Harmonic curves and the beauty of Projective Geometry

A conic curve or section is defined as the intersection of a circular cone with a plane. This concept is inherently metric. However, that conic curves somehow belong to projective geometry is well known, but not clear how. For example, the statement that ellipses are what we see when we look at a circle, which is how we should present them to children, makes many of us unconfortable because we didn’t hear it in class and don’t have a simple proof at hand. Trying to make sense of this incongruent state of affairs, we were lead to define harmonic curves in a purely projective style (they are basically the locus of points in the plane that see a quadrangle as a harmonic set) and study their properties without any reference to metric concepts. Yes, harmonic curves become conic sections in the euclidean plane, but they have a rich set of purely projective properties so that their study constitutes a beautiful approach to projective geometry. Such approach is what this paper is all about.

Motivated by the technics of perspective drawing, which were discovered by the painters of the renaissance to represent realisticly three dimensional scenes on flat canvases, Girard Desargues (1591-1661) initiated the development of Projective Geometry expanding the concept of space to include ideal points (also known as points at infinity), [6]. Mostly forgotten through the following two centuries, his visionary work was revived and recognized as fundamental during the first half of the XIX century, when Projective Geometry was firmly established as a field on its own. In 1872 Felix Klein opened his famous Erlangen Program, [9], with the statement “Among the advances of the last fifty years in the field of geometry, the development of Projective Geometry occupies the first place”. One of the mathematicians whose work deserved such praise is Karl Georg Christian von Staudt (1798-1867). In his treatise on the subject [13], he proves that the notions of harmonicity, i.e. the concepts of harmonic sets and pencils, are entirely projective and independent of metric ones like distances or angles. He also shows that conic curves may be defined through the abstract, and purely projective concept of polarity.

In an attempt to make these facts accesible to high-school teachers, and inspired by [11] where John Stillwell argues that such goal is not only worthwhile and culturally urgent, but also feasible, the authors of this article developed the dynamic geometry system ProGeo3D [1] which is specialized in projective geometry. In particular, it incorporates harmonicity as a construction tool. All the figures of this paper are snapshots of constructions in ProGeo3D. To appreciate their dynamic capacity and the 3D nature of some (e.g., Figure 2.a), we present them in a web site¹¹1https://arquimedes.matem.unam.mx/harmonic_curves/scenes.html. Playing with this system, we discovered several interesting constructions, proofs and new results that do not seem to be included in the known literature on the subject. This paper presents them to propose an alternative approach to projective geometry which is intuitive, synthetic and, in our subjective opinion, beautiful.

We start by reviewing von Staudt’s definition of harmonicity, emphasizing the duality it intrinsically carries. That leads us to a simple definition of what we call harmonic curves to differentiate them from the classic treatment of conic sections, although in euclidean space they are the same. In the following section, we relate them to von Staudt’s definition, which uses polarities, but now it must be stated as a theorem, the “Polarity Theorem” (Thm. 2). Its proof uses the classic idea in projective geometry of going out to 3D, finding a way to work out things there, and coming back to 2D. The first to use our explicit technic out in 3D, was Germinal Pierre Dandelin (1794–1847) to prove Pascal’s Hexagon Theorem in [5]. It is deeply related to ruled surfaces as defined by Hilbert and Cohn-Vossen in [7] using only incidence geometry. This suggests the formulation of a new axiom for Projective Geometry, which we call the Equipal Axiom. It is shown to be equivalent to Pappus’ Theorem and thus to other equivalent axioms frequently used in projective geometry. In the final section we discuss the axiomatic foundations of our approach in which the use of 3 dimensions is fundamental.

One of the seminal contributions of Karl von Staudt was to prove that harmonicity (the notion of harmonic conjugates which had been used since antiquity in terms of distances) only depends on incidence using quadrangles.

A quadrangle, $\mathcal{Q}$, is defined as four points in the projective plane in general position (i.e., no three of them are collinear), called its vertices, together with 4 lines, called its sides, such that their incidence relation is a 4-cycle (each object of one type is incident with two of the other); that is, $\mathcal{Q}$ is defined by its vertices (4 points in general position) together with a dihedral (i.e., a cyclic but non-oriented) order on them which yields the 4 sides.

The term “quadrangle” is adequate because at any vertex, its two incident lines (an “angle”) distinguish two adjacent vertices and thus, it also determines the opposite vertex (as the remaining one); note that the partition of vertices into opposite pairs determines the quadrangle. The center of the quadrangle $\mathcal{Q}$ is the intersection of its two diagonal lines (the lines that join opposite vertices); and its horizon is the line joining the intersection of opposite sides, which are called its two diagonal points.

