GEOMETRICAL INVESTIGATION OF SOME PROPERTIES OF QUADRIC SURFACES.

By W. M. Hicks, B.A., Scholar of St. John's College.

1. I Do not know that an attempt has ever been made to discuss the properties of quadrics from a purely geometrical point of view. The following pages do not by any means contain all that might be deduced by legitimate geometrical reasoning, but they are offered to the reader as a small specimen of what might be done in this direction. In Salmon's Solid Geometry many problems are solved without making the slightest use of analytical geometry, except in so far as previous resu ts are employed. Those previous results might have been obtained by ordinary geometry, and then his proofs would follow in a logical geometrical course of deduction. We have started with the modular definition of a quadric, which seemed more easy of application, and which also has the advantage of an analogy with the usual definition of a conic.

2. We begin then with the following definition of a quadric, from which we shall deduce all that follows.

Def. A quadric is the locus of a point which moves so that its distance from a given point bears a constant ratio to its distance from a given straight line, the distance being measured parallel to a given plane.

The fixed point is called the focus.

The fixed straight line, the directrix.

The fixed plane (for reasons to appear hereafter), the cyclic plane.

The ratio, the modulus.

3. If a cyclic plane be given, then the same quadric can be described with the same focus, directrix, and modulus, but with another cyclic plane.

For let (fig. 22) XY be the directrix, P any point, PM its distance from XY measured as above. Through P draw the

* As the quadric in geometry of three dimensions is the analogue of the conic in plane geometry, and as the name quadric is given from analytical considerations, it would seem more consistent to adopt the name conicoid which Messrs. Frost and Wolstenholme use in their Treatise on Solid Geometry, but as the word quadric is well established and it would be inconvenient to have two words for the same surface, we have employed the word quadric.

line in the given cyclic plane perpendicular to XY, and through this draw another plane making the same angle with XY that the cyclic plane does. Let it cut XY in N and join PN. Then evidently PMN= PNM, and, therefore, PM-PN. But the second plane is fixed in direction. Hence, if we take this other plane as a cyclic plane we shall get the same quadric.

4. The section of a quadric by a plane parallel to a cyclic plane is a circle.

For let the plane cut the directrix in Z, then the distance of a point in the plane from the line XY will be measured in the plane and therefore from Z.

Therefore the section is the same as the section of the locus of a point whose distances from the focus S and Z are in a constant ratio.

But this locus is a sphere.

Therefore the section is a circle.

Hence (and §3) all quadrics have two systems of circular sections.

5. LEMMA. The locus of the centres of the spheres in § 4 is a straight line parallel to the directrix.

Let (fig. 23) the sphere cut SZ in A, B, e=modulus, O the centre of the sphere, then

therefore locus of O is a straight line parallel to XY.

The centre of the circular section through Z is the foot of the perpendicular let fall from O on the plane through Z parallel to the cyclic plane.

6. We shall evidently get no circular section, and therefore no section at all if the radius of the corresponding sphere is less than the perpendicular from its centre on the plane of circular section.

Let a be the angle between the cyclic plane and the directrix, then 2a is the angle between the two cyclic planes.

In fig. 23 let M be centre of circle.

O centre of sphere.

OA OB radius of sphere.

=

Then if we have a circular section OA must be greater than OM.

Now as Z moves off to a great distance from X, SZ approaches to equality with XZ. Therefore if the quadric extends to an infinite distance we must have e> sin a.

If e<sina we have a closed surface which is called an ellipsoid.

If e=sina we have an ellipsoid which extends to infinity and is called the elliptic paraboloid.

If e> sina the surface is called an hyperboloid except in the case e = 1, which we shall see has peculiar properties and is called the hyperbolic paraboloid.

7. The locus of the centres of one system of circular sections of a quadric is a straight line (conjugate to cyclic plane).

