Projection, in mathematics. If from a fixed point S in space lines or rays be drawn to different points A, B, C, ... in space, and if these rays are cut by a plane in points A', B', C', ... the latter are called the projections of the given points on the plane. Instead of the plane another surface may be taken, and then the points are projected to that surface instead of to a plane. In this manner any figure, plane or in space of three dimensions, may be projected to any surface from any point which`is called the centre of projection. If the figure projected is in three dimensions then this projection is the same as that used in what is generally known as perspective. In modern mathematics the word projection is often taken with a slightly different meaning, supposing that plane figures are projected into plane figures, but three-dimensional ones into three-dimensional figures. Projection in this sense, when treated by co-ordinate geometry, leads in its algebraical aspect to the theory of linear substitution and hence to the theory of invariants and co-variants (see Algebraic Forms).
In this article projection will be treated from a purely geometrical point of view. References like (G. § 87) relate to the article Geometry, § Projective, in vol. xi.
§ I. Projection of Plane Figures. - Let us suppose we have in space two planes 7r and 7r'. In the plane 7 a figure is given having known properties; then we have the problem to find its projection from some centre S to the plane 7r', and to deduce from the known properties of the given figure the properties of the new one.
If a point A is given in the plane 7r we have to join it to the centre S and find the point A' where this ray SA cuts the plane 7r'; it is the projection of A. On the other hand if A' is given in the plane Tr', then A will be its projection in 7r. Hence if one figure in 7r' is the projection of another in 7, then conversely the latter is also the projection of the former. A point and its projection are therefore also called corresponding points, and similarly we speak of corresponding lines and curves, &c.
§ 2. We at once get the following properties: The projection of a point is a point, and one point only. The projection of a line (straight line) is a line; for all points in a line are projected by rays which lie in the plane determined by S and the line, and this plane cuts the plane 7r' in a line which is the projection of the given line.
If a point lies in a line its projection lies in the projection of the line. The projection of the line joining two points A, B is the line which joins the projections A', B' of the points A, B. For the projecting plane of the line AB contains the rays SA, SB which project the points A, B.
The projection of the point of intersection of two lines a, b is the point of intersection of the projections a', b' of those lines. Similarly we get The projection of a curve is a curve. The projections of the points of intersection of two curves are the points of intersection of the projections of the given curves. If a line cuts a curve in n points, then the projection of the line cuts the projection of the curve in n points. Or The order of a curve remains unaltered by projection. The projection of a tangent to a curve is a tangent to the projection of the curve. For the tangent is a line which has two coincident points in common with a curve.
The number of tangents that can be drawn from a point to a curve remains unaltered by projection. Or The class of a curve remains unaltered by projection. § 3. Two figures of which one is a projection of the other obtained in the manner described may be moved out of the position in which they are obtained. They are then still said to be one the projection of the other, or to be projective or homographic. But when they are in the position originally considered they are said to be in perspective position, or (shorter) to be perspective. All the properties stated in §§ I, 2 hold for figures which are projective, whether they are perspective or not. There are others which hold only for projective figures when they are in perspective position, which we shall now consider.
If two planes and 7r' are perspective, then their line of intersection is called the axis of projection. Any point in this line coincides with its projection. Hence All points in the axis are their own projections. Hence also - Every line meets its projection on the axis. § 4. The property that the lines joining corresponding points all pass through a common point, that any pair of corresponding points and the centre are in a line, is also expressed by saying that the figures are co-linear or co-polar; and the fact that both figures have a line, the axis, in common on which corresponding lines meet is expressed by saying that the figures are co-axal.
The connexion between these properties has to be investigated.
For this purpose we consider in the plane a triangle ABC, and let the lines BC, CA, AB be denoted by a, b, c. The projection will consist of three points A', B', C' and three lines a', b', c'. These have such a position that the lines AA', BB', CC' meet in a point, viz. at S, and the points of intersection of a and a', b and b', c and c' lie on the axis (by § 2). The two triangles therefore are said to be both co-linear and co-axal. Of these properties either is a consequence of the other, as will now be proved.
