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Originally appearing in Volume V22, Page 434 of the 1911 Encyclopedia Britannica.
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ABC AB AB. A'BC A'B _ — BA" or The constant ratio of corresponding areas is equal and opposite to the ratio in which the axis divides the segment joining two corresponding points. - § 18. Several special cases of parallel projection are of interest. Orthographic Projection.—If the two planes a and ar' have a definite position in space, and if a figure in r is projected to r' by rays perpendicular to this plane, then the projection is said to be orthographic. If in this case the plane a be turned till it coincides with a' so that the figures remain perspective, then the projecting rays will be perpendicular to the axis of projection, because any one of these rays is, and remains during the turning, perpendicular to the axis. The constant ratio of the area of the projection so that of the original figure is, in this case, the cosine of the angle between the two planes w and a', as will be seen by projecting a rectangle which has its base in the axis. Orthographic projection is of constant use in geometrical drawing. Shear.—If the centre of projection be taken at infinity on the axis, then the projecting rays are parallel to the axis; hence corresponding points will be equidistant from the axis. In this case, therefore, areas of corresponding figures will be equal. If A, A' and B, B' (fig. 7) are two pairs of corresponding points on the same line, parallel to the axis, then, as corresponding segments parallel to the axis are equal, it follows that AB = A'B', hence also AA'=BB'. If these points be joined to any point 0 on the axis, then AO and A'O will be corresponding lines; they will there-fore be cut by any line parallel to the axis in corresponding points. In the figure therefore C, C' and also D, D' will be pairs of corresponding points and CC' = DD'. As the ratio CC'/AA' equals the ratio of the distances of C and A from the axis, therefore Two corresponding figures may be got one out of the other by moving all points in the one parallel to a fixed line, the axis, through distances which are proportional to their own distances from the axis. Points in a line remain hereby in a line. Such a transformation of a plane figure is produced by a shearing stress in any section of a homogeneous elastic solid. For this reason Lord Kelvin gave it the name of shear. A shear of a plane figure is determined if we are given the axis and the distance through which one point has been moved; for in this case the axis, the centre, and a pair of corresponding points are given. § 19. Symmetry and Skew-Symmetry.—If the centre is not on the axis, and if corresponding points are at equal distances from it, they must be on opposite sides of it. The figures will be in involution (§ Ii). In this case the direction of the projecting rays is said to be conjugate to the axis. The conjugate direction may be perpendicular to the axis. If the line joining two corresponding points A, A' cuts the axis in B, then AB = BA'. Therefore, if the plane be folded over along the axis, A will fall on A'. Hence by this folding over every point will coincide with its corresponding point. The figures therefore are identically equal or congruent, and in their original position they are symmetrical with regard to the axis, which itself is called an axis of symmetry. If the two figures are considered as one this one is said to be symmetrical with regard to an axis, and is said to have an axis of symmetry or simply an axis. Every diameter of a circle is thus an axis; also the median line of an isosceles triangle and the diagonals of a rhombus are axes of the figures to which they belong. In the more general case where the projecting rays are not perpendicular to the axis we have a kind of twisted symmetry which may be called skew-symmetry. It can be got from symmetry by giving the whole figure a shear. It will also be easily seen that we get skew-symmetry if we first form a shear to a given figure and then separate it from its shear by folding it over along the axis of the shear, which thereby becomes an axis of skew-symmetry. Skew-symmetrical and therefore also symmetrical figures have the following properties: Corresponding areas are equal, but of opposite sense. Any two corresponding lines are harmonic conjugates with regard to the axis and a line in the conjugate direction. If the two figures be again considered as one whole, this is said to be skew-symmetrical and to have an axis of skew-symmetry. Thus the median line of any triangle is an axis of skew-symmetry, the side on which it stands having the conjugate direction, the other sides being conjugate lines. From this it follows, for in-stance, that the three median lines of a triangle meet in a point. For two median lines will be corresponding lines with regard to the third as axis, and must therefore meet on the axis. An axis of skew-symmetry is generally called a diameter. Thus every diameter of a conic is an axis of skew-symmetry, the conjugate direction being the direction of the chords which it bisects. § 20. We state a few properties of these figures useful in mechanics, but we omit the easy proofs: If a plane area has an axis of skew-symmetry, then the mass-centre (centre of mean distances or centre of inertia) lies on it. If a figure undergoes a shear, the mass-centre of it*. area remains the mass-centre; and generally In parallel projection the mass-centres of corresponding areas (or of groups of points, but not of curves) are corresponding points. 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 momental 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 Euc. 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 similarly E'F'G'H' = K'L'M'N skew-symmetry, hence of involution. Figures which are identically equal are of course projective, and they are perspective when placed so that they have an axis or a centre 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. § 21. 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 ii 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 (§ i4). 