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The elements of descriptive geometry Volume 1; being the first part of a treatise on descriptive geometry, and its application to ship building - Softcover
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Inhaltsangabe
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. 1850 Excerpt: ...made by meridian planes are similar and equal curves. Let A A' be the axis of the surface, A A'X, A A' X' two meridian planes, P Q R,P' Q' R' the curves in which they intersect the surface. Let 0 and 0' be the centres of the two circles P P', Q Q' of the surface, and P, Q, P' and Q' the corresponding points of section with the meridian planes of these circles. In each meridian plane draw the radii P 0, P' 0, Q 0, Q' 0', the trapeziums P Q00'; P'Q'00' will be equal, since having 0 0' common, and OP = 0P', and 0' Q = 0' Q', and the angles PO0', Q00', P'O0' and Q' 0 0' right angles, the sides P Q and P' Q' must be equal, and this is true for any points Q, Q'. Hence if any polygon be inscribed in the section PQR,a. similar and equal polygon may be inscribed in P' Q' R'. Hence the curve P Q R is similar and equal to the curve P' Q' R'. Cor. 1. All the tangent lines drawn to the different meridian sections at points corresponding to the same circle of the surface will pass through the same point of the axis; for each of the tangents Pt, P' t will make equal angles with the radii P0,P'0: whence it follows that the triangles P0t,P'Ot must be equal or common to both tangents. Cor. 2. Similarly all the normals drawn to the meridian section through the points corresponding to the same circle of the surface will pass through the same point in the axis of the surface. Obs. The sphere, some of whose properties we have before considered, belongs to this class of surfaces. Another example of the same class, whose properties we shall hereafter investigate, is the surface called the Twisted Surface of Revolution, or the Hyperboloid of Revolution, which is generated by the revolution of a straight line about an axis not in the same plane with it. We shall investigate its prop...
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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. 1850 Excerpt: ...made by meridian planes are similar and equal curves. Let A A' be the axis of the surface, A A'X, A A' X' two meridian planes, P Q R,P' Q' R' the curves in which they intersect the surface. Let 0 and 0' be the centres of the two circles P P', Q Q' of the surface, and P, Q, P' and Q' the corresponding points of section with the meridian planes of these circles. In each meridian plane draw the radii P 0, P' 0, Q 0, Q' 0', the trapeziums P Q00'; P'Q'00' will be equal, since having 0 0' common, and OP = 0P', and 0' Q = 0' Q', and the angles PO0', Q00', P'O0' and Q' 0 0' right angles, the sides P Q and P' Q' must be equal, and this is true for any points Q, Q'. Hence if any polygon be inscribed in the section PQR,a. similar and equal polygon may be inscribed in P' Q' R'. Hence the curve P Q R is similar and equal to the curve P' Q' R'. Cor. 1. All the tangent lines drawn to the different meridian sections at points corresponding to the same circle of the surface will pass through the same point of the axis; for each of the tangents Pt, P' t will make equal angles with the radii P0,P'0: whence it follows that the triangles P0t,P'Ot must be equal or common to both tangents. Cor. 2. Similarly all the normals drawn to the meridian section through the points corresponding to the same circle of the surface will pass through the same point in the axis of the surface. Obs. The sphere, some of whose properties we have before considered, belongs to this class of surfaces. Another example of the same class, whose properties we shall hereafter investigate, is the surface called the Twisted Surface of Revolution, or the Hyperboloid of Revolution, which is generated by the revolution of a straight line about an axis not in the same plane with it. We shall investigate its prop...
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