The Parabola: Section of a Cone or Locus of Points of a Plane? Tips for Teaching of Geometry from Some Writings by Mydorge and Wallis

Abstract

This article proposes a possible path devoted to upper secondary school and early university students, as well as training teachers, with the aim to build a conscious approach to the learning/teaching of the conics, which uses, for an educational purpose, the close relationship between conics as loci of points of a plane and conics as sections of a cone. In this path, we will refer to some elements taken from the history of mathematics relating to a particular conic: the parabola. These elements could help students to discover and realize the transition from a parabola considered as a curve in a plane to the same parabola considered on a cone of which it is a section, as well as the inverse passage, and to grasp the profound link between two presentations of the same geometric object. Both steps will be carried out through constructions made with the use of the GeoGebra dynamic geometry software. In addition, it will be highlighted how the construction of conics by points has allowed the creation of lenses and mirrors, which represents a practical application of geometry very relevant to physics and astronomy. Such a practical approach could help students to overcome the difficulty in understanding conics by making the argument less abstract. Moreover, this path could build up an environment in which teachers and students could explore some semiotic registers and their changes, through which Mathematics expresses itself. In the final part, an educational experiment of the path that was proposed will be shown to the students of the Master’s degree course of “mathematics education” at the Department of Mathematics and Computer Science of the University of Calabria. The results of this experiment are described in detail and seem to confirm that the twofold view of the parabola as a section of a cone and as locus of points of a plane helps the students in understanding its meaning in both theoretical and applicative fields.

1. Introduction

The didactic path proposed in this paper is designed for both upper secondary school and university students and is aimed at highlighting the existing connection between conics as loci of points of a plane and conics as sections of a cone.

Generally, when a parabola is presented in the classroom starting from a plane it is said that it is the locus of the points of the plane equidistant from a given point called focus and from a given straight line called directrix.

To present a parabola starting from a cone, on the other hand, implicit or explicit reference is made to proposition 11 of the first book of Apollonius’ Conics [1], for which a parabola is a section obtained by intersecting a cone with a plane passing through its axis and then with a second plane, such that its intersection with the base of the cone is perpendicular to the base of the triangle for the axis and the diameter of the section is parallel to one of the sides of the triangle for the axis.

In each of the two presentations, however, the ancient link between them often remains in the shadow. To highlight that they are relative to the same geometric object (parabola), it is necessary to show that, in the passage from the cone to the plane and from the plane to the cone, the parabola maintains its properties unchanged.

This problem has important consequences from an educational point of view. Several authors have evidenced many difficulties, for both the students and the teachers, with this kind of approach. Some of the difficulties come from the fact that the students often consider the parabola as an “abstract” concept, sharing no connection with everyday life and having nearly no use in it, and therefore as unimportant [2]. Other difficulties come from the fact that students identify conics merely in a graphical context, without recognizing their characteristic properties or their geometric/algebraic relations [3]. Lara Torres (2018) [4] underlines that the students mostly seem to have problems understanding the parabola as a locus of points and stress the fact that many teachers have similar problems themselves, in the context of Latin-American countries.

The latter remark was further developed by Advíncula Clemente et al. (2021) [5]. They insist on the fact that not only students, but even teachers have often problems in understanding basic properties of the conics and are not able to transmit those notions to the students because of this problem. They proposed the creation of questionnaires (following the MTSK approach) to evaluate the knowledge of Latin-American teachers on the parabola as a section of a cone. Interestingly enough, Montes and Gamboa (2018) [6] and Villareal-Villa et al. (2019) [7], all underline the teachers, especially older ones; although acknowledging the importance of technological tools in mathematics education, they seldom use them due to their inability to manage them.

All those examples taken from the literature highlight that both students and teachers perceive the parabola and its algebraic/geometric relations as an “abstract” concept, not easily identifiable in “real” life. On the other hand, the identification of the parabola as section of a cone might help to overcome these difficulties, since the cone is a 3D figure commonly encountered in everyday life. Unfortunately, the treatment of conics as section of the cone is limited, at both high school and university level, to the pure geometric representation of the intersection of the cone with a plane, in different configurations, without specifying that the same metric relations can be read in an algebraic context, and they hold unchanged when considering the conic both as a plane curve or a section of a cone.

The educational impact given by the use of different views of the same mathematical object was especially developed by Duval (2018) [8] in his base theory of registers of semiotic representation. Following this theory, Valbuena-Duarte et al. (2021) underline the importance of the changes in semiotic representation theory of registers as “a mediator for the study of conics and the acquisition of new knowledge that allows the mobilization of mathematical knowledge from perspectives other than graphic approach or common language” [9], p. 181. A similar conclusion is drawn by da Silva and Moretti (2018) [10], who reveal the importance of recognizing the same mathematical object in different semiotic registers.