In fig. 23 let XO' be the trace of plane SXZ on the cyclic plane through S, O'O the line of centres of the spheres, OM a perpendicular on the cyclic section corresponding to O, so that M is its centre. Draw O'M' perpendicular to the cyclic plane ZLM. Join MO'. Then M is always in the same plane, viz. that through 00′ and perpendicular to the cyclic plane.

Since O'M' is parallel to OM,

LM' : M'M= LO' : 00′ = SZ: SO.

Therefore, locus of M is a straight line through O', which

we will call a cyclic line.

8. Considering the other systems of circular sections, we get locus of their centres is another straight line through O", where SX'X= SXX', and in the same plane as the other and equally inclined to the line O'O". They, therefore, intersect unless they are parallel. If they intersect, the point is called the centre and the surface a central quadric; but if they are parallel, the centre will be at an infinite distance, and they are then called paraboloids.

If we draw a plane through S perpendicular to XX', the surface is evidently symmetrical on both sides, since the opposite systems of circular sections are similar. This plane will go through the centre of the surface. We also see that there are two other planes about which the surface is symmetrical, viz. that containing the two cyclic lines, and that perpendicular to this and bisecting the other angle between them, and therefore perpendicular to the first plane.

These three planes are perpendicular to each other, and are called the principal planes of the quadric.

9. We will now show that the plane section through the cyclic lines is a conic.

We know that the quadric can be generated by two systems of circles, whose centres lie on two intersecting straight lines.

Let (fig. 24) C be the centre, CD, CD, the cyclic conjugates, CP, CP' the lines in which the plane of section cuts the central cyclic planes.

Then CP CP,

=

= a given quantity.

Describe the conic in which CD, CP and CD1, CP, are two sets of conjugate diameters; CP, CP, will be equal semidiameters.

The axes will bisect the angles between CP and CP ̧.

Let them be CA, CB.

If CD and CP lie on opposite sides of CB, we get an ellipse; if on the same side, an hyperbola.

=

2

This conic will be the required section. For P is a point on the section. Draw PNP, parallel to CP, meeting CD, in N, and make Np, PN. PN. Then P1 is a point on the section, but it is also on the conic. Next draw p‚N2P2 parallel to CP cutting CD in N, and make NP2 = P1‚Ñ2 then P2 is a point on the section and also on the conic, and so on. By this means we can find any number of points on the section, which will all lie on the above conic.

Hence the section is the above conic.

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10. It easily follows from this that if we take any plane section through a cyclic line it is a conic.

Also that if we draw a plane through the extremity D parallel to a corresponding cyclic plane, then it will be a tangent plane to the surface, for the section of it by any plane through CD will cut it in a tangent to the section of the surface.

Also since any section through CD is a conic, whose centre is C, it is clear that if any straight line be drawn through the centre and cutting the quadric it will be bisected at C. This is the reason C is the called the centre.

11. All plane sections of a quadric are conics.

Draw the planes parallel to a cyclic plane that touch the section by the given plane, and let the plane of the paper be the plane through a cyclic line and one of the points of

contact B.

Let (fig. 25) CD be the cyclic line, BC'B' the trace of the cutting plane, A CA' the trace of a parallel plane through the centre, RSR' the central cyclic plane, CS its intersection with plane through ACA'.

Draw any cyclic plane cutting the plane of paper in QUQ, and the cutting plane in PU. Then, by the construction, it is evident that PUQ is a right angle.

=

Therefore, PU QU.Q'U by property of the circle.
Again, since DAD'A' is a conic,

where C'S' is parallel to CS.

Therefore, locus of P is a conic, whose diameter is in the direction of BB' and 2B'C', 2C'S' are conjugates.

C'S'

Since

C'B'

CS

CA'

a quadric are similar.

we see that all parallel sections of

12. We see that locus of C' is a straight line, and it is called the conjugate of the plane ASA'.

It has the following properties.

1. It is the locus of the centres of sections parallel to its conjugate plane.

2. Any straight line through it parallel to its conjugate plane is bisected by it.