The notion of a quadrangle is autodual, but in a quadrilateral the stress is given on the 4 sides, so that its dual-center is the horizon of the corresponding quadrangle whose center is the dual-horizon of the quarilateral. However, we will keep the “horizon” and “center” terminology: understood always as a line and a point, respectively. These terms are natural because if a square tile is drawn on a canvas, the center of the tile and the horizon of the square tiling of the plane which it generates should be drawn, respectively, at the center and the horizon of the quadrangle which such a drawing determines.

Four points in general position are the vertices of three quadrangles. Their corresponding centers and horizons form its diagonal triangle.

Four collinear points $A,C,B,D$, as in Figure 1.a, are said to be a harmonic set,²²2The terms “harmonic quadruple” or “harmonic range” are also used, but we stick to “harmonic set” as in the classic texts [12] and [4]. if there exists a quadrangle such that its diagonal points are $A$ and $B$ (hence the horizon of the quadrangle is their support line) and the other pair, $C$ and $D$, are incident with the diagonal lines. Dually, four concurrent lines are called a harmonic pencil³³3The term “harmonic set of lines” is also used, e.g. [12, 4]; but we will use “pencil” for simplicity and to distinguish them immediately from harmonic sets (of points). if there exists a quadrangle such that one pair of lines are the diagonal lines of the quadrangle (hence, its center is the concurrency point of the pencil, also called its center) and the other two lines are incident to the diagonal points of the quadrangle, see Figure 1.b.

As stated, the pairs of elements in the definitions play a different role but, as we will see, they are interchangeable, so that both notions include an explicit dihedral order of the four elements involved, which coincides with their geometric placement (points within a projective line or lines about a point).

Let us now show that these definitions are sound. Given a collinear triple $A,C,B$ with $C$ distinguished, two auxiliary points out of the support line and collinear with $C$ determine a unique quadrilateral $\mathcal{Q}$ as in Figure 1.a, and therefore produce the point $D$ as the intersection of the other diagonal line with the horizon; this construction, called the harmonic fourth, has as outcome the point $D$, called the harmonic conjugate of $C$ with respect to $A$ and $B$. Since for the triple $A,D,B$, one can choose the other opposite pair of vertices of $\mathcal{Q}$ as auxiliary points and then obtain $C$ as outcome, we can further say that the (unordered) pair of points $C,D$ are harmonic conjugates with respect to $A,B$, [4, 12].

The proof is well known and follows from Desargues’ Theorem. It may also be directly proved in 3D using only incidence arguments (see Figure 2.a), and deducing the planar case imediatly from it. We omit the details for brevity.

Finally, to see that the definition of harmonic set is symmetric with regard to the role played by the two pairs of points, extend the quadrangle $\mathcal{Q}$ to a 2 by 2 tiling drawn in perspective, as in Figure 2.b (the know-how comes from the renaissance artists and the coincidences follow from the Harmonic Theorem). Then, the quadrilateral of (dashed) diagonals not incident with its center $O$ proves that $A,B$ are harmonic conjugates with respect to $C,D$.

Since the point $O$ in Figure 2.b may be chosen to be any point not on the support line of the harmonic set, we obtain that any such point sees them as harmonic, that is, the lines to them with their dihedral order is a harmonic pencil. Dually, there is also a harmonic fourth construction for lines and any line not through the center of a harmonic pencil cuts it in a harmonic set. Thus, harmonic sets and pencils are preserved by projections.

The harmonic fourth construction also makes sense in the singular cases when $C=A$ or $C=B$. In them, the outside quadrangle collapses to a line, but the diagonals do not collapse and so the construction holds in the sense of not becoming ambiguous, and it yields $D=A$ or $D=B$, respectively. Therefore, given two (distinct) points $A$ and $B$ in a line $\ell$ we get a well defined map

called the harmonic reflection of $\ell$ with respect to $A$ and $B$: it fixes these two points and it gives the harmonic conjugate elsewhere. $\rho_{A,B}$ is an involution which interchanges the two segments in which the points $A$ and $B$ divide their projective line. And in particular, it interchanges its ideal point at infinity with the (euclidean) midpoint of $A$ and $B$, making the harmonic fourth construction a very useful tool for perspective drawing.

