The projection of a straight line is made by drawing a plane through it and through the centre of projection. Thus if we draw the plane cab and produce it to meet the plane of projection in a'b', this line a'b' will be the projection of ab. In parallel projection we must draw through the line a plane parallel to the projecting lines, like the plane ab a'b' in the second figure. We see in this way that the projection of a straight line is always a straight line, and that, since the line and its projection are in the same plane, they must either meet at a finite distance or be parallel (meet at an infinite distance).

In parallel projection, parallel lines are projected into parallel lines, and the ratio of their lengths is unaltered. Through the parallel lines ab, cd

we must draw the planes aba'b', cdc'd' both parallel to the projecting lines, and therefore parallel to each other. These planes will consequently be cut by the plane of projection in the parallel lines ab', c'd'. Moreover the triangles pbb', qdd, having their respective

Ст

a

sides parallel, are similar; therefore pb : qd = pb': qd', and so also ab : cd=a'b' : c'ď.

The orthogonal projection of a finite straight line on a straight line or plane is equal in length to the length of the projected line multiplied by the cosine of its inclination to the straight line or plane. If pq is the

projection of PQ, draw pq equal and parallel to PQ. Then Qg is parallel to Pp and therefore perpendicular to pq; therefore the plane Qqq' is perpendicular to pq, and therefore q'q is perpendicular to pq. Hence pq=pq' cos 1pqPQ x cosine of angle between PQ and pq.

The orthogonal projection of an area on a plane is equal to the area multiplied by the cosine of its inclination to the

PROJECTION OF AN AREA.

A

A

B

D

27

plane. This is clearly true for a rectangle ABCD, one of whose sides is parallel to the line of intersection of the planes. For the side AB is unaltered, and the other, BC, is altered into Bc, which is BC cos 0. Hence it is true for any area which can be made up of such rectangles. But any area A can be divided into such rectangles together with pieces over, by drawing lines across it at equal distances perpendicular to the intersection of the two planes, and then lines parallel to the intersection through the points when they meet the boundary. All these pieces over, taken together, are less than twice the strip whose height PQ is the difference in height between the lowest and highest point of the area; for those on either side of it can be slid sideways into that strip so as not to fill it. And by increasing the number of strips, and diminishing their breadth, we can make this as small as we like. Let then A' be the sum of the rectangles, then A' can be made to differ from A as little as we like. Now the projection of A' is A' cos 0, and this can be made to differ from the projection of A as little as we like. Therefore there can be no finite difference between the projection of A and Acos 0, because A' cos can be made to differ as little as we like from both of them.

PROPERTIES OF THE ELLIPSE.

The ellipse may be defined in various ways, but for our purposes it is most convenient to define it as the parallel projection of a circle. This definition leads most easily to those properties of the curve which are chiefly useful in dynamic.

Centre. The centre of a circle bisects every chord passing through it; such a chord is called a diameter.

The projection of the centre of the circle is a point having the same property in regard to the ellipse, which is therefore called the centre of the ellipse. For let aca' be the projection of ACA'; then ac ca AC: CA'; but AC CA', therefore acca'. It follows also that if any two chords bisect one another, their intersection is the centre.

=

:

Conjugate Diameters. The tangents at the extremity of a diameter of a circle, AA', are perpendicular to that diameter; if we draw another diameter BB' perpendicular to AA', and therefore parallel to these tangents, the tangents at the extremities of BB' will be perpendicular to BB', and therefore parallel to AA'. It follows that in the ellipse, if we draw a diameter bb' parallel to the tangents at the ends of aa' the projection of AA', this line bb' will be the projection of BB', for parallel lines project into parallel lines; therefore also the tangents at the extremities of bb' will be parallel to aa'. Such diameters are called conjugate diameters; they are projections of perpendicular diameters of the circle.

Each of the diameters AA', BB' bisects all chords parallel to the other; thus AA' bisects PQ in the point R. Now PQ is projected into a chord pq parallel to bb', and the middle point R is projected into the middle point r. Hence also in the ellipse, each of two conjugate diameters bisects all chords parallel to the other.

γ.

The assumption here made, that a tangent to the circle projects into a tangent to the ellipse, may be justified as follows. If we take a line PQ cutting the circle in two points, and move it away from the centre until these two points coalesce into one, as at A, the

PROPERTIES OF THE ELLIPSE.

29

line becomes a tangent. Now when these two points. coalesce, their projections must also coalesce; therefore when the line becomes a tangent to the circle, its projection also becomes a tangent to the ellipse.

Relation between ordinate and abscissa. In the circle, if PM, PL be drawn parallel to CB, CA respectively, we know that CPCM2 + MP2, and since CP = CA = CB, it follows that

L

C

B

B'

M

b

C

la

Hence it is equally true in the ellipse that +

=

1.

cm2 mp2 ca2 cb2 For the ratio of parallel lines being unaltered by parallel projection, cm: ca= CM: CA, and mp: cb= MP: CB. The line mp is called an ordinate or standing-up line, and cm is called an abscissa or part cut off. If we write x for cm, y for mp, a for ca, b for cb, the equation becomes

The same relation may be expressed in another form which is sometimes more useful. Namely, observing that the rectangle a'm. ma = (ca + cm) (ca — cm) = ca3 — cm3, we find that mp3: cb=a'm.ma: ca. This may also be proved directly by observing that it is true for the circle and that the ratios involved are ratios of parallel lines.

This relation shews that when two conjugate diameters are given in magnitude and position, the ellipse is completely determined. For through every point m in aa' we can draw a line parallel to bb', and the points p, p' where this line meets the ellipse are fixed by the equation

mp2 (or mp'2): cb' a'm. ma: ca2.

=

Axes. The longest and shortest diameters of an ellipse are conjugate and perpendicular to each other. We may shew in general that if the distance of a curve from a fixed point o increases up to a point a and then decreases,

the tangent at a, if any, will be perpendicular to oa. We say if any, because the curve might have a sharp point at a, and then there would be properly speaking no tangent at Since the distance from o increases up to a and then decreases, we can find two points p, q, one on each side of a, such that the lengths op, oq are equal. Then the perpen

a.

p

dicular from o on the line pq will fall midway between and q. Now suppose p and q to move up towards a, keeping always the lengths op, og equal; then the foot of the perpendicular on pq will always lie between p and q. When therefore the line pq moves on until Ρ and q coalesce at a, the foot of the perpendicular will coalesce with them, or oa is perpendicular to the tangent at a.

The length oa is called a maximum value of the distance from o. It need not be absolutely the greatest value, but it must be greater than the values immediately close to it on either side. A similar demonstration applies to a point where the distance, after decreasing, begins to increase; that is, to a minimum value of the distance.

Applying these results to the Ellipse, we see that the tangents at the extremities of the longest and shortest diameters (which of course must be points of greatest and least distance from the centre)

B

are perpendicular to those diameters. Let bb' be the shortest diameter, and draw aa' perpendicular to it, and therefore parallel to the tangents at b, b'; then aa' is conjugate to bb', and consequently the tangents at a, a' are parallel to bb', and therefore perpendicular to aa'.

Therefore aa' and bb' are conjugate diameters per