If two triangles, whether in the same plane or not, are co-linear they are co-axal. Or If the lines AA', BB', CC' joining the vertices of two triangles meet in a point, then the intersections of the sides BC and B'C', CA and C'A', AB and A'B' are three points in a line. Conversely If two triangles are co-axal they are co-linear. Or If the intersection of the sides of two triangles ABC and A'B'C', 'viz.' of BC and B'C', of CA and C'A', and of AB and A'B', lie in a line, then the lines AA', BB',' and CC' meet in a point. Proof - Let us first suppose the triangles to be in different places. By supposition the lines AA', BB', CC' (fig. I) meet in a point S. But three intersecting lines determine three planes, SCB, SCA and SAB. In the first lie the points B, C and also B', C'. Hence the lines BC and B'C' will intersect at some point P, because any two lines in the same plane intersect. Similarly CA and C'A' will intersect at some point Q, and AB and A'B' at some point R. These points P, Q, R lie in the plane of the triangle ABC because they are points on the sides of this triangle, and similarly in the plane of the triangle A'B'C'. Hence they lie in the intersection of two planes - that is, in a line. This line (PQR in fig. I) is called the axis of perspective or homology, and the AA', BB', CC', i.e. S' in the figure, the centre of per Secondly, if the triangles ABC and A'B'C' lie both in the same plane the above proof does not hold. In this case we may consider the plane figure as the projection of the figure in space of which we have just proved the theorem. Let ABC, A'B'C' be the co-linear triangles with S as centre, so that AA', BB', CC' meet at S. Take now any point in space, say your eye E, and from it draw the rays projecting the figure. In the line ES take any point Si, and in EA, EB, EC take points A 1, B 1, C i respectively, but so that Si, A1, B1, C1 are not in a plane. In the plane ESA which projects the line S,A1 lie then the line S1A1 and also EA'; these will therefore meet in a point A1', of which A' will be the projection. Similarly points B1', C1' are found. Hence we have now in space two triangles A,B 1 C 1 and AI'BI'Cl' which are co-linear. They are therefore coaxal, that is, the points PI, Qi, R I, where A,Bi, &c., meet will lie in a line. Their projections therefore lie in a line. But these are the points P, Q, R, which were to be proved to lie in a line.
This proves the first part of the theorem. The second part or converse theorem is proved in exactly the same way. For another proof see (G. § 37).
§ 5. By aid of this theorem we can now prove a fundamental property of two projective planes.
Let s be the axis, S the centre, and let A, A' and B, B' be two pairs of corresponding points which we suppose fixed, and C, C' any other pair of corresponding points. Then the triangles ABC and A'B'C' are co-axal, and they will remain co-axal if the one plane Tr' be turned relative to the other about the axis. They will therefore, by Desargue's theorem, remain co-linear, and the centre will be the point S', where AA' meets BB'. Hence the line joining any pair of corresponding points C, C' will pass through the centre S'. The figures are therefore perspective. This will remain true if the planes are turned till they coincide, because Desargue's theorem remains true.
If two planes are perspective, then if the one plane be turned about the axis through any angle, especially if the one plane be turned till it coincides with the other, the two planes will remain perspective; corresponding lines will still meet on a line called the axis, and the lines joining corresponding points will still pass through a common centre S situated in the plane. Whilst the one plane is turned this point S will move in a circle whose centre lies in the plane 7r, which is kept fixed, and whose plane is perpendicular to the axis. The last part will be proved presently. As the plane 7r' may be turned about the axis in one or the opposite sense, there will be two perspective positions possible when the planes coincide.
§ 6. Let (fig. 2) 7r, ir' be the planes intersecting in the axis s whilst S is the centre of projection. To project a point A in it we join A to S and see where this line cuts 7. This gives the point A'. But if we draw through S any line parallel to 7r, then this line will cut 7r' in some point I', and if all lines through S be drawn which are parallel to 7r these will form a plane parallel to 7r which will cut the plane in a line i' parallel to the axis s. If we say that a line parallel to a plane cuts the latter at an infinite distance, we may say that all points at an infinite distance in 7 are projected into points which lie in a straight line i', and conversely all points in the line are projected to an infinite distance in 7r, whilst all other points are projected to finite points. We say therefore that all points in the plane it at an infinite distance may be considered as lying in a straight line, because their projections lie in a line. Thus we are again led to consider points at infinity in a plane as lying in a line (cf. G. §§ 2-4).