The natural generalization to the projective plane (space) is the harmonic reflection $\rho_{C,m}$ with respect to a point $C$, called the center, and a non-incident line (plane) $m$, called the mirror.⁴⁴4In the plane, Coxeter calls it harmonic homology in [4]. It is defined on every line $\ell$ through $C$ as the harmonic reflection with respect to $C$ and the intersection of $\ell$ with $m$. If we denote by $\vee$, “join”, and $\wedge$, “meet”, the basic projective operations of linear span and intersection, respectivelly, we may write for $X\neq C$:

where note that we write the action of maps or functions on the right. This notion amalgamates two classic euclidean examples: the central inversions, when the mirror is the line (plane) at infinity, and the reflections when the center is the ideal point in the direction perpendicular to the mirror.

Harmonic reflections are collineations (i.e., they send lines to lines). They act in the dual plane as harmonic reflections in the sense that if $\ell$ is a line different from the mirror $m$, then $\ell,m,{\ell\cdot\rho_{C,m}},{(\ell\wedge m)\vee C}$ is a harmonic pencil centered at $\ell\wedge m$.

We define the harmonic curve, $\mathcal{C}_{\mathcal{Q}}$, of a quadrangle $\mathcal{Q}$ as the locus of points that are centers of harmonic pencils transversal to $\mathcal{Q}$. By transversal we mean that each vertex of $\mathcal{Q}$ is incident to a line of the pencil and this correspondence preserves the dihedral orders. Dually, the harmonic bundle of a quadrilateral consists of the lines that support a harmonic set transversal to its sides and with corresponding dihedral orders.

Consider a quadrangle $\mathcal{Q}$ with vertices $A,C,B,D$. First observe that the vertices are points of its harmonic curve $\mathcal{C}_{\mathcal{Q}}$. Indeed, for each vertex, the harmonic conjugate of its diagonal with respect to its sides completes a harmonic pencil centered at it which is transversal to $\mathcal{Q}$, see Figure 4.a. These new lines are the tangent lines to $\mathcal{C}_{\mathcal{Q}}$ at the vertices and will be denoted by the corresponding lower case letter. The harmonic bundle of the quadrilateral $a,c,b,d$ is called the tangent bundle of $\mathcal{C}_{\mathcal{Q}}$ and will be denoted $\mathcal{C}_{\mathcal{Q}}^{*}$.

Now consider a point $Z\in\mathcal{C}_{\mathcal{Q}}$ different from the vertices, we call it generic, see Figure 4.b. By definition, the four lines from $Z$ to the vertices are a harmonic pencil centered at $Z$. Let $X=(A\vee B)\wedge(C\vee Z)$ and $Y=(A\vee B)\wedge(D\vee Z)$. Then $A,X,B,Y$ is a harmonic set. Therefore, we can recover $Z$ from $X\in A\vee B$ by defining

But this makes sense for $X$ ranging over all of $A\vee B$ and gives the four vertices (at a harmonic set), so that $\mathcal{C}_{\mathcal{Q}}$ is parametrized by $X\in(A\vee B)$ via this construction. We call this the HC-construction.

A polarity in the plane (in space) is a bijective correspondence between points and lines (planes) that preserves incidence; the terms polar of a point, pole of a line (plane) or a polar pair are used⁶⁶6An extra hypothesis is required in [4]. Namely, that for some line, the map to the line pencil of its pole be a projectivity. But we do not need to stress this issue..

We have already seen a part of item (i) as Lemma 2 because tangent lines to points in a harmonic curve are defined as their polar lines. The rest of the proof will be given in the next section as a consequence of an analogue in 3D, Theorem 4. For the moment, let us make two remarks about this theorem and, assuming it is true, explore some of its profound consequences.

The two mathematicians that father this theorem are Jean-Victor Poncelet (1788-1867) and Karl G. C. von Staudt (1798-1867). Poncelet proved the relation of poles and polars for conic sections using harmonicity (in its metric version), and soon after, von Staudt developed polarities as a general concept and used it as an alternative way to define conic curves within projective geometry with no metric or algebraic considerations, [13]. This definition via polarities is the one Coxeter uses in his influential book [4], and calls it “extraordinarily natural and symmetrical” because it has duality built into it. In general, there are two types of polarities: euclidian in which no point is incident with its polar line, and hyperbolic when there exist pole and polar incident pairs. The terms used are related to the groups generated by harmonic reflections of non-incident polar pairs. So that von Staudt’s definition of a conic curve is equivalent to item (i) of the theorem for a hyperbolic polarity, while Poncelet’s results can be rephrased as item (ii).