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 a 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 (§ ii), 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 off by the chord AB we have therefore (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 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 l 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 =In' 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 = z ABC, and ADE = BFD = 4 ABC; hence a ABC =m ABC, or m= g. The area of a parabolic segment equals two-thirds of the area of 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 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 tab. For the circle of radius r the inscribed parallelogram becomes the square of area 2r2 and the circle has the area rear; the constant ratio of an ellipse to the inscribed parallelogram has therefore also the value Ir. Hence The area of an ellipse equals abs-. § 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. The former are called projective, or descriptive, the latter metrical properties of figures, because 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 Al, B1, Cl. If we now take on BC the harmonic conjugate A2 to Ai and similarly on CA and AB the harmonic conjugates to Bi and Ci respectively, then the lines AA2, BB2, CC2 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 Proprittes 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. Thus in projecting two circles which lie in different planes it depends on the accidental position of the centre of projection whether the projections be two conics which do or do not meet. Poncelet introduced the principle of continuity in order to make theorems general and independent of those accidental positions which depend analytically on the fact that the equations used have real or imaginary roots. But the correctness of this principle remained without a proof. Von Staudt has, however, shown how it is possible to introduce imaginary elements by purely geometrical reasoning, and we shall now try to give the reader some idea of his theory. § 25. Imaginary Elements.—If a line cuts a curve and if the line be moved, turned for instance about a point in it, it may happen that two of the points of intersection approach each other till they coincide. The line then becomes a tangent. If the line is still further moved in the same manner it separates from the curve and two points of intersection are lost. Thus in considering the relation of a line to a conic we have to distinguish three cases—the line cuts the conic in two points, touches it, or has no point in common with it. This is quite analogous to the fact that a quadratic equation with one unknown quantity has either two, one, or no roots. But in algebra it has long been found convenient to express this differently by sayipg a quadratic equation has always two roots, but these may be either both real and different, or equal, or they may be imaginary. In geometry a similar mode of expressing the fact above stated is not less convenient. We say therefore a line has always two points in common with a conic, but these are either distinct, or coincident, or invisible. The word imaginary is generally used instead of invisible; but, as the points have nothing to do with imagination, we prefer the word " invisible " recommended originally by Clifford. Invisible points occur in pairs of conjugate points, for a line loses always two visible points of intersection with a curve simultaneously. This is analogous to the fact that an algebraical equation with real coefficients has imaginary roots in pairs. Only one real line can be drawn through an invisible point, for two real lines meet in a real or visible point. The real line through an invisible point contains also its conjugate. Similarly there are invisible lines—tangents, for instance, from a point within a conic—which occur in pairs of conjugates, two con ugates having a real point in common. The introduction of invisible points would be nothing but a play upon words unless there is a real geometrical property indicated which can be used in geometrical constructions—that it has a definite meaning, for instance, to say that two conics cut a line in the same two invisible points, or that we can draw one conic through three real points and the two invisible ones which another conic has in common with a line that does not actually cut it. We have in fact to give a geometrical definition of invisible points. This is done by aid of the theory of involution (G. § 76 seq.). An involution of points on a line has (according to G. § 77 [2]) either two or one or no foci. Instead of this we now say it has always two foci which may be distinct, coincident or invisible. These foci are determined by the involution, but they also determine the involution. If the foci are real this follows from the fact that conjugate points are harmonic conjugates with regard to the foci. That it is also the case for invisible foci will presently appear. If we take this at present for granted we may replace a pair of real, coincident or invisible points by the involution of which they are the foci. Now any two pairs of conjugate points determine an involution (G. § 77 [61). Hence any point-pair, whether real or invisible, is completely determined by any two pairs of conjugate points of the involution which has given the point-pair as foci and may therefore be replaced by them. Two pairs of invisible points are thus said to be identical if, and only if, they are the foci of the same involution. We know (G. § 82) that a conic determines on every line an in-volution in which conjugate points are conjugate poles with regard to the conic—that is, that either lies on the polar of the other. This holds whether the line cuts the conic or not. Furthermore, in the former case the points common to the line and the conic are the foci of the involution. Hence we now say that this is always the case, and that the invisible points common to a line and a conic are the invisible foci of the involution in question. If then we state the problem of drawing a conic which passes through two points given as the intersection of a conic and a line as that of drawing a conic which determines a given involution on the line, we have it in a form in which it is independent of the accidental circumstance of the intersections being real or invisible. So is the solution of the problem, as we shall now show. § 26. We have seen (§ 21) that