798 PROJECTION The moment of inertia, of a plane figure does not change if the figure undergoes a shear in the direction of the axis with regard to which the moment has been taken. If a figure has an axis of skew-symmetry, then this axis and the conjugate direction are conjugate diameters of the momcntal ellipse for every point in the axis. If a figure has an axis of symmetry, then this is an axis of the momental ellipse for every point in it. The truth of the last propositions follows at once from the fact that the product of inertia for the lines in question vanishes. It is of interest to notice how a great many propositions of Euclid are only special cases of projection. The theorems Eucl. I. 35-41 about parallelograms or triangles on equal bases and between the same parallels are examples of shear, whilst I. 43 gives a case of skew-symmetry, hence of involution. Figures which are identi cally equal are of course protective, and they are perspective when placed so that they have an axis or a centre of symmetry (comp. Henrici, Elementary Geometry, Congruent Figures). In this case again the relation is that of involution. The importance of treat ing 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. 21. THEOREM. 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 tJie tangents at two of them on the other. Proof. If u and u are any two conies, 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 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 contact of two of them as corresponding to similar elements on the other. 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 an hyper bola. 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 projecting the circle form such a cone. Any conic may be projected 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 regard to the centre and the axis. We therefore have involution ( 11), 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 sq.). Two conies 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 tangent at B and at infinity of the other. This completely deter mines 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, Cf 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 off by the chord AB we have therefore (AB)/ABC-(A B )/A B C j 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 tangents 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 I will be bisected at D by the parabola (G. 74), and the tangent at D will be parallel to AB. Let this tangent cut AC in E and CB in F, then by the last theorem (AB) (AD) (BD) ABC ADE BFD where m is some number to be determined. The figure gives (AB)-ABD + (AD) + (BD). Combining both equations, we have ABD = m (ABC - ADE - BFD). But we have also ABD =4 ABC, and ADE = BFD = * ABC ; hence i ABC = HI (1 - - i) ABC, or HI = . The area of a parabolic segment equals two thirds of the area if the triangle formed by the chord and the tangents at the end points of the chord. 23. ELLIPTIC AREAS. To consider one ellipse a parallel projec tion 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 , I) ( 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 pro jected 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 arc 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 ivhosc sides are parallel to conjugate diameters are equal in area. If a, b are the length of the semi-axes of the ellii se, then the area of the circumscribed parallelogram will be 4ab and of the inscribed one lab. For the circle of radius r the inscribed parallelogram becomes the square of area 2/- 2 and the circle has the arear 2 ir ; the constant ratio of an ellipse to the inscribed parallelogram has therefore also the value -K. Hence The area of an ellipse equals abir. 24. PHOJECTIVE PROPERTIES. The properties of the projection of a figure depend partly on the relative position of the planes of the figure and the centre of projection, but principally on the pro perties 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 pro perties which remain unaltered by projection, whilst others change. The former are called projective or descriptive, the latter metrical properties of figures, becau-e the latter all depend on measurement. To a triangle and its median lines correspond a triangle and three lines which meet in a point, but which as a rule are not median lines. 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 A 1( BJ, Cj. If we now take on BC the harmonic conjugate A 2 to Aj and similarly on CA and AB the harmonic conjugates to Bj and C, respectively, then the lines AA 2 , BB 2 , CG, 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 projeuted 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