PROBLEM X. Fig. 91. Through a given point on a cylinder of revolution to draw a tangent plane to the surface, when the axis of the cylinder is parallel to the ground line. Let abcd, ef'g'h' be the projections of a right circular cylinder, and p the horizontal projection of a point on the surface; it is required to draw a tangent plane at the point P. Construction. Draw LR,, the traces of a plane cutting the cylinder at right angles to the axis. Let that plane be rabatted on the horizontal plane, and find the position of 0, the centre of the circular section of the cylinder, and of P, the given point on the surface, in the rabatment of the plane....... (Prob. XXIV. Chapter II.) These are the points O, and P, respectively. 1 1 1 Describe the circle O,P,; draw the tangent PL, meeting ay in R; and make KR1 = KR. LM and RN, parallel to xy, are the traces of the plane required. Proof. The line LR is the rabatment of the tangent to the circular section of the cylinder OP, by construction; that is to say, the line whose traces are L and R1 is a tangent line to the surface at the point P. But the plane LM R,N contains the line LR,, and therefore passes through P; and passing through P it contains the line through that point parallel to its traces, which is the generator PS. Therefore LM RN is a tangent plane to the cylinder at the point P. PROBLEM XI. Fig. 91. Through a given external point to draw a tangent plane to a cylinder of revolution, when the axis is parallel to the ground line. Let abcd, ef'g'h' be the projections of the cylinder and the projections of the point; it is required to draw through a tangent plane to the cylinder. Construction. Draw, as in Prob. X., LR, the traces of a plane containing Q and perpendicular to the axis of the given cylinder. Rabat that plane on the horizontal, and determine O, and Q,, the rabatments of O and Q. With 0, as centre describe a circle equal to the circular section of the cylinder, and draw QR a tangent to the circle. The remaining part of the solution is the same as in the preceding problem. PROBLEM XII. Fig. 92. To find the development of a right circular cylinder. Let ABC, a'c'f'd be the projections of a right circular cylinder; it is required to find its development. A cylinder may be considered as a prism the faces of which are indefinitely small, that is to say, it is the limit towards which the prism approaches as the number of its faces is indefinitely increased. When the cylinder is right, as in this case, the generators being all at right angles to the circular base, each of the small faces is a rectangle. Hence the following: Construction. Divide the circumference of the circle ABC into a number of equal parts C.1, 12, &c., and set off on a straight line c'C, distances equal to C1, 12, &c., or in any other way make c'C, equal to the circumference ABC. The rectangle c'C,F,f' is the development required. 1 Cor. To find the position of any point P on the development, when its projections 4p' on the cylinder are given. Find the development of the generator passing through the point, and make 4P, 4'p'. In this way the development of any curve on the surface may be found. = Any curve on a right cylinder such that its development is a straight line oblique to the generators is called a helix. c'g'f' is the projection of such a curve, c'F, being its development. It manifestly cuts the generators of the cylinder at equal angles, and is the well-known form of a screw |