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On the Method of Projections.
1. DEFINITIONS. The Orthogonal projection of a point on a plane is the foot of the perpendicular drawn from the point to the plane.
The plane on which the projection is made is called the plane of projection.
The orthogonal projection of a line on a plane is the line traced on the plane by a straight line which passes through all successive points of the given line and is always perpendicular to the plane of projection.
In the present treatise when projections are spoken of it may be understood that orthogonal projections are intended.
If a straight line be drawn through two adjacent points of a curve, and the two points move towards one another on the curve until they coincide, the straight line in its final position is said to be a tangent to the curve at the point which it has finally in common with it.
2. In order to project a straight line AB (fig. 8 5) on a plane abc, we must draw the perpendicular Aa from some
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point A of AB to the plane of projection (Euclid xi. 11), then make a plane pass through AB, Aa; it will be perpendicular to the plane of projection (xi. 18). Hence the perpendicular drawn from any point P of AB to ab the intersection of the planes will be perpendicular to the plane of projection; and the straight line ab will be the projection of AB. Hence the projection of a straight line is also a straight line.
3. The projections of parallel straight lines are also parallel.
From points A and C (fig. § 5) in the parallel straight lines AB, CD draw Aa, Cc perpendicular to the plane of projection, these lines will be parallel (Euclid xi. 6), and the planes BAa, D Cc will be parallel (xi. 15): also the plane of projection will cut these planes in parallel lines ab, cd (XI. 16), and these are the projections of AB, CD. Hence the projections of parallel straight lines are parallel.
4. A line of finite length and its projection are cut in the same ratio by any point and its projection.
Let a, p, 6 be the projections of the points A, P, B in the straight line AB (fig. § 5), draw AE perpendicular to Bb cutting Pp in Q, then Ap, Qb are parallelograms, and AQ, QE equal to ap, pb. Also PQ is parallel to BE the side of the triangle BAE, and therefore AQ : QE :: AP : PB, or ap : pb :: AP : PB. The line AB and its projection ab are cut in the same ratio by the point P and its projec
5. Parallel straight lines of finite lengths are diminished by projection in the same ratio.
Let the parallel straight lines AB, CD of finite length have projections ab, cd. From A and C draw AE, CF perpendicular to Bb, Dd: these lines are equal to ab, cd: they are also parallel to these parallel straight lines and are therefore parallel to one another (Euclid xi. 9). The lines AB, CD are also parallel, hence the angles BAE, DCF are equal
(XI. 10): the angles at E and F are right angles, and the triangles BAE, DCF are similar. Hence
BA: AE :: DC: CF or BA : ab :: DC :cd ; AB, CD are diminished by projection in the same ratio.
Hence it appears that the ratio in which any line is altered by projection from one plane to another depends only upon its inclination to the line of intersection of the planes. It may be seen that lines parallel to the line of intersection are unaltered by projection, and that lines more inclined to it are diminished in a greater ratio than those less so: while lines at right angles to the line of intersection are diminished in the greatest ratio. The ratio which a line at right angles to the line of intersection of its plane with the plane of projection bears to its projection may conveniently be called the maximum ratio for the two planes, and it will increase from unity to a ratio as great as we please when we increase the inclination of the planes from zero to a right angle.
In other words, a line at right angles to the line of intersection may have its projection as nearly equal to itself or as small as we please by taking the plane of projection at first very slightly inclined to the plane of the line, and then increasing the inclination till it becomes a right angle. The projection is the base of a right-angled triangle of which the line is the hypotenuse, and the inclination of the planes the angle at the base. By increasing the angle at the base from zero to a right angle we diminish the base from the length of the hypotenuse to zero.
6. Angles will be in general increased or diminished in magnitude by projection, except those contained by lines parallel and right angles perpendicular to the line of intersection.
Of the four angles contained by two intersecting lines, that which is subtended by the line of intersection and the vertical opposite angle will be increased, the other two diminished by projection; and these angles may be increased and diminished to any extent by making the inclination of the plane of projection sufficiently great. If, for example, it be required to increase the acute angle ACB subtended by AB the line of intersection to a right angle; on AB describe the semicircle AcB, and draw CD perpendicular to AB cutting the semicircle in c. Then let the inclination be so chosen that the maximum ratio shall be CD : D: then CD will have its projection equal to cD, and the angle ACB will project into an angle equal to AcB, i.e. to a right angle.
This only holds, however, when D lies between A and B. If D fell outside AB, the angle ACB might be increased or diminished by projection, and would be greatest when the circle circumscribing ABC touched CD, and in that case DC would be a mean proportional between DA and DB.
7. The area of any figure will be reduced by projection in the maximum ratio of the planes.
For every triangle may be divided by a line through one of the angular points parallel to the line of intersection into two, each having its base parallel and its altitude at right angles to the line of intersection: the base of each will be unaltered by projection, the altitude diminished in the maximum ratio. Hence the area of each and therefore of the two together will be diminished in the maximum ratio. Hence, whatever its magnitude or position, every triangle is diminished by projection in the maximum ratio of the planes.
But every figure, whether rectilinear or curvilinear, may be as nearly occupied by triangles as we please, by making them sufficiently numerous. Each of these will be diminished by projection in the maximum ratio of the planes; and therefore the figure composed of them will be diminished in the same ratio.
8. The projection of the tangent to a curve at any point is the tangent to the projection of the curve at the projection of the point.
If p, q be the projections of two points P and of a curve, the straight line pq is the projection of the straight line PQ; and if p, q move so as to be always the projections of P and Q as they approach one another on the curve until they coincide and Q is merged in P, p and q will also approach one another on the projected curve, and will finally