PROJECTION respond parallel lines of another direction meeting at I . Further, in any two corresponding rows the two points at infinity are cor responding 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 ( 10). 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 lie 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 hare a constant ratio. We prove this first for parallelograms. Let ABCD and EFGH be any two parallelograms in IT, A B C D and E F G H the corre sponding parallelograms in IT . Then to the parallelogram KLMN which lies (fig. 6) be tween the lines AB, CD and EF, GH will correspond a parallelo gram 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 Fig. 6. rT- = 77-T> KLMN KL But AB/KL = A B /K L Hence i . , i A B C D A B nd similarly ==,. ,,>,-, = f^n J K L M N k L as the rows AB and A B are similar. ABCD A B C D K L M N Hence also KLMN . . ., , ,, and similarly EFGH _KLMN
- K L M N
ABCD A B C D = EFGH 1 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 then by the well-known method of exhaustion extended 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 indi cated 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 sh:ill 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 state ment : - If A, A are two corresponding points, if the line A A 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 or The constant ratio of corresponding areas is equal and opposite to the ratio in which the axis divides the segment joining two corre sponding points. 18. Several special cases of parallel projection arc of interest. ORTHOGRAPHIC PROJECTION. If the two planes ir and IT have a definite position in space, and if a figure in ir is projected to w by rays perpendicular to this plane, then the projection is said to be orthographic. If in this case the plane ir be turned till it coincides with ir 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 tJie projection to that of the original figure is, in this case, the canine of the angle between the two planes ir and IT , as will be seen by projecting a rectangle which has its base in the axis. Orthographic projection is of constant use in geometrical drawing and will be treated of fully later on in this article ( 28 4-7.). SHEAR. If the centre of projection be taken at infinity on the axis, then the projecting rays aie parallel to the axis ; hence cor responding points will be equidistant from the axis. In this case therefore areas of corresponding figures will be cqnal. If A, A and B, B (fig. 7) are two pairs of corresponding points on the same line, parallel to the axis, then, as correspond ing 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 on the axis, then AO and A O will be corresponding lines ; they will therefore be cut by any
Fig 7. 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 dis tances of C and A from the axis, therefore Tivo corresponding figures may be got one out of the other ly 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 Sir William Thomson hns given it the name of shear. A shear of a plane figure is determined if we are giveri 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 ( 11). 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 per pendicular 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 arc harmonic Conjugates with regard to the axis and a line in tlic 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] arc corresponding points.