Similarly there is a line j in it which is projected to infinity in 7r'; this projection will be denoted by j so that i and j are lines at infinity.
§ 7. If we suppose through S a plane drawn perpendicular to the axis s cutting it at T, and in this plane the two lines SI' parallel to 7 and SJ parallel to 7r', then the lines through I' and J intersection of spective.
FIG. I.
FIG. 2.
parallel to the axis will be the lines i' and j. At the same time a parallelogram Sjti'S has been formed. If now the plane ir' be turned about the axis, then the points I' and J will not move in their planes; hence the lengths TJ and TI', and therefore also SI' and SJ, will not change. If the plane 7 is kept fixed in space the point J will remain fixed, and S describes a circle about J as centre and with SJ as radius. This proves the last part of the theorem in § 5.
s
T
.1
J
I'
§ 8. The plane 7r' may be turned either in the sense indicated by the arrow at Z or in the opposite sense till ir' falls into 7r. In the first case we get a figure like fig. 3; i' and j will he on the same side of the axis, and on this side will also lie the centre S; and J J FIG._3. FIG. 4.
then ST =SJ +SI' or SI' = JT, SJ = I'T. In the second case (fig. 4) i' and j will be on opposite sides of the axis, and the centre S will lie between them in such a position that I'S=TJ and I'T =SJ. If I'S=SJ, the point S will lie on the axis.
It follows that any one of the four points S, T, J, I' is completely determined by the other three: if the axis, the centre, and one of the lines i' or j are given the other is determined; the three lines s, i', j determine the centre; the centre and the lines i', j determine the axis.
§. 9. We shall now suppose that the two projective planes 7r, 7r' are perspective and have been made to coincide.
If the centre, the axis, and either one pair of corresponding points on a line through the centre or one pair of corresponding lines meeting on the axis are given, then the whole projection is determined. Proof. - If A and A' (fig. 1) are given corresponding points, it has to be shown that we can find to every other point B the corresponding point B'. Join AB to cut the axis in R. Join RA'; then B' must lie on this line. But it must also lie on the line SB. Where both meet is B'. That the figures thus obtained are really projective can be seen by aid of the theorem of § 4. For, if for any point C the corresponding point C' be found, then the triangles ABC and A'B'C' are, by construction, co-linear, hence co-axal; and s will be the axis, because AB and AC meet their corresponding lines A'B' and A'C' on it. BC and B'C' therefore also meet on s. If on the other hand a, a' are given corresponding lines, then any line through S will cut them in corresponding points A, A' which may be used as above.
§ 10. Rows and pencils which are projective or perspective have been considered in the article Geometry (G. §§ 12-40). All that has been said there holds, of course, here for any pair of corresponding rows or pencils. The centre of perspective for any pair of corresponding rows is at the centre of projection 5, whilst the axis contains coincident corresponding elements. Corresponding pencils on the other hand have their axis of perspective on the axis of projection whilst the coincident rays pass through the centre.
We mention here a few of those properties which are independent of the perspective position: - The correspondence between two projective rows or pencils is completely determined if to three elements in one the corresponding ones in the other are given. If for instance in two projective rows three pairs of corresponding points are given, then we can find to every other point in either the corresponding point (G. §§ 29-36).
If A, B, C, D are four points in a row and A', B', C', D' the corresponding points, then their cross-ratios are equal (AB, CD) = (A'B', C'D') - where (AB, CD) =AC/CB :AD/DB.
If in particular the point D be at infinity we have (AB, CD) = - AC/CB =AC/BC. If therefore the points D and D' are both at infinity we have AC/BC =AD/BD, and the rows are similar (G. § 39). This can only happen in special cases. For the line joining corresponding points passes through the centre; the latter must therefore lie at infinity if D, D' are different points at infinity. But if D and D' coincide they must lie on the axis, that is, at the point at infinity of the axis unless the axis is altogether at infinity. Hence In two perspective planes every row which is parallel to the axis is similar to its corresponding row, and in general no other row has this property. But if the centre or the axis is at infinity then every row is similar to its corresponding row. In either of these two cases the metrical properties are particularly simple. If the axis is at infinity the ratio of similitude is the same for all rows and the figures are similar. If the centre is at infinity we get parallel projection; and the ratio of similitude ,changes from row to row (see §§ 16, 17).