a conic may always be projected into itself by taking any point S as centre and its polar s as axis of projection, corresponding points being those in which a line through S cuts the conic. If then (fig. 9) A, A' and B, B' are pairs of corresponding points so that the lines AA' and BB' pass through S, then the lines AB and A'B', as corresponding lines, will meet at a point R on the axis, and the lines AB' and A'B will meet at another point R' on the axis. These points R, R' are conjugate points in the involution which the conic determines on the line s, 433 because the triangle RSR' is a polar triangle (G. § 62), so that R' lies on the polar of R. This gives a simple means of determining for any point Q on the line s its conjugate point Q'. We take any two points A, A' on the conic which lie on a line through S, join Q to A by a line cutting the conic 'again in C, and join C to A'. This line will cut s in the point Q' required. To draw some conic which shall determine on a line s a given involution. We have here to reconstruct the fig. 9, having given on the line s an involution. Let Q, Q' _ and _R, R' (fig. 9) be two pairs of conjugate points in this C involution. We take any point B and join it to R and R', and another point C to Q and Q'. Let BR and CQ meet at A, and BR' and CQ' at A'. If now a point P be moved along s its conjugate point P' will also move and the two points will describe projective rows The two rays AP and A'P' will therefore describe projective pencils, and the intersection of corresponding rays will lie on a conic which basses through A, A', B and C. This conic determines on s the given involution. Of these four points not only B and C but also the point A may be taken arbitrarily, for if A, B, C are given, the line AB will cut s in some point R. As the involution is supposed known, we can find the point R' conjugate to R, which we join to B. In the same way the line CA will cut s in some point Q. Its conjugate point Q' we join to C. The line CQ' will cut BR' in a point A', and then AA' will pass through the pole S (cf. fig. 9). We may now interchange A and B and find the point B'. Then BB' will also pass through S, which is thus found. At the same time five points A, B, C, A', B' on the conic have been found, so that the conic is completely known which determines on the line s the given involution. Hence Through three points we can always draw one conic, and only one, which determines on a given line a given involution, all the same whether the involution has real, coincident or invisible foci. In the last case the theorem may now also be stated thus: It is always possible to draw a conic which passes through three given real points and through two invisible points which any other conic has in common with a line. § 27. The above theory of invisible points gives rise to a great number of interesting consequences, of which we state a few. The theorem at the end of § 21 may now be stated Any two conics are similar and similarly situated if they cut the line at infinity in the same two points—real, coincident or invisible. It follows that Any two parabolas are similar; and they are similarly situated as soon as their axes are parallel. The involution which a circle determines at its centre is circular (G. § 79); that is, every line is perpendicular to its conjugate line. This will be cut by the line at infinity in an involution which has the following property: The lines which join any finite point to two conjugate points in the involution are at right angles to each other. Hence all circular involutions in a plane determine the same involution on the line at infinity. The latter is therefore called the circular involution on the line at infinity; and the involution which a circle determines at its centre is called the circular involution at that point. All circles determine thus on the line at infinity the same involution; in other words, they have the same two invisible points in common with the line at infinity. Ml circles may be considered as passing through the same two points at infinity. These points are called the circular points at infinity, and by Professor Cayley the absolute in the plane. They are the foci of the circular involution in the line at infinity. Conversely—Every conic which passes through the circular points is a circle; because the involution at its centre is circular, hence conjugate diameters are at right angles, and this property only circles possess. We now see why we can draw always one and only one circle through any three points; these three points together with the circular points at infinity are five points through which one conic only can be drawn. Any two circles are similar and similarly situated because they have the same points at infinity (§ 21). Any two concentric circles may be considered as having double contact at infinity, because the lines joining the common centre to the circular points at infinity are tangents to both circles at the circular points, as the line at infinity is the polar of the centre. Any two lines at right angles to one another are harmonic conjugates with regard to the rays joining their intersection to the circular points, because these rays are the focal rays of the circular involution at the intersection of the given lines. To bisect an angle with the vertex A means (G. § 23) to find two rays through A which are harmonic conjugates with regard to the limits of the angle and perpendicular to each other. These rays are therefore harmonic with regard to the limits of the given angle and with regard to the rays through the circular points. Thus perpendicularity and bisection of an angle have been stated in a projective form. It must not be forgotten that the circular points do not exist at all; but to introduce them gives us a short way of making a statement which would otherwise be long and cumbrous. We can now generalize any theorem relating to metrical properties. For instance, the simple fact that the chord of a circle is touched by a concentric circle at its mid point proves the theorem If two conics have double contact, then the points where any tangent to one of them cuts the other are harmonic with regard to the point of contact and the point where the tangent cuts the chord of contact. (0. H.)
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