What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

§8. CONICS AND QUADRIC SURFACES

1. Elementary Metric Geometry of Conics

Until now we have been concerned only with points, lines, planes, and figures formed by a number of these. If projective geometry were nothing but the study of such “linear” figures, it would be of relatively little interest. It is a fact of fundamental importance that projective geometry isnot confined to the study of linear figures, but includes also the whole field of conic sections and their generalizations in higher dimensions. Apollonius’ metric treatment of the conic sections—ellipses, hyperbolas, and parabolas—was one of the great mathematical achievements of antiquity. The importance of conic sections for pure and applied mathematics (for example, the orbits of the planets and of the electrons in the hydrogen atom are conic sections) can hardly be overestimated. It is little wonder that the classical Greek theory of conic sections is still an indispensable part of mathematical instruction. But Greek geometry was by no means final. Two thousand years later the important projective properties of the conics were discovered. In spite of the simplicity and beauty of these properties, academic inertia has so far prevented their introduction into the high school curriculum.

We shall begin by recalling the metric definitions of the conic sections. There are various such definitions whose equivalence is shown in elementary geometry. The usual ones refer to the foci. An ellipse is defined as the geometrical locus of all points P in the plane the sum of whose distances, r₁, r₂, from two fixed points F₁, F₂, the foci, has a constant value. (If the two foci coincide the figure is a circle.) The hyperbola is defined as the locus of all points P in the plane for which the absolute value of the difference r₁ – r₂ is equal to a fixed constant. The parabola is defined as the geometrical locus of all points P for which the distance r to a fixed point F is equal to the distance to a given line l.

In terms of analytic geometry these curves can all be expressed by equations of the second degree in the coördinates x, y. It is not hard to prove, conversely, that any curve defined analytically by an equation of the second degree:

ax² + by² + cxy + dx + ey + f = 0,

is either one of the three conics, a straight line, a pair of straight lines, a point, or imaginary. This is usually proved by introducing a new and suitable coördinate system, as is done in any course in analytic geometry.

These definitions of the conic sections are essentially metric, since they make use of the concept of distance. But there is another definition that establishes the place of the conic sections in projective geometry: The conic sections are simply the projections of a circle on a plane. If we project a circle C from a point O, then the projecting lines will form an infinite double cone, and the intersection of this cone with a plane π will be the projection of C. This intersection will be an ellipse or a hyperbola according as the plane cuts one or both portions of the cone. The intermediate case of the parabola occurs if π is parallel to one of the lines through O (see Fig. 94).

The projecting cone need not be a right circular cone with its vertex O perpendicularly above the center of the circle C; it may also be oblique. In all cases, as we shall here accept without proof, the intersection of the cone with a plane will be a curve whose equation is of second degree; and conversely, every curve of second degree can be obtained from a circle by such a projection. It is for this reason that the curves of second degree are called conic sections.

When the plane intersects only one portion of a right circular cone we have stated that the curve of intersection E is an ellipse. We may prove that E satisfies the usual focal definition of the ellipse, as given above, by a simple but beautiful argument given in 1822 by the Belgian mathematician G. P. Dandelin. The proof is based on the introduction of the two spheres S₁ and S₂ (Fig. 95), which are tangent to π at the points F₁ and F₂, respectively, and which touch the cone along the parallel circles K₁ and K₂ respectively. We join an arbitrary point P of E with F₁ and F₂and draw the line joining P to the vertex O of the cone. This line lies entirely on the surface of the cone, and intersects the circles K₁ and K₂ in the points Q₁ and Q₂ respectively. Now PF₁ and PQ₁ are two tangents from P to S₁, so that

Fig. 94. Conic sections.

PF₁ = PQ₁.

Similarly,

PF₂ = PQ₂.

Adding these two equations we obtain

PF₁ + PF₂ = PQ₁ + PQ₂.

But PQ₁ + PQ₂ = Q₁Q₂ is just the distance along the surface of the cone between the parallel circles K₁ and K₂ and is therefore independent of the particular choice of the point P on E. The resulting equation,

PF₁ + PF₂ = constant

Fig. 95. Dandelin’s spheres.

for all points P of E, is precisely the focal definition of an ellipse. E is therefore an ellipse and F₁, F₂ are its foci.

Exercise: When a plane cuts both portions of the cone, the curve of intersection is a hyperbola. Prove this fact, using one sphere in each portion of the cone.