The previously cited references to the literature, and the evidence concerning the problems of the teachers, have supplied further motivations to the development of this work. Boero and Guala (2008) [11] illustrate and validate the central role, in the training of teachers, of their competence concerning the Cultural Analysis of the Content (CAC), with the goal to put forward the need to consider epistemological, historical and anthropological aspects of the mathematical content. The practice of CAC suggests not only to take into account how the content is addressed in current mathematicians’ work, but also its historical evolution and how it is addressed in both academic and non-academic contexts, according to different traditions.

The didactic path shown in this paper reflects some cultural and practical instances that, in some moments of history of mathematics, have given significance and sense to the study of the conics and, in fact, can be adequately taken into consideration in a didactic context and could motivate the interest of both students and teachers towards this argument.

Jankvist, in his excellent article [12], develops critical tools to characterize the ways in which one can organize and structure the discussion on the “whys” and “hows” to use history of mathematics in its teaching and learning, as well as the interrelations among the arguments supporting the use of the history and the different approaches of such use. In respect to his categorization, the material and the path presented here propose a “modules” approach (“instructional units devoted to history, … based on cases”, p. 246) to the mathematical concepts, through the use of original sources, where history of mathematics takes the role of a tool to convey a conscious knowledge, with the help of empirical support, to point out that the students can effectively gain advantage when making use of the history of mathematics in the classroom. Jankvist himself introduces this terminology as equivalent to the “historical packages” introduced by Tzanakis et al. (2002) in their “milestone” paper [13], namely as “a collection of materials narrowly focused on a small topic, with strong ties to the curriculum, suitable for two or three class periods, ready for use by teachers in their classroom” (p. 217). The approach followed in the suggested path is indeed an instructional unit, based on historical sources, devoted to clarifying the relations between the concept of parabola as both section of a cone and plane curve which, through the use of GeoGebra, can be directly employed by the teacher in the classroom “as is”, without the additional use of other tools.

As regards the passage from the parabola as a section of a cone to the parabola as a curve in a plane, an example in this direction can be found in the treatise Prodromi catoptricorum et dioptricorum sive Conicorum operis ad abdita radii reflexi et refracti mysteria praevij et facem praeferentis. Libri quatuor priores [14] of 1639 by Claude Mydorge (1585–1647).

Instead, concerning the inverse passage from the parabola as a curve in a plane to the parabola as a section of a cone, an example in this direction can be found in the De Sectionibus Conicis Tractatus [15] of 1655 by John Wallis (1616–1703).

The knowledge of how these steps have historically occurred can help us to understand them better [16] and to promote awareness of a misperception of a lack of relationship between them, thus avoiding the persistence of any misconceptions relating to mathematical concepts [17]. Furthermore, knowing how a concept was born and how it has evolved over time enhances the knowledge of that concept itself, with useful feedback in teachers’ training [18,19]. In this way, teachers can enrich their cultural background [20] and find in the history of mathematics suggestions on specific contents and on appropriate ways of communicating the same concepts [21,22]. Many authors have highlighted the importance of the history of mathematics for teaching, both for its cultural role [23,24] and for the material it provides [25]. It was just from this material that the two examples of the proposed path have been drawn. Moreover, Tzanakis et al. (2002) [13] stress the educational importance of the history of mathematics “as a bridge between mathematics and other subjects”, because “history exposes interrelations among different mathematical domains, or, of mathematics with other disciplines” and it “may help to bring out connections between domains which at first glance appear unrelated”, as well as “it also provides the opportunity to appreciate that fruitful research in a scientific domain does not stand in isolation from similar activities in other domains”, whilst “on the contrary, it is often motivated by questions and problems coming from apparently unrelated disciplines and having an empirical basis” (p. 205).

The geometric constructions in these examples have been made using the GeoGebra dynamic geometry software. The use of Dynamic Geometry Software (DGS) favors the constructive aspect of geometry while maintaining deductive accuracy, clarity of hypotheses and consequences relating to the discipline [26,27,28]. Thanks to the DGS, the graphic-construction phase, both before the acquisition of some geometric concepts and properties, and, subsequently, as a verification or in-depth study, can greatly help teaching, as it lends itself to both visualization and exemplification or exploration. The use of traditional tools, such as a ruler and compass, can be simulated by the DGS in order to facilitate the geometric intuition and at the same time increase and stimulate the interest and imagination of the students, helping them in speculation, which is easily verifiable thanks to the immediate computer feedback [27,29,30,31,32].

The article is organized as follows: in the next section, starting from the Proposition I 11 of the Conics of Apollonius [1], with the help of the GeoGebra software, a cone and a parabola will be built on it. With the help of a slider, it will be possible to verify that the points of the parabola built on the cone and considered on a plane all enjoy the same property; that is, they are equidistant from a given point and a given straight line. We will also reflect on how Apollonius derived the metric relationship that expresses the parabola, read geometrically, from the equality of two areas. This allows us not only to introduce the geometric object as the mere intersection of the two 3D objects (the cone and the plane), but also introduces the metric relation which characterizes the parabola and that brings to its equation in the “canonical” form. It is worth noting that none such passage is present in the majority of textbooks, which merely say that the parabola can be obtained as the intersection of the cone with a plane, and then immediately define the curve in the plane as the locus of points which are equidistant from the focus and the directrix. Explaining that the two different definitions are totally equivalent could help to understand the true nature of the object under study in the sense of Duval’s theory [8].