As examples of polar pairs, we have named lines and points in Figure 5 according to the upper and lower case rule for poles and polars with respect to the displayed harmonic curve $\mathcal{C}_{\mathcal{Q}}$. Indeed, a point $Z\in\mathcal{C}_{\mathcal{Q}}$ and the corresponding line in $z\in\mathcal{C}_{\mathcal{Q}}^{*}$ described in Lemma 2 constitute a polar pair satisfying (i).

We now prove that von Staudt’s definition of conic curves with mild extra hypothesis gives harmonic curves.

The following proof of the Polarity Theorem (2) is inspired by Dandelin’s proof of Pascal’s Hexagonal Theorem. Given a conic curve, Dandelin constructs, in [5], a hyperboloid of revolution that has it as a plane section; then, using the fact that this surface is doubly-ruled, he obtains a configuration of 6 lines in three dimensional space associated to the six points of the hexagon in the conic, and argues with the geometric-combinatorial properties of the configuration to conclude the proof. We use the same general idea and get to the same configuration of 6 lines, but instead of hyperboloids of revolution we can now use general ruled surfaces following Hilbert and Cohn-Vossen’s construction of ruled surfaces in [7], which appeared in print almost a century after Dandelin’s proof, and made clear that they can be constructed by simple incidence arguments.

Consider two lines $a$ and $b$ in three dimensional projective space. They touch if and only if they are coplanar. If this is not the case, they can be called a generating pair because for any point $X$ not in them, there exists a unique line through $X$ transversal (i.e., with a common point) to $a$ and $b$; namely:

Now consider three lines $a$, $b$, $c$ in general position (i.e., each pair is generating, or equivalently, no pair of them is coplanar). The transversal ruling to $a,b,c$, denoted $\mathcal{R}(a,b,c)$, is the set of lines that are transversal to them (i.e., that touch all three); any such set of lines will be called a ruling and its elements are called its rules, see Figure 8.a. If we denote $\mathcal{R}=\mathcal{R}(a,b,c)$, the above observation implies that $\mathcal{R}$ is parametrized by incidence with the points in any of the three generating lines (through any point in them there passes a unique rule). It will be important to note that, dually, $\mathcal{R}$ is also parametrized by planes containing one of the lines; if we denote planes by greek letters (points and lines are, respectivelly, upper and lower case latin) we have, for example, that

Every pair of rules in $\mathcal{R}$ is generating, otherwise, their three transversal lines $a,b,c$ would be coplanar. Thus, for any triplet $a^{\prime},b^{\prime},c^{\prime}\in\mathcal{R}$ we get a transversal rulling $\mathcal{R}(a^{\prime},b^{\prime},c^{\prime})$ that contains the original three lines, $a,b,c$; this ruling is an extension of $a,b,c$, see Figure 8.b. In real projective space there is only one extension to a ruling of three lines in general position, but this is not true in all projectives spaces. In what follows we will assume that our projective space does have this property, that is, we will assume it as an axiom:

Later on we will prove it is equivalent to Pappus’ Theorem and thus to other classic statements that are commonly adopted as axioms in projective geometry. Another name for it could be the “Double-ruling Axiom” because it immediately implies that rulings are matched or paired: any ruling has an opposite ruling which is the transversal ruling to any three of its rules. The doubly-ruled surface (we also refer to it simply as a ruled surface) obtained as the union of the rules in any such ruling is also the union of the rules in its opposite ruling.

Hence, every point on a ruled surface has a tangent plane, the one generated by the unique rules through the point in the opposite rulings of the surface.

What we have done in this paper works verbatim in the abstract setting of projective geometry. The included images, which are designed to help the reader to understand intuitively the abstract arguments are, of course, drawn in euclidean 2 and 3 dimensional spaces. However no euclidean arguments are used except for a few statements, e.g., when showing that harmonic curves are conic sections and for the model of the hyperbolic plane (the plane not the group!). Since Projective Geometry has always been ground for considerations about math foundations, we think it is appropriate to close with a few remarks concerning axiomatics and the rigorous mathematical content of our proofs.

The axioms on which our presentation is based and all its theorems are proved are the following. A projective space consists of a ground set, or space, of points with a well defined family of subsets called lines, satisfying:

1. 1.

   Any two distinct points $A$ and $B$ lie on a unique line $A\vee B$.

2. 2.

