The elements of descriptive geometry - Softcover

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1841 Excerpt: ...whatever to be traced upon the surface, and if from the point under consideration we take infinitely small portions of these curve lines, we may regard these portions as straight lines, which produced will be tangents to the respective curves of which ultimately they form a part; but all these small lines will also be in the tangent plane, and therefore when produced will also be in that plane. Hence we see that the tangent plane to a surface contains all the tangents, drawn through the point of contact, to all the lines which can be traced through the point upon the surface. 40. This proposition is of great importance in determining the position of a tangent plane, since as it is only necessary to have two lines given in order to find a plane, it follows, that in order to construct the tangent plane to a given point of a surface, we have merely to draw tangents to two lines traced upon the surface, and to make the plane pass through these tangents. Our choice of the lines on the surface will, of course, be directed to those whose tangents are most readily found. Dej. A normal is a perpendicular to the tangent plane, drawn through the point of contact. Tangent Planes To Surfaces. THE CYLINDEB. Problem 1. Given the horizontal trace of a cylinder and the direction of fhe generatrixes, to draw a tangent plane to a given point of the cylinder. 41. Let aecf be the horizontal trace of the cylinder; and in this instance suppose aecf to be a circle. Draw ab, cd, tangents to aecf, and parallel to the horizontal projections of the generatrixes; these tangents will be also the limits of the horizontal projection of the cylinder, since all the generatrixes must be projected between these two lines. Draw the tangents ee', ff,-l to xy, and e'g', fh', parallel to the vert...