In the next section, a parabola will be constructed by points following the example provided by Claude Mydorge, highlighting how that simple construction was used to make mirrors and lenses of significant use in the physical and astronomical fields. This is a very practical approach that again could help students to overcome that lack of visualization of conics in the real life. Apropos of this, it is important to underline that Mydorge, in its construction of the parabola uses only the notions of focus and vertex, which are easily visible for both students and teachers, much more than the directrix of the parabola. Mydorge uses, in doing this, the same metric relations obtained by Apollonius, thus giving a clear connection between the parabola as a section of a cone and its pointwise construction.

The choice of those two constructions, among the many known in the literature, has been made because they are among the simplest that can be presented to the students, not only with the help of GeoGebra, as shown here, but also with other simple instruments such as cones built in different materials on which the conic sections are drawn (for the Apollonius construction) and with rule and compass (for the points of Mydorge’s parabola). Moreover, it is easier in Mydorge’s construction to directly compare the metric relation with the one found by Apollonius and verify that the properties of Mydorge’s parabola in the plane are the same as that of Apollonius on the cone.

In the third section, starting from a parabola, we will build a cone of which that parabola is a section following the indications of John Wallis.

In the fourth section, the details of an experiment conducted among the students of the Master’s degree course on “mathematics education” of the University of Calabria will be presented, during which they were administered two questionnaires and a series of lectures. The first questionnaire had the aim of providing information on the initial knowledge on conics possessed by the students. Subsequently, the suggested path about the parabola as a section of a cone and as a plane curve was presented during 8 h of lectures. Finally, in the second questionnaire, the opinion of the students about the opportunity to use the double representation of the parabola as a section of a cone and as a plane curve was gathered, as well as their satisfaction level about the lectures. Then, the results of that experiment were analyzed, which seemed to confirm the effectiveness of the proposed approach.

Finally, some conclusions will be drawn.

2. From a Parabola on a Cone to the Plane of Which It Is the Locus of Points

As is known, the conics are presented by Apollonius in the first book of the Conics, in three Propositions (11, 12 and 13) in which it is indicated how to build them on a cone and which metric relation their points must satisfy. In particular, Proposition 11 is devoted to the parabola [1], pp. 14–15:

Proposition I 11. If a cone is cut by a plane through its axis and also cut by another plane cutting the base of the cone in a straight line perpendicular to the base of the axial triangle, and if further, the diameter of the section is parallel to one [lateral] side of the axial triangle, and if any straight line is drawn from the section of the cone to its diameter such that this straight line is parallel to the common section of the cutting plane and of the cone’s base, then this straight line dropped to the diameter will be equal in square to [the rectangular plane] under the straight line from the section’s vertex to [the point] where the straight line dropped to the diameter cuts it off and under another straight line to the straight line between the angle of the cone and the vertex of the section as the square on the base of the axial triangle to [the rectangular plane] under the remaining two sides of the triangle. I call such a section a parabola.

In this proposition, Apollonius provides the indications for building a particular section of a cone, the metric relation that expresses it and gives the name “parabola” to the section.

Following Apollonius’ directions, we build a parabola on a cone using the GeoGebra dynamic geometry software.

Consider, for example, on the plane $z=0$ the circle with center $O\left(0,0,0\right)$ and radius $r$. On it we build the cone with vertex $V\left(0,0,v\right)$ (see Figure 1).

We want to build a parabola on this cone.

We intersect the cone with a plane passing through the axis, for example with the plane $y=0$, and we obtain the triangle $VBC$ as a section.

We then build a second plane that intersects the base of the cone along the $DE$ segment, perpendicular to $BC$ and such that the axis of the section (intersection between this second plane and the cone) is parallel to $VB$ (Figure 2).

The intersection of this second plane with the surface of the cone is the parabola $DFE$ of vertex $F$.

With the help of GeoGebra, teachers will be able to show their students what they might otherwise only imagine. Once the construction of the parabola on a cone has been completed, teachers could make students reflect on how Apollonius comes to determine the metric relationship that expresses the constructed conic (see Figure 3).

We know that $ED\perp BC$ and $FG\parallel VB$.

From point P of the section, we draw the $PH$ segment, perpendicular to the axis of the parabola. It will be $HP\parallel ED$.

We want to prove that $P{H}^2=FH·FM$, where $FM$ is a segment such that $\frac{FM}{VF}=\frac{B{C}^2}{VC·VB}$.

We remind that: $PFH$ is the ordinate to the diameter or ordinate to the axis; $FH$ the distance of $PH$ from $F$; $FM$ the right side of the parabola (perpendicular to the plane of the parabola), the one that Mydorge will call “parameter”; $FG$ the axis or main diameter; $F$ the vertex of the section.

