiro, L: A Survey of the Riordan Group, lecture document from the author at Nankai University (2005), availableat http : %20 Group.pdf (link verified in July 2019).[18] Volcic, A.:
A three-point solutions to Hammer’s X-ray problem , J. Lodon Math. Soc. 34 (1986).[19] Wirsing, E.: Zur Analytisitat von Doppelspeichkurven, Arch. Math. 9, 300307 (1958).[20] Zuccheri,L: Characterization of the circle by equipower properties, Arch. Math. 58, 199-208 (1992)., J. Lodon Math. Soc. 34 (1986).[19] Wirsing, E.: Zur Analytisitat von Doppelspeichkurven, Arch. Math. 9, 300307 (1958).[20] Zuccheri,L: Characterization of the circle by equipower properties, Arch. Math. 58, 199-208 (1992).