Elements of Descriptive Geometry: With Its Applications to Spherical Projections, Shades and Shadows, Perspective and Isometric Projections

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A.S. Barnes & Company, 1865 - Geometry, Descriptive - 192 pages
 

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Page 8 - ... plane a meridian plane ; and it is also evident that all meridian lines of the same surface are equal, and that the surface may be generated by revolving any one of these meridian lines about the axis.
Page 128 - Gr is the tangent of Co', the complement of Co, or is the co-tangent of Co. Hence the poles of any circle of the sphere are projected into the line of measures, the one furthest from the point of sight at a distance from the centre of the primitive circle equal to the tangent of half the inclination, and the other at a distance equal to the co-tangent of half the inclination of the given to the primitive circle. 215. PROBLEM 60. To project the sphere upon the plane of any of its great circles, as...
Page 41 - H, and z'x' will be its vertical projection. Since the angle which a tangent to the helix makes with the horizontal plane is constant, and since each element of the curve is equal to the hypothenuse of a right-angled triangle of which the base is its horizontal projection, the angle at the base, the constant angle, and the altitude, the ascent of the point while generating the element ; it follows, that when the helix is rolled out on its tangent, the sum of the elements, or length...
Page 2 - The solution of problems relating to these magnitudes in space. These drawings are so made as to present to the eye, situated at a particular point, the same appearance as the magnitude or object itself, were it placed in the proper position. The representations thus made are the projections of the magnitude or object. The planes upon which these projections are usually made are the planes of projection. The point, at which the eye is situated, is the point of sight.
Page 38 - ... perpendicular to either plane of projection, its projection on that plane will be a point, and its projection on the other plane will be perpendicular to the ground line.
Page 36 - An hyperbola is a curve which is the locus of a point that moves in a plane so that the difference of its distances from two fixed points in the plane is constant.
Page 7 - If the plane of the curve be parallel to either plane of projection, its projection on that plane will be equal to itself, since each element of the curve will be projected into an equal element, Art.
Page 83 - ... contact will intersect the given line in a point, and the required tangent plane in a right line drawn from this point tangent to the circle. The plane of this tangent and the given line will be the required plane. Without constructing the cylinder, we have then simply to pass a plane through the center of the sphere perpendicular to the given line, and from the point in which it intersects the line, to draw a tangent to the circle cut from the sphere by the same plane, and pass a plane through...
Page 118 - The line of measures of a circle evidently contains the projections of both poles of the circle, Art. (195) ; and since the arc which measures the distance of either pole from the pole of the primitive circle, measures also the inclination of the two circles, Art.
Page 26 - ... to determine the projection of one point on the plane, and through this to draw a line parallel to th'e given line, Art. (14). 46. PROBLEM 10. Through a given point, to pass a plane perpendicular to a given right line. Let M, Fig. 25, be the given point, and NO the given line. Analysis. Since the plane is to be perpendicular to the line, its traces must be respectively perpendicular to the projections of the line, Art.

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