In both cases the mid-points of corresponding segments will be corresponding points. § 11. Involution. - If the planes of two projective figures coincide, then every point in their common plane has to be counted twice, once as a point A in the figure 7, once as a point B' in the figure 7r'. The points A' and B corresponding to them will in general be different points; but it may happen that they coincide. Here a theorem holds similar to that about rows (G. §§ 7 6 seq.).
that is to say, to every point will correspond the same point in the first as in the second plane. In this case the figures are said to be in involution. Proof. - Let (fig. 5) 5 be the centre, s the axis of projection, and let a point denoted by A in the first plane and by B' in the second have the property that the S points A' and B corresponding to them again coincide. Let C and D' be the names which some other point has in the two planes. If the line AC cuts the axis in X, then the point where the line XA' cuts SC will be the point C' corresponding to C (§ 9). The line B'D' also cuts the axis in X, and therefore the point D corresponding to D' is the point where XB cuts SD'. But this is the same point as C'.
This point C' might also be got by drawing CB and joining its intersection Y with the axis to B'. Then C' must be the point where B'Y meets SC. This figure, which now forms a complete quadrilateral, shows that in order to get involution the corresponding points A and A' have to be harmonic conjugates with regard to S and the point T where AA' cuts the axis.
If two perspective figures be in involution, two corresponding points are harmonic conjugates with regard to the centre and the point where the line joining them cuts the axis. Similarly Any two corresponding lines are harmonic conjugates with regard to the axis and the line from their point of intersection to the centre. Conversely If in two perspective planes one pair of corresponding points be harmonic conjugates with regard to the centre and the point where the line joining them cuts the axis, then every pair of corresponding points has this property and the planes are in involution. § 12. Projective Planes which are not in perspective position. - We return to the case that two planes 7r and 7r' are projective but not in perspective position, and state in some of the more important cases the conditions which determine the correspondence between them. Here it is of great advantage to start with another definition which, though at first it may seem to be of far greater generality, is in reality equivalent to the one given before.
We call two planes projective if to every point in one corresponds a point in the other, to every line a line, and to a point in a line a point in the corresponding line, in such a manner that the cross-ratio of four points in a line, or of four rays in a pencil, is equal to the cross-ratio of the corresponding points or rays. The last part about the equality of cross-ratios can be proved to be a consequence of the first. As space does not allow us to give an exact proof for this we include it in the definition.
If one plane is actually projected to anott _r .ve get a correspondence which has the properties required in the new definition. This shows that a correspondence between two planes conform to this definition is possible. That it is also definite we have to show. It follows at once that Corresponding rows, and likewise corresponding pencils, are projective in the old sense (G. §§ 25, 30). Further If two planes are projective to a third they are projective to each other. The correspondence between two projective planes and 7r' is determined if we have given either two rows u, v in 7r and the corresponding rows u', v' in 7r', the point where u and v meet corresponding to the points where u' and v' meet, or two pencils U, V ir and the corresponding pencils U', V' in 7r', the ray UV joining the centres of the pencils in corresponding to the ray U'V'. It is sufficient to prove the first part. Let any line a cut u, v in the points A and B. To these will correspond points A' and B' in u' and v' which are known. To the line a corresponds then the line A'B'. Thus to every line in the one plane the corresponding line in the other can be found, hence also to every point the corresponding point.