Let us draw $IHL\parallel BC$.

The plane generated by $HP$ and $IL$ is parallel to the plane on which the base of the cone lies, that is to the plane generated by $BC$ and $ED$, and cuts the cone identifying the circle with diameter $IL$.

Since in each right triangle the square built on the height relative to the hypotenuse is equivalent to the rectangle having sides equal to the projections of the two catheti onto the hypotenuse, in the right triangle $IPL$ we have: $H{P}^2=IH·HL$.

By hypothesis we know that $\frac{FM}{VF}=\frac{B{C}^2}{VC·VB}=\frac{BC}{VC}·\frac{BC}{VB}$.

For the similarity of the triangles $IFH$ and $VBC$ we have: $\frac{IH}{IF}=\frac{BC}{VC}$ and $\frac{IH}{FH}=\frac{BC}{VB}$.

For the parallelism of $FG$ and $VB$ we have: $\frac{IH}{IF}=\frac{HL}{VF}$.

It follows that $\frac{HL}{VF}=\frac{BC}{VC}$.

Then: $\frac{FM}{VF}=\frac{HL}{VF}·\frac{IH}{FH}$, i.e., $\frac{FM·FH}{VF·FH}=\frac{HL·IH}{VF·FH}$ and then $FM·FH=HL·IH$.

Since $HL·IH=P{H}^2$, we obtain $FM·FH=P{H}^2$.

Note that $PH$ indicates a distance, for which $P{H}^2$ indicates an area.

This section definition takes into account the first meaning of parabola as an area applied to a segment.

Apollonius in this way derives the relationship that characterizes the parabola (read geometrically) as the equality of two areas.

If we put $PH=x$, $FH=y$ and $FM=k$ we get $x^2=k·y$, which is the equation of a parabola.

3. A simple Construction of Claude Mydorge’s Parabola by Points

Claude Mydorge is a little-known author, whom little is known about, although his name and references to his treatise are frequent in the writings of mathematicians after him. Mydorge was a well-known figure in the cultural and scientific Paris of the years 1625–1645, in particular for his presence and for his skills within the Circle of Father M. Mersenne which, in the same years, gathered the most culturally evolved people at the Convent of the Minimal Friars. Despite this, Mydorge’s person and activity do not find ample space in current literature [33].

Claude Mydorge (1585–1647) was born, lived and died in Paris. He was born into one of the most illustrious and richest families in France, occupying important roles in public administration (Conseiller au Châtelet, Trésorier de France en la Généralité de Amiens).

In fact, he preferred to devote time to the study of mathematics and other related disciplines, such as astronomy and optics.

In Paris he met Descartes, probably in 1625, and in a short time established a solid relationship with him that lasted throughout his life. Descartes considered Mydorge as a “prudent and faithful friend” and the “first mathematician of France in his time”. Mydorge’s knowledge and skills on optical problems and the construction of parabolic, hyperbolic and elliptical lenses were very useful to Descartes.

In 1639, Mydorge published the first part of his treatise on conics, consisting of four books and considered by his contemporaries to be superior to other books on the same topic. The second part of the treatise, also made up of four books, remained handwritten until his death and subsequently its traces have been lost.

A simple construction of the parabola by points is found in Proposition 19 of the second book. The following proposition is premised on it [14], p. 94:

Proposition II 17. Let any triangle ABC be given, right angle at B. Divide AC into two equal parts at point D and draw the perpendicular to AC from D. From point C take the parallel to AB, which intersects the perpendicular to AC, conducted through D, at point E. Divide AB equally at point F. Then E is on a parabola with vertex F and focus A (see Figure 4).

We know that: $\widehat{ABC}=90°$, $AD=DC$, $DE\perp AC$, $CE\parallel AB$, $AF=FB$.

We denote by $G$ the point of intersection of $DE$ with $AB$.

Let us draw $FH\parallel BC$.

$FH$ passes through point $D$ and the angles in $F$ and $H$ are right.

The triangles $FDA$ and $HDC$ are congruent for the second congruence criterion of the triangles (Proposition I 26 of Euclid’s Elements), therefore: $FD=DH$.

For the same criterion, the triangles $FGD$ and $HDE$ are congruent as well, therefore: $HE=FG$.

Let us draw $EI\parallel BC$; then: $HE=IF=FG$.

In the triangle $ADG$, rectangle in $D$, we have: $F{D}^2=AF·FG$, that is $F{D}^2=AF·IF$, that is $4\cdot F{D}^2=4\cdot AF·IF$, which we can write as ${\left(2FD\right)}^2=IF·4AF$.

Since $2FD=FH=EI$, we obtain: $E{I}^2=IF·4AF$.

Therefore, the point $E$ is on the parabola with vertex $F$ and focus $A$.

If $A$ is the focus of a parabola and $F$ its vertex, from the classical definition of parabola given by Apollonius we can say that the square of the applied by the section to the diameter or axis is equal to the product of the distance of the point of application of the ordinate from the vertex by the right side of the section.

