Page:EB1911 - Volume 22.djvu/447

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skew-symmetry, hence of involuticn. Figures which are identically equal are of course projective, and they are perspective when placed so that they have an axis or acentre of symmetry (cf. Henrici, Elementary Geometry, Congruent Figures). In this case again the relation is that of involution. The importance of treating similar figures when in perspective position has long been recognized; we need only mention the well-known proposition about the centres of similitude of circles. Applications to Conics.

§ 2I. Any conic can be projected into any other conic. This may be done in such a manner that three points on one conic and the tangents at two of them are projected to three arbitrarily selected points and the tangents at two of them on the other.

If u and u' are any two conics, then we have to prove that we can project u in such a manner that five points on it will be projected to points on u'. As the projection is determined as soon as the projections of any four points or four lines are selected, we cannot project any five points of u to any five arbitrarily selected points on u'. But if A, B, C be any three points on u, and if the tangents at B and C meet at D, if further A', B', C' are any three points on u', and if the tangents at B' and C' meet at D', then the plane of u may be projected to the plane of u' in such a manner that the points A, B, C, D are projected to A', B', C', D'. This determines the correspondence (§ 14). The conic u will be projected into a conic, the points A, B, C and the tangents BD and CD to the points A', B', C' and the lines B'D' and C'D', which are tangents to u at B' and C'. The projection of u must therefore (G. § 52) coincide with u', because it is a conic which has three points and the tangents at two of them in common with u'.

Similarly we might have taken three tangents and the points of coh1;tact of two of them as corresponding to similar elements on the ot r.

If the one conic be a circle which cuts the line j, the projection will cut the line at infinity in two points; hence it will be a hyperbola. Similarly, if the circle touches j, the projection will be a parabola; and, if the circle has no point in common with j, the projection will be an ellipse. These curves appear thus as sections of a circular cone, for in case that the two planes of projection are separated the rays project in the circle form such a cone. Any conic may be projec/ei into itself.

If we take any point S in the plane of a conic as centre, the polar of this point as axis of projection, and any two points in which a line through S cuts the conic as corresponding points, then these will be harmonic conjugates with re ard to the centre and the axis. We therefore have involution (§ IIB, and every point is projected into its harmonic conjugate with regard to the centre and the axis hence every point A on the conic into that point A' on the conic in which the line SA' cuts the conic again, as follows from the harmonic properties of pole and polar (G. § 62 seq.). Two conics which cut the line at infinity in the same two points are similar figures and similarly situated—the centre of similitude being in general some finite point.

To prove this, we take the line at infinity and the asymptotes of one as corresponding to the line at infinity and the asymptotes of the other, and besides a tangent to the first as corresponding to a parallel tangent to the other. The line at infinity will then correspond to itself point for point; hence the figures will be similar and similarly situated.§

22. Areas of Parabolic Segments.-One parabola may always be considered as a parallel projection of another in such a manner that any two points A, B on the one correspond to any two points A', B' on the other; that is, the points A, B and the point at infinity on the one may be made to correspond respectively to the points A', B' and the point at infinity on the other, whilst the tangents at A and at infinity of the one correspond to the tangents at B and at infinity of the other. This completely determines the correspondence, and it is parallel projection because the line at infinity corresponds to the line at infinity. Let the tangents at A and B meet at C, and those at A', B' at C'; then C, C will correspond, and so will the triangles ABC and A'B'C' as well as the parabolic segments cut off by the chords AB and A'B'. If (AB) denotes the area of the segment cut oli by the chord AB we have (AB)/ABC = (A'B')/A'B'C'; or

The area of a segment of a parabola stands in a constant ratio to the area of the triangle formed by the chord of the segment and the tangent: at the end points of the chord.


If then (fig. 8) we join the point C to the mid-point M of AB, then this line l will be bisected at D by the parabola (G. § 74), and M the tangent at D will be parallel to AB. Let F this tangent cut AC in E and CB in F, then B by the last theorem

& f§ l Q§ 1D (B D)


where m is some number to be determined. The Figure gives (AB) =ABD-l-(AD) +(BD).

Fig. 8.

