If we join a point in space to all points on a circle, we get a (circular) cone of the second order (§ 43). Every plane section of this cone is a conic. This conic will be an ellipse, a parabola, or an hyperbola, according as the line at infinity in the plane has no, one or two points in common with the conic in which the plane at infinity cuts the cone. It follows that our curves of second order may be obtained as sections of a circular cone, and that they are identical with the “Conic Sections” of the Greek mathematicians.
§ 61. Any two tangents to a parabola are cut by all others in projective rows; but the line at infinity being one of the tangents, the points at infinity on the rows are corresponding points, and the rows therefore similar. Hence the theorem—
The tangents to a parabola cut each other proportionally.
Pole and Polar
§ 62. We return once again to fig. 21, which we obtained in § 55.
If a four-side be circumscribed about and a four-point inscribed in a conic, so that the vertices of the second are the points of contact of the sides of the first, then the triangle formed by the diagonals of the first is the same as that formed by the diagonal points of the other.
Such a triangle will be called a polar-triangle of the conic, so that PQR in fig. 21 is a polar-triangle. It has the property that on the side p opposite P meet the tangents at A and B, and also those at C and D. From the harmonic properties of four-points and four-sides it follows further that the points L, M, where it cuts the lines AB and CD, are harmonic conjugates with regard to AB and CD respectively.
If the point P is given, and we draw a line through it, cutting the conic in A and B, then the point Q harmonic conjugate to P with regard to AB, and the point H where the tangents at A and B meet, are determined. But they lie both on p, and therefore this line is determined. If we now draw a second line through P, cutting the conic in C and D, then the point M harmonic conjugate to P with regard to CD, and the point G where the tangents at C and D meet, must also lie on p. As the first line through P already determines p, the second may be any line through P. Now every two lines through P determine a four-point ABCD on the conic, and therefore a polar-triangle which has one vertex at P and its opposite side at p. This result, together with its reciprocal, gives the theorems—
All polar-triangles which have one vertex in common have also the opposite side in common.
All polar-triangles which have one side in common have also the opposite vertex in common.
§ 63. To any point P in the plane of, but not on, a conic corresponds thus one line p as the side opposite to P in all polar-triangles which have one vertex at P, and reciprocally to every line p corresponds one point P as the vertex opposite to p in all triangles which have p. as one side.
We call the line p the polar of P, and the point P the pole of the line p with regard to the conic.
If a point lies on the conic, we call the tangent at that point its polar; and reciprocally we call the point of contact the pole of tangent.
§ 64. From these definitions and former results follow—
| The polar of any point P not on the conic is a line p, which has the following properties:— | The pole of any line p not a tangent to the conic is a point P, which has the following properties:— |
| 1. On every line through P which cuts the conic, the polar of P contains the harmonic conjugate of P with regard to those points on the conic. | 1. Of all lines through a point on p from which two tangents may be drawn to the conic, the pole P contains the line which is harmonic conjugate to p, with regard to the two tangents. |
| 2. If tangents can be drawn from P, their points of contact lie on p. | 2. If p cuts the conic, the tangents at the intersections meet at P. |
| 3. Tangents drawn at the points where any line through P cuts the conic meet on p; and conversely, | 3. The point of contact of tangents drawn from any point on p to the conic lie in a line with P; and conversely, |
| 4. If from any point on p, tangents be drawn, their points of contact will lie in a line with P. | 4. Tangents drawn at points where any line through P cuts the conic meet on p. |
| 5. Any four-point on the conic which has one diagonal point at P has the other two lying on p. | 5. Any four-side circumscribed about a conic which has one diagonal on p has the other two meeting at P. |
The truth of 2 follows from 1. If T be a point where p cuts the conic, then one of the points where PT cuts the conic, and which are harmonic conjugates with regard to PT, coincides with T; hence the other does—that is, PT touches the curve at T.