In this case, $EI$ is the applied from the section to the diameter or axis, $IF$ is the distance of the point of application of the ordinate from the vertex, $4AF$ is the right side of the section (called by Mydorge “parameter”), therefore $E{I}^2=IF·k$. This relation corresponds to that which was written in the first paragraph as $FM·FH=P{H}^2$, with $FM=k$.

In this case $EI$ corresponds to $PH$ and $IF$ corresponds to $FH$.

As it is easy to understand, introducing the concept of parameter and reflecting on the relationship between parameter and variable leads us to imagine dynamic transformations of the figure. Students can be guided to consciously make the transition from one particular case to another and from a particular case to the generalized case, highlighting the linguistic transformation, effects and descriptive efficacy, gradually acquiring familiarity with forms of increasing complexity and flexibility in interpreting, reflecting, analyzing.

After proving Proposition II 17, we can follow Mydorge in the construction of the parabola, which he presented in Proposition II 19 and that we build up with GeoGebra.

Proposition II 19. Given by position the focus (umbilicum) and the vertex of a parabola, describe the parabola by points in the same plane.

[14], p. 95

Let $A$ be the focus and $F$ the vertex of a parabola (see Figure 5).

Let us draw $AF$ and let us take the point $B$ on it, in such a way that $AF=FB$ holds.

From point $F$ we draw the first perpendicular to $AB$.

From point $B$ we draw the second perpendicular to $AB$.

On the second perpendicular we choose how many points $C_i$ we want.

Let us join each $C_i$ with $A$.

Each segment $A{C}_i$ intersects the first perpendicular to $AB$ (the one conducted by point $F$) in $D_i$.

From the points $C_i$ we draw the parallels to $AB$.

From the points $D_i$ we draw the perpendiculars to the segments $A{C}_i$.

These perpendiculars intersect the parallels to $AB$, conducted by the points $C_i$, at the points $E_i$.

For the first preliminary proposition the points $E_i$ are found on the same parabola with vertex $F$ and focus $A$.

Following Mydorge’s indications, we can build points that are on the same parabola but, once these points have been built, how do we connect them? How do we go from one point to another? These points, Mydorge says, can be joined by drawing a curve aequabili manus ductu, we could say “with a uniform hand”, “with a steady hand”, “continuously”. Using GeoGebra, this tracking can be simulated with the help of a slider.

Once the parabola has been constructed by points, it will be possible to verify that its properties in the plane are the same as those on the cone.

One can verify that the point $E$, built according to Mydorge’s prescriptions, is equidistant from the point $A$ and from the straight line $BC$, namely that $EA=EC$ (the quadrilateral $ECGA$ is a rhombus).

We can use Proposition II 17 by Mydorge to identify the focus and the directrix of the parabola built by Apollonius.

With the help of GeoGebra, one can also show to the students that the points of the parabola built on the cone, according to Apollonius prescriptions, are equidistant from both one same point and one same straight line.

Given the parabola with vertex $F$ and axis $FG$ previously constructed, from any point $P$ on the parabola we can draw the perpendicular to the axis, that will intersect it in the point $W$.

On the extension of $FG$ on the side of $F$, we can take a point $T$ in such a way that $TF=WF$. Let us join $P$ with $T$ and draw the axis of the segment $TP$ which intersects the axis of the parabola in the point $A$.

Let us construct the symmetric of $A$ with respect to $F$ and call it $S$. Then, we draw the straight line $r$, passing by $S$ and perpendicular to the axis of the parabola.

With the help of a slider in GeoGebra, it can be verified that each point $P$ of the parabola is equidistant from the point $A$ and the straight line $r$, which are the focus and the directrix of the parabola, respectively (Figure 6).

Mydorge’s analysis can be considered one of the first moments in which the attention on conics is centered on their construction as plane curves, although we find examples of pointwise constructions of the parabola in ancient authors. For instance, in prop. IV of the book On Burning Mirrors, Diocles presents a pointwise construction of the parabola, and in prop. X, he uses the same method to construct two parabolas [34]. In the passage from the first book of his treatise (in which the conics are presented as sections of the cone) to the second book (in which he gives a description of the conics on the plane by various methods) the author points out that the properties enjoyed by the conic sections and those enjoyed by their respective curves in the plane are preserved. When he makes the passage from space to plane explicit, Mydorge generalizes the classical results, essentially those of Apollonius (the conics are considered on a generic cone, the conic elements of Apollonius constitute the theoretical foundation indispensable to be able to trace a conic in the plane [33]) and considers the fact that the diameter of a conic in the plane is the same as that on the surface of the cone and that the properties of conics as curves in a plane are the same as the properties of conics as sections of a cone. Having made this explicit, he considers conics only as plane curves.