§ 13. If the planes of two projective figures coincide, and if either four points, of which no three lie in a line, or else four lines, of which no three pass through a point, in the one coincide with their corresponding points, or lines, in the other, then every point and every line coincides with its corresponding point or line so that the figures are identical. If the four points A, B, C, D coincide with their corresponding points, then every line joining two of these points will coincide with s T FIG. 5.
its corresponding line. Thus the lines AB and CD, and therefore also their point of intersection E, will coincide with their corresponding elements. The row AB has thus three points A, B, E coincident with their corresponding points, and is therefore identical with it (§ io). As there are six lines which join two and two of the four points A, B, C, D, there are six lines such that each point in either coincides with its corresponding point. Every other line will thus have the six points in which it cuts these, and therefore all points, coincident with their corresponding points. The proof of the second part is exactly the same. It follows § 14. If two projective figures, which are not identical, lie in the same plane, then not more than three points which are not in a line, or three lines which do not pass through a point, can be coincident with their corresponding points or lines. If the figures are in perspective position, then they have in common one line, the axis, with all points in it, and one point, the centre, with all lines through it. No other point or line can therefore coincide with its corresponding point or line without the figures becoming identical.
It follows also that The correspondence between two projective planes is completely determined if there are given - either to four points in the one the corresponding four points in the other provided that no three of them lie in a line, or to any four lines the corresponding lines provided that no three of them pass through a point. To show this we observe first that two planes 7, 7' ".may be made projective in such a manner that four given points A, B, C, D in the one correspond to four given points A', B', C', D' in the other; for to the lines AB, CD will correspond the lines A'B' and C'D', and to the intersection E of the former the point E' where the latter meet. The correspondence between these rows is therefore determined, as we know three pairs of corresponding points. But this determines a correspondence (by § 12). To prove that in this case and also in the case of § 12 there is but one correspondence possible, let us suppose there were two, or that we could have in the plane two figures which are each projective to the figure in 7 and which have each the points A'B'C'D' corresponding to the points ABCD in 7r. Then these two figures will themselves be projective and have four corresponding points coincident. They are therefore identical by § 13.
Two projective planes will be in perspective if one row coincides with its corresponding row. The line containing these rows will be the axis of projection. As in this case every point on s coincides with its corresponding point, it follows that every row a meets its corresponding row a' on s where corresponding points are united. The two rows a, a' are therefore perspective (G. § 30), and the lines joining corresponding points will meet in a point S. If r be any one of these lines cutting a, a' in the points A and A' and the line s at K, then to the line AK corresponds A'K, or the ray r corresponds to itself. The points B, B' in which r cuts another pair b, b of corresponding rows must therefore be corresponding points. Hence the lines joining corresponding points in b and b also pass through S. Similarly all lines joining corresponding points in the two planes 7r and 7r meet in S; hence the planes are perspective.
The following proposition is proved in a similar way: Two projective planes will be in perspective position if one pencil coincides with its corresponding, one. The centre of these pencils will be the centre of perspective. In this case the two planes must of course coincide, whilst in the first case this is not necessary.
§ 15. We shall now show that two planes which are projective according to definition (§ 12) can be brought into perspective position, hence that the new definition is really equivalent to the old. We use the kilo wing property: If two coincident planes and 7' are perspective with S as centre, then any two corresponding rows are also perspective with S as centre. This therefore is true for the row j and j' and for i and i', of which i and j are the lines at infinity in the two planes. If now the plane 7' be made to slide on 7 so that each line moves parallel to itself, then the point at infinity in each line, and hence the whole line at infinity in, remains fixed. So does the point at infinity on j, which thus remains coincident with its corresponding point on j', and therefore the rows j and j' remain perspective, that is to say the rays joining corresponding points in them meet at some point T. Similarly the lines joining corresponding points in i and i will meet in some point T'. These two points T and T' originally coincided with each other and with S.
Conversely, if two projective planes are placed one on the other, then as soon as the lines j and i are parallel the two points T and T' can be found by joining corresponding points in j and j', and also in i and i'. If now a point at infinity is called A as a point in and B' as a point in 7r', then the point A' will lie on i' and B on j, so that the line AA' passes through T' and BB' through T. These two lines are parallel. If then the plane 7' be moved parallel to itself till T' comes to T, then these two lines will coincide with each other, and with them will coincide the lines AB and A'B'. This line and similarly every line through T will thus now coincide with its corresponding line. The two planes are therefore according to the last theorem in § 14 in perspective position.