That 4 is true follows thus: If we draw from a point H on the polar one tangent a to the conic, join its point of contact A to the pole P, determine the second point of intersection B of this line with the conic, and draw the tangent at B, it will pass through H, and will therefore be the second tangent which may be drawn from H to the curve.
§ 65. The second property of the polar or pole gives rise to the theorem—
| From a point in the plane of a conic, two, one or no tangents may be drawn to the conic, as its polar has two, one, or no points in common with the curve. | A line in the plane of a conic has two, one or no points in common with the conic, according as two, one or no tangents can be drawn from its pole to the conic. |
Of any point in the plane of a conic we say that it was without, on or within the curve according as two, one or no tangents to the curve pass through it. The points on the conic separate those within the conic from those without. That this is true for a circle is known from elementary geometry. That it also holds for other conics follows from the fact that every conic may be considered as the projection of a circle, which will be proved later on.
The fifth property of pole and polar stated in § 64 shows how to find the polar of any point and the pole of any line by aid of the straight-edge only. Practically it is often convenient to draw three secants through the pole, and to determine only one of the diagonal points for two of the four-points formed by pairs of these lines and the conic (fig. 22).
These constructions also solve the problem—
From a point without a conic, to draw the two tangents to the conic by aid of the straight-edge only.
For we need only draw the polar of the point in order to find the points of contact.
§ 66. The property of a polar-triangle may now be stated thus—
In a polar-triangle each side is the polar of the opposite vertex, and each vertex is the pole of the opposite side.
 |
| Fig. 23. |
If P is one vertex of a polar-triangle, then the other vertices, Q and R, lie on the polar p of P. One of these vertices we may choose arbitrarily. For if from any point Q on the polar a secant be drawn cutting the conic in A and D (fig. 23), and if the lines joining these points to P cut the conic again at B and C, then the line BC will pass through Q. Hence P and Q are two of the vertices on the polar-triangle which is determined by the four-point ABCD. The third vertex R lies also on the line p. It follows, therefore, also—
If Q is a point on the polar of P, then P is a point on the polar of Q; and reciprocally,
If q is a line through the pole of p, then p is a line through the pole of q.
This is a very important theorem. It may also be stated thus—
If a point moves along a line describing a row, its polar turns about the pole of the line describing a pencil.
This pencil is projective to the row, so that the cross-ratio of four poles in a row equals the cross-ratio of its four polars, which pass through the pole of the row.
To prove the last part, let us suppose that P, A and B in fig. 23 remain fixed, whilst Q moves along the polar p of P. This will make CD turn about P and move R along p, whilst QD and RD describe projective pencils about A and B. Hence Q and R describe projective rows, and hence PR, which is the polar of Q, describes a pencil projective to either.
§ 67. Two points, of which one, and therefore each, lies on the polar of the other, are said to be conjugate with regard to the conic; and two lines, of which one, and therefore each, passes through the pole of the other, are said to be conjugate with regard to the conic. Hence all points conjugate to a point P lie on the polar of P; all lines conjugate to a line p pass through the pole of p.
If the line joining two conjugate poles cuts the conic, then the poles are harmonic conjugates with regard to the points of intersection; hence one lies within the other without the conic, and all points conjugate to a point within a conic lie without it.
Of a polar-triangle any two vertices are conjugate poles, any two sides conjugate lines. If, therefore, one side cuts a conic, then one of the two vertices which lie on this side is within and the other without the conic. The vertex opposite this side lies also without, for it is the pole of a line which cuts the curve. In this case therefore one vertex lies within, the other two without. If, on the other hand, we begin with a side which does not cut the conic, then its pole lies within and the other vertices without. Hence—
Every polar-triangle has one and only one vertex within the conic.
We add, without a proof, the theorem—
The four points in which a conic is cut by two conjugate polars are four harmonic points in the conic.
§ 68. If two conics intersect in four points (they cannot have more points in common, § 52), there exists one and only one