This simple construction by points of the parabola, like those of the ellipse and the hyperbola, were designed to help craftsmen in the construction of lenses and mirrors. Mydorge intended to give a treatment of the conics on the plane, considering them as “loci of points”, capable of solving well-identified problems, as he states in the title of his treatise. Its purpose was to draw up an edition, which had as its reference point the text of Apollonius, in the reading of which the lovers of catoptric and dioptric did not encounter difficulties of any kind, enriching their knowledge on conics by deepening their study and proceeded with linearity in solving problems.

The pointwise constructions of the conics, built by following Mydorge’s indications, can help students to become aware that in the geometric field the figures perform the task of visualizing the objects on which the problems are centered, according to the specific properties that are attributed to them, and on which one can reflect to perceive by intuition and demonstrate others. The same constructions can favor the recognition of the object we are talking about. This recognition [35] is fundamental for the realization of learning [36,37] through a diversification of the way in which this object “is seen” and then “told” through a new sign [38,39]. They also constitute an example of connection between geometric and algebraic knowledge, considering the knowledge as a potential that emerges from human activity in the process in which it materializes into knowing.

4. From a Parabola on a Plane to the Cone of Which It Is a Section

In the previous paragraphs we have considered a parabola as a section of a cone and then thought of it as a curve on a plane. Now we ask ourselves the inverse problem, that is: given a parabola in a plane, is it possible to construct the cone of which it is a section?

While teaching, if we do not want to limit ourselves to a single level, that is to consider conics only as plane curves, but we want to compare the way in which the ancients identified conics with a treatment of them as loci of points of a plane, we have to show that following the two paths (from the cone to the plane and from the plane to the cone) it is possible to obtain the same geometric entity.

While the first problem (which concerns the passage from a conic as a section of a cone to the conic as a curve in a plane) was a consequence of the knowledge of Apollonius’ conic elements and related treatments, the second problem (concerning the passage from a conic as a curve in a plane to a conic as a section of a cone) took some time to work out. An attempt in this direction was made by John Wallis in the De Sectionibus Conicis Tractatus of 1655 [15]. We remark here that the problem of constructing the cone which a given parabola belongs to was also addressed by Apollonius in Conics I 52. However, there is a difference between the case studied by Apollonius and the one studied by Wallis: the first uses this construction to solve the problem of drawing the parabola given the axis and the vertex; the latter shows that, given a generic parabola on the plane, that parabola is the same as the one of Apollonius.

Wallis was born in Ashford, Kent in 1616. In his manuscript sheets there is no mention of a university or higher education in mathematics [40]. Taking into account his elaborations and productions, probably the study of mathematics accompanied him throughout his training period in Cambridge (1633–1640). In 1649 he was called as a professor of geometry at the University of Oxford. He died in 1703.

In his treatise on conics, inside the Proposition 12 [15], pp. 309–312, in which he shows how to determine the right side of the parabola, Wallis shows that “for every parabola built on a plane it is possible to construct the cone of which it is a section”.

Following his speech, we carry out the construction with GeoGebra in order to visualize well each element that composes it.

Suppose we have a generic parabola in a plane (see Figure 7).

Let $F$ be its vertex, $d$ its diameter or axis and $k$ its straight side or parameter.

Let us draw its ordinate to the diameter, i.e., a segment conducted from any point of the parabola perpendicularly to the axis.

Let us draw the segment $PB=PF$ in such a way that it is $\widehat{\mathrm{FPB}}=90°$ and, on the extension of $BP$ we take a point $S$ such that $PS=k$.

From point $S$ we draw the parallel to $PF$ (Figure 8).

This parallel intersects the line passing through $F$ and $B$ at point $V$.

In this way the triangle $VBS$ is identified.

Let us construct the circle of diameter $BS$, which forms $VBS$ an angle equal to $\widehat{FPE}$ with the triangle VBS.

The cone $VBES$, through whose axis passes the triangle $VBS$, is the cone of which the given parabola is a section (see Figure 9).

5. Educational Experiment and Results

In order to understand whether the proposed path from the parabola as section of a cone to plane curve and vice versa can gain potential interest among students, an educational experiment setup was prepared, which was submitted to the students of the course in “mathematics education” for the Master’s degree in Mathematics at the University of Calabria. Two questionnaires have been prepared: the first one aimed to understand the background level of the students concerning the parabola, the second one to understand the effectiveness of the path proposed. In between, it was proposed to the students, during 8 h of the course, after a historical foreword on the mathematicians Mydorge and Wallis and the period in which they lived, Apollonius’ construction of the parabola, as explained in Section 2, Mydorge’s pointwise construction of the parabola in the plane, reported in Section 3, and Wallis’ construction of the cone of which a given parabola is a section, explained in Section 4. All the constructions have been carried out through the use of the DGS GeoGebra, as previously illustrated.

The sample of students was made of 25 plus 23 students, in total 48 persons, the first set of students belonging to the course of the current academic year (2021–2022), the second belonging to the same course of the previous academic year (2020–2021).

The first step of the experimental procedure was to administer the students a test to verify their current background knowledge about the parabola. The questionnaire was composed of some questions that can be synthetized as follows:

(1)

Did the students previously meet, during their studies, the concept of parabola and how it was eventually presented, as a section of the cone or as a plane curve or both ways?