It will be noticed that the plane -ir' may be placed on 7 in two different ways, viz. if we have placed -ir' on we may take it off and turn it over in space before we bring it back to 7r, so that what was its upper becomes now its lower face. For each of these positions we get one pair of centres T, T', and only one pair, because the above process must give every perspective position. It follows - In two projective planes there are in general two and only two pencils in either such that angles in one are equal to their corresponding angles in the other. If one of these pencils is made coincident with its corresponding one, then the planes will be perspective. This agrees with the fact that two perspective planes in space can be made coincident by turning one about their axis in two different ways (§ 8).
In the reasoning employed it is essential that the lines j and i' are finite. If one lies at infinity, say j, then i and j coincide, hence their corresponding lines i' and j' will coincide; that is, i' also lies at infinity, so that the lines at infinity in the two planes are corresponding lines. If the planes are now made coincident and perspective, then it may happen that the lines at infinity correspond point for point, or can be made to do so by turning the one plane in itself. In this case the line at infinity is the axis, whilst the centre may be a finite point. This gives similar figures (see § 16). In the other case the line at infinity corresponds to itself without being the axis; the lines joining corresponding points therefore all coincide with it, and the centre S lies on it at infinity. The axis will be some finite line. This gives parallel projection (see § 17). For want of space we do not show how to find in these cases the perspective position, but only remark that in the first case any pair of corresponding points in 7r and 7' may be taken as the points T and T', whilst in the other case there is a pencil of parallels in r such that any one line of these can be made to coincide point for point with its corresponding line in 7r', and thus serve as the axis of projection. It will therefore be possible to get the planes in perspective position by first placing any point A' on its corresponding point A and then turning 7r' about this point till lines joining corresponding points are parallel.
If the axis is at infinity every line is parallel to its corresponding line. Corresponding angles are therefore equal. The figures are similar, and (§ io) the ratio of similitude of any two corresponding rows is constant.
If similar figures are in perspective position they are said to be similarly situated, and the centre of projection is called the centre of similitude. To place two similar figures in this position, we observe that their lines at infinity will coincide as soon as both figures are put in the same plane, but the rows on them are not necessarily identical. They are projective, and hence in general not more than two points on one will coincide with their corresponding points in the other (G. § 34). To make them identical it is either sufficient to turn one figure in its plane till three lines in one are parallel to their corresponding lines in the other, or it is necessary before this can be done to turn the one plane over in space. It can be shown that in the former case all lines are, or no line is, parallel to its corresponding line, whilst in the second case there are two directions, at right angles to each other, which have the property that each line in either direction is parallel to its corresponding line. We also see that If in two similar figures three lines, of which no two are parallel, are parallel respectively to their corresponding lines, then every line has this property and the two figures are similarly situated; or Two similar figures are similarly situated as soon as two corresponding triangles are so situated. If two similar figures are perspective without being in the same plane, their planes must be parallel as the axis is at infinity. Hence - Any plane figure is projected from any centre to a parallel plane into a similar figure. If two similar figures are similarly situated, then corresponding points may either be on the same or on different sides of the centre. If, besides, the ratio of similitude is unity, then corresponding points will be equidistant from the centre. In the first case therefore the two figures will be identical. In the second case they will be identically equal but not coincident. They can be made to coincide by turning one in its plane through two right angles about the centre of similitude S. The figures are in involution, as is seen at once, and they are said to be symmetrical with regard to the point S as centre. If the two figures be considered as part of one, then this is said to have a centre. Thus regular polygons of an even number of sides and parallelograms have each a centre, which is a centre of symmetry.