2. Projective Properties of Conics

On the basis of the facts stated in the preceding section we shall adopt the tentative definition: a conic is the projection of a circle on a plane. This definition is more in keeping with the spirit of projective geometry than is the usual focal definition, since the latter is entirely based on the metric notion of distance. Even our present definition is not free from this defect, since “circle” is also a concept of metric geometry. We shall in a moment arrive at a purely projective definition of the conics.

Since we have agreed that a conic is merely the projection of a circle (i.e., that the word “conic” is to mean any curve in the projective class of the circle; see p. 186), it follows that any property of the circle that is invariant under projection will also be possessed by any conic. Now a circle has the well-known (metric) property that a given arc subtends the same angle at every point O on the circle. In Figure 96, the angle AOB subtended by the arc AB is independent of the position of O. This fact can be brought into relation with the projective concept of cross-ratio by considering not two points A, B but four points A, B, C, D on the circle. The four lines a, b, c, d joining them to a fifth point O on the circle will have a cross-ratio (a b c d) which depends only on the angles subtended by the arcs CA, CB, DA, DB. If we join A, B, C, D to another point O′ on the circle, we obtain four rays a′, b′, c′, d′. From the property of the circle just mentioned, the two quadruples of rays will be “congruent.”† Hence they will have the same cross-ratio: (a′ b′ c′ d′) = (a b c d). If we now project the circle into any conic K, we shall obtain on K four points, again called A, B, C, D, two other points O, O′, and the two quadruples of lines a, b, c, d and a′, b′, c′, d′. These quadruples will not be congruent, since equality of angles is in general destroyed by projection. But since cross-ratio is invariant under projection, the equality (a b c d) = (a′ b′ c′ d′) will still hold. This leads to a fundamental theorem: If any four given points A, B, C, D of a conic K are joined to a fifth point O of K by lines a, b, c, d, then the value of the cross-ratio (a b c d) is independent of the position of O on K (Fig. 97).

Fig. 96. Cross-ratios on a circle.

Fig. 97. Cross-ratios on an ellipse.

This is indeed a remarkable result. We already knew that any four given points on a straight line appear under the same cross-ratio from any fifth point O. This theorem on cross-ratios is the basic fact of projective geometry. Now we learn that the same is true of four points on a conic, with one important restriction: the fifth point is no longer absolutely free in the plane, but is still free to move on the given conic.

It is not difficult to prove a converse of this result in the following form: if there are two points O, O′ on a curve K such that every quadruple of four points A, B, C, D on K appears under the same cross-ratio from both O and O’, then K is a conic, (and therefore A, B, C, D appear under the same cross-ratio from any third point O″ of K). The proof is omitted here.

These projective properties of the conics suggest a general method for constructing such curves. By a pencil of lines we shall mean the set of all straight lines in a plane which pass through a given point O. Now consider the pencils through two points O and O′ which are chosen to lie on a conic K. Between the lines of pencil O and those of pencil O′ we may establish a biunique correspondence by coupling a line a of O with a line a′ of O′ whenever a and a′ meet in a point A of the conic K. Then any four lines a, b, c, d of the pencil O will have the same cross-ratio as the four corresponding lines a′, b′, c′, d′ of O′. Any biunique correspondence between two pencils of lines which has this property is called a projective correspondence. (This definition is obviously the dual of the definition given on p. 178 of a projective correspondence between the points on two lines.) Pencils between which there is defined a projective correspondence are said to be projectively related. With this definition we can now state: The conic K is the locus of the intersections of corresponding lines of two projectively related pencils. This theorem provides the basis for a purely projective definition of the conics: A conic is the locus of the intersections of corresponding lines in two projectively related pencils.† It is tempting to follow the path into the theory of conics opened by this definition, but we shall confine ourselves to a few remarks.

Pairs of projectively related pencils can be obtained as follows. Project all the points P on ε, straight line l from two different centers O and O″; in the projecting pencils let lines a and a″ which intersect on l correspond to each other. Then the two pencils will be projectively related. Now take the pencil O″ and transport it rigidly into any position O′. The resulting pencil O′ will be projectively related to O. Moreover, any projective correspondence between two pencils can be so obtained. (This fact is the dual of Ex. 1 on p. 179.) If the pencils O and O′ are congruent, we obtain a circle. If angles are equal but with opposite sense, the conic is an equilateral hyperbola (see Fig. 99).

Fig. 98. Preliminary to construction of projectively related pencils.

Note that this definition of conic may yield a locus which is a straight line, as in Figure 98. In this case the line O O″ corresponds to itself, and all its points are counted as belonging to the locus. Hence the conic degenerates into a pair of lines, which agrees with the fact there are sections of a cone (those obtained by planes through the vertex) which consist of two lines.