   If $A,B,C,D$ are four distinct points and lines $A\vee B$ and $C\vee D$ meet, then the lines $A\vee C$ and $B\vee D$ also meet.

3. 3.

   There are two lines that do not meet.

4. 4.

   Lines have more than two points.

5. 5.

   The harmonic fourth of three collinear points is neither of them.

6. 6.

   The Equipal Axiom.

These axioms are a variation of those commonly used (e.g., the ones suggested by Stillwell in [11]). The main difference being the replacement of Pappus’s Theorem by the Equipal (or Double-rulings) Axiom. Axioms 1 and 2 are the fundamental incidence axioms. The statement of Axiom 2 is attributed to Pash and Veblen; it cleverly says that two lines meet if and only if they are coplanar without the need of having planes previously defined. Axiom 3 means the space is at least three dimensional (more than a plane), and it is known to be equivalent to Desargues’ Theorem. Axiom 4 is required for geometry to become interesting and not simply some trivialized set theory.

The two final axioms depend on some further development of the theory; they are not primitive. Axiom 5 guarantees that the ground field is of characteristic different from $2$ or, equivalently, that the geometry does not contain the Fano Plane. The characteristic of a projective space can be defined geometrically using the harmonic fourth construction; essentially from how far can one go in a harmonic sequence without returning. It is needed here to make sense of harmonic curves (and that harmonic reflections are not the identity) because it implies that harmonic sets that have exactly four points do exist. Axiom 6 is a required additional principle for geometry to be rich enough to have a deep relation to other classic branches of mathematics like analysis and topology; bellow, we will discuss the several versions it may adopt.

From the first 4 axioms, flats can be defined as the closed subsets under the operation of taking lines, and then, the dimension of a flat is obtained as one less than the number of points needed to generate it; so that planes are defined as flats of dimension 2, [12]. The incidence properties of planes and lines in a space of dimension 3 are obtained from this; and the Hilbert-Cohn Vossen construction of ruled surfaces follows, making sense of the statement of the Equipal Axiom.

Since the Equipal Axiom is equivalent to Pappus’ Theorem, the arithmetization of projective space yields as ground field a commutative one. In [11] commutativity is proved from Pappus’ Theorem and the ground field is described from scratch. The necessity of the axiom is proved by constructing a projective space over a non-commutative field like the quaternions.

Another widely used version of Axiom 6 is as the uniqueness part of the Fundamental Theorem of Projective Geometry. It is usually stated in the context of planar geometry where Axioms 1 and 2 become appealingly dual (and Axiom 3 is false but replaced by Desargues’ Theorem). A projectivity is defined as the composition of projections between (points in) lines or (lines in) concurrent pencils; they are always bijections. It is not hard to construct a projectivity determined by it’s (arbitrary) effect on (any) three elements of it’s domain. This is the existence part of the Fundamental Theorem. However, the uniqueness is proved to be equivalent to Pappus’ Theorem, so one must be assumed to prove the other, see [4], [12].

We think that Axiom 3 is natural because, among other things, it responds and gives credit to the original motivation for the creation of Projective Geometry which is renaissance perspective, for which dimension 3 is quintessential. But then, if one thinks about projectivities between non coplanar lines in a three dimensional projective space, one is naturally lead to consider ruled surfaces. Indeed, given a projectivity from a line $a$ to a non coplanar line $b$, the set of lines joining a point in $a$ to its image in $b$ turns out to be a ruling. So, the Equipal Axiom is intimately related to the uniqueness of projectivities given by three arbitrary values (the projectivity is determined by the extended ruling of three lines). Moreover, this association of a set of lines to a map between lines is also a classic idea. It is the dual of how Jakob Steiner (1796–1863) defined conic curves in a purely projective manner; and is a natural, visual way of presenting them, see e.g., [12].

Projective geometry is remarkable in many ways. One of them is the importance of some mathematical notions that were worked and experimented within it long before their abstract general acknowledgement. For example, projectivities were masterly used almost one century before the notion of sets and the language of abstract functions was stablished; moreover, together they constitute what we now call a groupoid (defined in the mid XX century within category theory). And of course, there is the leading role it played in broadening our notion of geometry and its influence on the dawn of topology. There are many ways to approach it and present it. We hope this paper contributes to the awareness of its cultural significance and the convenience and possibility of finding “its way down into secondary schools”, [8], or at least into early undergraduate courses.

Acknowledgments J. Bracho acknowledges support from PAPIIT-UNAM Project IN109023.

The authors declare that there is no conflict of interest.