(2)

Are they able to describe (even in words) a parabola and give a precise definition of it in mathematical terms?

(3)

Do they know the meaning of the expression “parabola as a section of a cone” and its precise definition?

(4)

Do they know how to obtain the equation of the parabola: $y=a{x}^2$?

(5)

Did they meet before the metric relation of the parabola (namely the relation that identifies a parabola as the equality of two areas, given in Section 2 above) and its relationship with the algebraic equation of the curve?

(6)

How would they make a pointwise construction of the parabola?

(7)

Are they able, given a parabola on a cone, to identify the plane it belongs to and, given a parabola on a plane, build the cone it belongs to?

The answers given by the students can be summarized as follows:

(1)

75% of the students reported that they have met the parabola since the third year of the high school courses (the remaining students said the second year, when studying the second-degree equations, or at the fourth year, or even that they could not remember). All the students said that the parabola had been presented to them as a plane curve, albeit about the 42% of all the students reported that they had seen the parabola as section of a cone, but without giving any definition, just a graphical representation;

(2)

All the students answered that they would have been able to describe the parabola as a plane curve only, except for 17% of them who described it as a curve given by the intersection of a cone with a plane, without any specification about the position of the plane itself. However, when asked to give a mathematical definition of a parabola, the 50% of the students defined it as the locus of the points of the plane equidistant from the focus and the directrix, 25% of them were able to define it only in terms of its Cartesian equation; the remaining 25% were not able to give any definition at all;

(3)

At the question 3, if they knew the meaning of “parabola as a section of a cone”, the 75% of the students answered “yes”, though only about 40% were able to provide its exact definition;

(4)

About the 58% of the students admitted that they did not know how to obtain the algebraic equation of the parabola $y=a{x}^2$, while the remaining percentage just said that it could be obtained by posing the $b=c=0$ in the general equation: $y=a{x}^2+bx+c$, without giving any derivation of it;

(5)

None of the students have ever heard about the Apollonius’ metric relation of the parabola, nor they were able to see any connection with its Cartesian equation;

(6)

Again, none of the students had previously ever encountered a pointwise construction of the parabola;

(7)

Finally, none of the students were able to build the cone a given parabola belongs to and only about 40% were able to identify the plane to which the given parabola belongs on a cone.

From those answers, one can easily conclude that the majority of the students participating in the questionnaire know the parabola as section of a cone only in the qualitative sense that it can be obtained by intersecting a plane with a cone, but few of them know the exact definition or the properties of such a plane and nobody had seen before how to build the cone of which the parabola is a section. Another important note concerns the fact that though the students have seen the parabola mainly as a curve on the plane, they could not remember how to obtain its equation, albeit they had studied such a topic during the university courses, as well as at the high school. This seems to be an indication of the fact that the topic is conceived as too theoretical, with few relations with reality and not easy to remember. The last statement was later confirmed by the opinions of the students in the second questionnaire.

After administering the first questionnaire, the students have been taught for a total of 8 h the construction of the parabola by Apollonius, the metric relation and the construction of the parabola as a section of the cone with GeoGebra. Then, the pointwise construction of the parabola by Mydorge and how to go back to the cone of which a parabola is a section according to Wallis’ construction was seen, reproduced with GeoGebra as indicated in the previous sections of the article.

After the eight-hour course, the second questionnaire was submitted to the students. The questions included in the test were:

• How has the reading of the historical materials that have been presented modified and enriched your knowledge on the parabola?
• Which parts in the studied material have added new elements (at a geometric or algebraic level) that were unknown to you before, concerning the way of thinking a parabola and its usefulness?
• Imagining yourself as a teacher in a classroom, which parts of the proposed material you think are suitable to be taught in a lecture for your students to introduce the study of the parabola?
• Do you think that presenting the equivalence between the parabola as a plane curve and as a section of a cone has improved your understanding of this geometrical object?
• Do you think that the description of the parabola as a section of a cone, with respect to the classic definition as locus of the points of a plane, helps in better visualizing the object, making it, in such a way, less “abstract”?
• Imagining yourself as a teacher in a classroom, would you decide to begin from the discussion of the parabola in space or on the plane? Moreover, would you choose the geometric or algebraic path? Eventually, would you try to combine the two and in which form?
• Would you think it would be useful to make the students work in the physics laboratory to a “construction” of a lens as presented by Mydorge?