Fig. 99. Circle and equilateral hyperbola generated by projective pencils.

Exercises: 1) Draw ellipses, hyperbolas, and parabolas by means of projective pencils. (The reader is strongly urged to experiment with such constructions. They will contribute greatly to his understanding.)

2) Given five points, O, O′, A, B, C, of an unknown conic K. It is required to construct the point D where a given line d through O intersects K. (Hint: Consider through O the rays a, b, c given by OA, OB, OC, and similarly through O′ the rays a′, b′, c′. Draw through O the ray d and construct through O′ the ray d′ such that (a, b, c, d) = (a′, b′, c′, d′). Then the intersection of d and d′ is necessarily a point of K.)

3. Conics as Line Curves

The concept of tangent to a conic belongs to projective geometry, for a tangent to a conic is a straight line that touches the conic in only one point, and this property is unchanged by projection. The projective properties of tangents to conics are based on the following fundamental theorem: The cross-ratio of the points of intersection of any four fixed tangents to a conic with a fifth tangent is the same for every position of the fifth tangent.

The proof of this theorem is very simple. Since a conic is a projection of a circle, and since the theorem concerns only properties which are invariant under projection, a proof for the case of the circle will suffice to establish the theorem in general.

Fig. 100. A circle as a set of tangents.

For the circle, the theorem is a matter of elementary geometry. Let P, Q, R, S be any four points on a circle K with the tangents a, b, c, d; T another point with the tangent o, intersected by a, b, c, d in A, B, C, D. If M is the center of the circle, then obviously , and is equal to the angle subtended by the arc TP at a point of K. Similarly, TMB is the angle subtended by the arc TQ at a point of K. Therefore , where is the angle subtended by the arc PQ at a point of K. Hence the points A, B, C, D are projected from Mby four rays whose angles are given by the fixed positions of P, Q, R, S. It follows that the cross-ratio (A B C D) depends only on the four tangents a, b, c, d and not on the particular position of the fifth tangent o. This is exactly the theorem that we had to prove.

Fig. 101. The tangent property of the circle.

In the preceding section we have seen that a conic may be constructed by marking the points of intersection of corresponding lines in two projectively related pencils. The theorem just proved enables us to dualize this construction. Let us take two tangents a and a′ of a conic K. A third tangent t will intersect a and a′ in two points A and A′ respectively. If we allow t to move along the conic, this will set up a correspondence

A ↔ A′

between the points of a and those of a′. This correspondence between the points of a and those of a′ will be projective, for by our theorem any four points of a will have the same cross-ratio as the corresponding four points of a′. Hence it appears that a conic K, regarded as the set of its tangents, consists of the lines which join corresponding points of the two projectively related ranges† of points on a and a′.

Fig. 102. Projective point ranges on two tangents of an ellipse.

This fact may be used to give a projective definition of a conic as a “line curve.” Let us compare it with the projective definition of a conic given in the preceding section:

I

A conic as a set of points consists of the points of intersection of corresponding lines in two projectively related pencils of lines.

II

A conic as a set of lines consists of the lines joining corresponding points in two projectively related ranges of points.

Fig. 103. A parabola defined by congruent point ranges.

Fig. 104. A parabola defined by similar point ranges.

If we regard the tangent to a conic at a point as the dual element to the point itself, and if we consider a “line curve” (the set of all its tangents) as the dual of a “point curve” (the set of all its points), then the complete duality between these two statements is apparent. In the translation from one statement to the other, replacing each concept by its dual, the word “conic” remains the same: in one case it is a “point conic,” defined by its points; in the other a “line conic,” defined by its tangents. (See Fig. 100, p. 206.)

An important consequence of this fact is that the principle of duality in plane projective geometry, originally stated for points and lines only, may now be extended to cover conics. If, in the statement of any theorem concerning points, lines, and conics, each element is replaced by its dual(keeping in mind that the dual of a point on a conic is a tangent to the conic), the result will also be a true theorem. An example of the working of this principle will be found in Article 4 of this section.

The construction of conics as line curves is shown in Figures 103–104. If, on the two projectively related point ranges, the two points at infinity correspond to each other (as must be the case with congruent or similar† ranges), the conic will be a parabola; the converse is also true.

Exercise: Prove the converse theorem: On any two fixed tangents of a parabola a moving tangent cuts out two similar point ranges.

4. Pascal’s and Brianchon’s General Theorems for Conics

One of the best illustrations of the duality principle for conics is the relation between the general theorems of Pascal and of Brianchon. The first was discovered in 1640, the second only in 1806. Yet one is an immediate consequence of the other, since any theorem involving only conics, straight lines, and points must remain true if replaced by its dual statement.