Combining both equations, we have


But we have also ABD = % ABC, and ADE = BFD =§ - ABC; hence

%ABC=m (1-%-é) ABC, or m=~§ .

The area of a parabolic segment equals two-thirds of the area of tie tgliarégle formed by the chord and the tangents at the end points of t e c or

§ 2 3. Elliptic Areas.-To consider one ellipse a parallel projection of another we may establish the correspondence as follows. If AC, BD are any pair of conjugate diameters of the one and A'C', B'D' any pair of conjugate diameters of the other, then these may be made to correspond to each other, and the correspondence will be completely determined if the parallelogram formed by the tangents at A, B, C, D is made to correspond to that formed by the tangents at A', B', C', D' (§§ 17 and 21). As the projection of the first conic has the four points A', B', C', D' and the tangents at these points in common with the second, the two ellipses are projected one into the other. Their areas will correspond, and so do those of the parallelograms ABCD and A'B'C'D'. Hence-The area of an ellipse has a constant ratio to the area of any inscribed parallelogram whose diagonals are conjugate diameters, and also to every circumscribed parallelogram whose sides are parallel to conjugate diameters.

It follows at once that-All

parallelograms inscribed in an ellipse whose diagonals are conjugate diameters are equal in area; and All parallelograms circumscribed about an ellipse whose sides are parallel to conjugate diameters are equal in area.-If a, b are the length of the semi-axes of the ellipse, then the area of the circumscribed parallelogram will be 4ab and of the inscribed one 2ab.

For the circle of radius r the inscribed parallelogram becomes the square of area 2r2 and the circle has the-area r21r; the constant ratio of an ellipse to the inscribed parallelogram has therefore also the value érr. Hence-The

area of an ellipse equals abvr.

§ 24. Projective Properties.-The properties of the projection of a figure depend partly on the relative position of the planes-of the figures and the centre of projection, but principally on the properties of the given figure. Points in a line are projected into points in a line, harmonic points into harmonic points, a conic into a conic; but parallel lines are not projected into parallel lines nor right angles into right angles, neither are the projections of equal segments or angles again equal. There are then some properties which remain unaltered by projection, whilst others change. he former are called projective or descriptive, the latter metrical properties of figures, because the latter all depend on measurement. o a triangle and its median lines correspond a triangle and three jines which meet in a point, but which as a rule are not median mes. »

In this case, if we take the triangle together with the line at infinity, we get as the projection a triangle ABC, and some other line j which cuts the sides a, b, c of the triangle in the points Al, Bl, Cl. If we now take on BC the harmonic conjugate A2 to A1 and similarly on CA and AB the harmonic conjugates to B1 and C1 respectively, then the lines AA2, BBQ, CC; will be the projections of the median lines in the given figure. Hence these lines must meet in a point.

As the triangle and the fourth line we may take any four given lines, because any four lines may be projected into any four given lines (§ 14). This gives a theorem:-

If each vertex of a triangle be joined to that point in the opposite side which is, with regard to the vertices, the harmonic conjugate of the point in which the side is cut by a given line, then the three lines thus obtained meet in a point.

We get thus out of the special theorem about the median lines of a triangle a more general one. But before this could be done we had to add the line at infinity to the lines in the given figure. In a similar manner a great many theorems relating to metrical properties can be generalized by taking the line at infinity or points at infinity as forming part of the original figure. Conversely special cases relating to measurement are obtained by projecting some line in a figure of known properties to infinity. This is true for all properties relating to parallel lines or to bisection of segments, but not immediately for angles. It is, however, possible to establish for every metrical relation the corresponding projective property. To do this it is necessary to consider imaginary elements. These have originally been introduced into geometry by aid of co-ordinate geometry, where imaginary quantities constantly occur as roots of equations.

Their introduction into pure geometry is due principally to Poncelet, who by the publication of his great work Traite des Propriétés Projectives des Figures became the founder of projective geometry in its widest sense. Monge had considered parallel projection and had already distinguished between permanent and accidental properties of figures, the latter being those which depended merely on the accidental position of one part to another. hus in projecting two circles which lie in different planes it