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 A any two parallelograms in -/r,
A'B'C'D' and E'F'G'H' the
° B corresponding parallelograms
F in 1r'. Then to the parallel-K
E gram 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 Klalxllfi
are between the same para e s
their areas are as the bases. Hence-ABCD AB . A'B'C'D' A'B
% =mf Hd S'““'=="'Y =r<T' e
HBut AB/KL=A'B'/K'L', as the rows AB and A'B' are similar. ence
ABCD . . 1
m@=mW °“<' mls* Y
M |, H 9.
F IG. 6.
KLMN EFGH KLMN
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, end 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 /Q .
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 1r and ar' have a definite position in space, and if a figure in 1r is projected to -/r' by rays perpendicular to this plane, then the projection is said to be orthographic. If in this case the plane 1r 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 olg these rays is, and remains during the turning, perpendicular to L e axis.
The constant ratio of the area of the projection to that of the original figure is, in this case, the cosine of the angle between the two planes ar and 1r', 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, t erefore, 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 O on the axis, then
AO and A'O will be corresponding
lines; they will therefore
be cut by any line parallel
to the axis in corresponding
points. In the figure therefore
C, C' and also D, D' will be
CC'=DD'. As the ratio CC'/AA' equals the ratio of the distances of C and A from the axis, therefore-Two corresponding jigures may be got one out of the other by moving all points in the one parallel to a jixed 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 (§ 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, nd is said to have an axis of symmetry or simply an axis. Every Biameter of a circle is thus an axis; also the median line of an isosceles triangle En? the diagonals of a rhombus are axes of the figures to which they e ong.
In the more general case where the projecting rays are not perpendicular to the axis we have a kind of twisted symmetry which maybe 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 an triangle is an axis of skew-symmetry, the side on which it stand; 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:-
I f '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 jigure 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 C C' oo
/L 2- §
pairs of corresponding points and