The theorems stated in §5 under the same name are degenerate cases of the following more general theorems:

Pascal’s theorem: The opposite edges of a hexagon inscribed in a conic meet in three collinear points.

Brianchon’s theorem: The three diagonals joining opposite vertices of a hexagon circumscribed about a conic are concurrent.

Both theorems are clearly of a projective character. Their dual nature becomes obvious if they are formulated as follows:

Pascal’s theorem: Given six points, 1, 2, 3, 4, 5, 6, on a conic. Join successive points by the lines (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 1). Mark the points of intersection of (1, 2) with (4, 5), (2, 3) with (5, 6), and (3, 4) with (6, 1). Then these three points of intersection lie on a straight line.

Brianchon’s theorem: Given six tangents, 1, 2, 3, 4, 5, 6, to a conic. Successive tangents intersect in the points, (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 1). Draw the lines joining (1, 2) with (4, 5), (2, 3) with (5, 6), and (3, 4) with (6, 1). Then these lines go through a point.

Fig. 105. Pascal’s general configuration. Two cases are illustrated: one for the hexagon 1, 2, 3, 4, 5, 6, and one for the hexagon 1, 3, 6, 2, 6, 4.

The proofs can be given by a specialization similar to that used in the degenerate cases. To prove Pascal’s theorem, let A, B; C, D, E, F be the vertices of a hexagon inscribed in a conic K. By projection we can make AB parallel to ED and FA parallel to CD, so that we obtain the configuration of Figure 107. (For convenience in representation the hexagon is taken as self-intersecting, although this is not necessary.) Pascal’s theorem now reduces to the simple statement that CB is parallel to FE; in other words, the line on which the opposite edges of the hexagon meet is the line at infinity. To prove this, let us consider

Fig. 106. Brianchon’s general configuration. Again two cases are illustrated. the points F, A, B, D, which, as we know, are projected by rays having a constant cross-ratio k from any other point of K, e.g., from C or E. Project these points from C; then the projecting rays intersect AF in four points, F, A, Y, ∞, which have the cross-ratio k. Hence YF: YA = k. (See p. 185.) If the same points are now projected from E onto BA, we obtain

Fig. 107. Proof of Pascal’s theorem.

k = (XAB ∞) = BX: BA.

Hence we have

BX: BA = YF: YA,

which establishes the parallelism of YB and FX. This completes the proof of Pascal’s theorem.

Brianchon’s theorem follows either by the duality principle or by direct reasoning dual to the above. The reader will find it a good exercise to carry out the details of the argument.

5. The Hyperboloid

In three dimensions the figures that correspond to the conics in the plane are the “quadric surfaces”; of these the sphere and the ellipsoid are special cases. These surfaces offer more variety and considerably more difficulty than do the conics. Here we shall discuss briefly and without giving proofs one of the more interesting quadrics, the “one-sheeted hyperboloid.”

This surface may be defined in the following manner. Choose any three lines, l₁, l₂, l₃, in general position in space. By this we mean that no two of the lines are to lie in the same plane nor are they all to be parallel to any one plane. It is a rather surprising fact that there will be infinitely many lines in space each of which intersects all three of the given lines. To see this, let us take any plane π through l₁. Then π will intersect l₂ and l₃ in two points, and the line m joining these two points will intersect l₁, l₂, and l₃. As the plane π rotates about l₁, the line m will move, always intersecting l₁, l₂, l₃, and will generate a surface of infinite extent. This surface is the one-sheeted hyperboloid; it contains an infinite family of straight lines of the type m. Any three of these lines, m₁, m₂, m₃, will also be in general position, and all the lines in space that intersect these three lines will also lie in the surface of the hyperboloid. This is the fundamental fact concerning the hyperboloid: it is made up of two different families of straight lines; every three lines of the same family are in general position, while each line of one family intersects all the lines of the other family.

An important projective property of the hyperboloid is that the cross-ratio of the four points where any four given lines of one family intersect a given line of the other family is independent of the position of the latter line. This follows directly from the method of construction of the hyperboloid by a rotating plane, as the reader may show as an exercise.

Fig. 108. Construction of lines intersecting three fixed lines in general position.

Fig. 109. The hyperboloid.

One of the most remarkable properties of the hyperboloid is that although it contains two families of intersecting straight lines, these lines do not make the surface rigid. If a model of the surface is constructed from wire rods, free to rotate at each intersection, then the whole figure may be continuously deformed into a variety of shapes.