If, instead of the axis, the centre be moved to infinity, all the projecting rays will be parallel, and we get what is called parallel projection. In this case the line at infinity passes through the centre and therefore corresponds to itself - but not point for point as in the case of similar figures. To any point I at infinity corresponds therefore a point I' also at infinity but different from the first. Hence to parallel lines meeting at I correspond parallel lines of another direction meeting at I'. Further, ir k any two corresponding rows the two points at infinity are corresponding points; hence the rows are similar. This gives the principal properties of parallel projection To parallel lines correspond parallel lines; or To a parallelogram corresponds a parallelogram. The correspondence of parallel projection is completely determined as soon as for any parallelogram in the one figure the corresponding parallelogram in the other has been selected, as follows from the general case in § 14. [Corresponding rows are similar (§ io).] The ratio of similitude for these rows changes with the direction: If a row is parallel to the axis, its corresponding row, which is also parallel to the axis, will be equal to it, because any two pairs AA' and BB' of corresponding points will form a parallelogram. Another important property is the following: The areas of corresponding figures have a constant ratio. We prove this first for parallelograms. Let ABCD and Efgh be any two parallelograms in A'B'C'D' and E'F'G'H' the corresponding parallelograms in Then to the parallelogram Klmn which lies (fig. 6) between the lines AB, CD and EF, GH will correspond a parallelogram K'L'M'N' formed in exactly the same manner. As ABCD and Klmn are between the same parallels their areas are as the bases. Hence ABCD AB A'B'C'D' A'B' Klmn -KL' and similarly K'L'M'N' - K'L' But AB/KL=A'B'/K'L', as the rows AB and A'B' are similar. Hence ABCD Klmn Efgh Klmn A'B'C'D' - K'L'M'N' Hence also ABCD Efgh A'B'C'D' - E'F'G'H' This proves the theorem for parallelograms and also for their halves, that is, for any triangles. As polygons can be divided into triangles the truth of the theorem follows at once for them, and is extended (by the method of exhaustion) to areas bounded by curves by inscribing polygons in, and circumscribing polygons about, the curves.
Just as (G. § 8) a segment of a line is given a sense, so a sense may be given to an area. This is done as follows. If we go round the boundary of an area, the latter is either to the right or to the left. If we turn round and go in the opposite sense, then the area will be to the left if it was first to the right, and vice versa. If we give the boundary a definite sense, and go round in this sense, then the area is said to be either of the one or of the other sense according as the area is to the right or to the left. The area is generally said to be positive if it is to the left. The sense of the boundary is indicated either by an arrowhead or by the order of the letters which denote points in the boundary. Thus, if A, B, C be the vertices of a triangle, then ABC shall denote the area in magnitude and sense, the sense being fixed by going round the triangle in the order from A to B to C. It will then be seen that ABC and ACB denote the same area but with opposite sense, and generally ABC = BCA = CAB = - ACB = - BAC = - CBA; that is, an interchange of two letters changes the sense. Also, if A and A' are two points on opposite sides of, and equidistant from, the line BC, then ABC = -A'BC.
Taking account of the sense, we may make the following statement: If A, A' are two corresponding points, if the line AA' cuts the axis in B, and if C is any other point in the axis, then the triangles ABC and A'BC are corresponding, and ABC AB AB.
_ A'BC - A'B 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.
If the two planes and have a definite position in space, and if a figure in is projected to r' by rays perpendicular to this plane, then the projection is said to be orthographic. If in this case the plane be turned till it coincides with 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 *o that of the original figure is, in this case, the cosine of the angle between the two planes and as will be seen by projecting a rectangle which has its base in the axis.
Orthographic projection is of constant use in geometrical drawing.
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 corre sponding lines; they will there fore be cut by any line parallel to the axis in corresponding FIG. 7.
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.
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 instance, 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 its 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 FIG. 6.
and 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 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 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 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 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.).
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.
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 =m' 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 = e ABC;. hence ABC =m (I - i - B) ABC, or m= 3.
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 areconjugate 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 2r 2 and the circle has the area r 2 2r; the constant ratio of an ellipse to the inscribed parallelogram has therefore also the value fr. Hence The area of an ellipse equals abr. § 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 A 1, B 1, C I. If we now take on BC the harmonic conjugate A2 to A 1 and similarly on CA and AB the harmonic conjugates to B 1 and C 1 respectively, then the lines AA 2, BB 2, CC 2 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 linen 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 Propriete's 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 FIG. 8. where m is some 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.
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 saying 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 t 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 conjugates 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 12]) 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 [6 Hen a n y 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 involution 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, 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 Q R' Q 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 passes 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.
All 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.
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 FIG. 9.
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.)
From Britannica 11th Edition (1911) 
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