I summarize the answers of the students below:

• Concerning the first question, more than 75% of the students admitted that they know none of the topics treated together during the lectures; less than 25% just knew Apollonius’s construction, since they had studied it in the course of history of mathematics. However, all the other topics were unknown to them. All the students have shown enthusiasm for those topics and declared to have been enriched by apprehending them;
• The second question is quite close to the first, therefore the answers were rather similar. One important point raised by the students in their answers is the comment concerning the fact that they found the geometrical approach to the conics is more “practical” and easier to visualize with respect to the algebraic one.
• When thinking themselves as teachers (since they could become future teachers in mathematics) the students imagined organizing their own lessons by including all the topics treated during my lectures (about 25% of them), the parts concerning the parabola as a section of a cone (about 50%), while another 25% did not give any answer. Regardless, all those who answered stressed the importance of the better visibility of the geometrical aspects of the parabola as a section of a cone, with respect to the algebraic approach. Moreover, all the students pointed out the effectiveness of the use of GeoGebra when approaching the problem from an educational point of view;
• All the students positively answered the fourth question, whether presenting the equivalence of the parabola as both a section of a cone and a plane curve improved their understanding of this geometrical object. Again at least 25% of the students stressed the importance of the use of the DGS during the lectures;
• Again, all the students agreed on the fact that presenting the parabola as a section of a cone, as well as a locus of points of a plane, makes the geometric curve a less “abstract” object, especially through the use of GeoGebra. In addition, some students imagined some alternative methods, such as modeling a cone in “play dough” and then cutting it with a knife, to obtain practical representations of the parabola on the cone and a more “realistic” representation of the geometrical object;
• Almost 100% of the students answered the sixth question by saying that if they were teachers and they had to explain the parabola to the students they would surely begin from the geometrical description, which they see as easier to visualize and more “practical”, and later eventually shift to the algebraic description, effectively combining the two approaches (about the 50% of the students);
• Finally, all the students agreed on the fact that the practical construction of a lens in the physics laboratory could be beneficial for the students, to improve their understanding of the topic, especially to link it with interdisciplinary aspects of the knowledge related to the practical use of theoretical acquisition on curves. In such direction, some of them (in the subsequent collegial discussion) have pointed out as the activity of building a lens, as indicated by Mydorge, could be a possible initial approach to the introduction of the parabola. According to the students’ observations, such an approach underlines the real need to define a theoretical object, just the parabola, whose properties have to be transferred to the physical object to be built in order to ensure its correct working and takes on the fundamental role of validating the correctness of the operations to be executed on the manipulated material.

All the reported answers seem to indicate that the approach based on the historical path followed during the lectures was most appreciated by the students and considerably improved, according to their own admissions, their understanding of the parabola. Moreover, the students underlined the educational value of this approach and the fact that they would repropose it to their future high school students. Finally, the answers point out that the students have been particularly stimulated by the use of GeoGebra for the step-by-step construction of the geometric objects to improve their understanding of both the geometric and algebraic properties of the parabola and they found this approach more “practical” with respect to the more “usual” ones. Furthermore, the improved geometrical visualization stimulated their creativity and fantasy by trying to suggest alternative ways to visualize the parabola in 3D space.

6. Conclusions

The examples presented in this path have been taken from the history of mathematics and have been proposed as a moment of reflection on the fact that, often, conics are presented only as loci of points of a plane, without making any reference to the connection they have with conics as sections of a cone, thus also leaving it unclear why we continue to call those curves in a plane “conics”. Although the work definitely has strong roots in the history of mathematics, its final aim is of didactic type. To investigate the effectiveness of the proposed path, an educational experiment was proposed to the students of “mathematics education” of the second year of the Master’s degree in Mathematics of the Department of Mathematics and Computer Science of the University of Calabria. Two questionnaires were submitted to a group of 48 students, one before and the other after an intermediate period of 8 h of lectures in which the proposed path was explained to the students. The answers to the questionnaires showed that the students had met the parabola before mainly as a plane curve with some generic reference to its view as a section of a cone and pointed out as such a notion was generally considered “too abstract” to attract their interest. Moreover, the answers to the second questionnaire indicate a general appreciation of the students for the historical approach used and its educational value.

The use of a DGS has been proposed to allow, in addition to constructing the figure by simulating drawing tools, us to make the same figure dynamic through the drag function, to measure (lengths of segments, amplitudes of angles, etc.), to trace and animate some parts, making the evolution of models visible, and to integrate different registers of representation, such as the geometric and algebraic registers, and also in order to model problematic situations. For instance, the use of sliders in GeoGebra could allow us to highlight the interactions between geometric register and algebraic register, by giving the student the possibility to measure areas, distances and showing that, for example, the metric relation found by Apollonius is satisfied, as well as the definition of the parabola in terms of points equidistant from the focus and from the directrix. Moreover, it could allow teachers to guide students to identify and develop new relationships between objects and to project experiments conceived in a two-dimensional and three-dimensional geometric environment. According to the path suggested in the present work, and through the use of the DGS, students (and teachers, as well) could thus realize the change of semiotic registers between the 2D and 3D view of the parabola, check, construct, prove, conjecture and generalize properties with constructions that cannot be realized on paper and verify complex situations more easily than with traditional tools. The use of the construction protocol can make students retrace the individual steps of each construction. Finally, it becomes meaningful to compare traditional “manual” tools with the automatic tools that the software makes available, looking for differences, points of contact, and explanations.