Elements of Descriptive Geometry: With Its Applications to Spherical Projections, Shades and Shadows, Perspective and Isometric Projections |
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Analysis auxiliary plane centre circumference consecutive points curve of intersection determined dicular diedral angle directrix draw a tangent ecliptic equal equator find the intersection generatrix given line given point given right line given surface ground line hence hori horizontal plane horizontal projection horizontal trace hyperbola hyperbolic paraboloid hyperboloid intersects the cone Let MN line of measures line pierces meridian curve meridian plane parallel plane pass a plane perpen pierces H pierces the horizontal pierces the plane plane be passed plane directer plane intersects plane of projection plane parallel plane perpendicular plane tangent point of contact point of sight polar distance pole preceding problem primitive circle primitive plane projecting plane radius rectilinear elements required curve required plane required points Revolve this plane revolved position right line perpendicular single curved surface sphere surface of revolution tangent plane true position vertex vertical plane vertical projection vertical trace warped surface
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Page 117 - 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. (194), it follows that either pole of a circle is orthographically projected in its line of measures, at a distance from the centre of the primitive circle equal to the fine of its inclination, Art. (196). 199. PROBLEM 59. To project the...
Page 7 - ... 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 40 - 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...
Page 37 - ... 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 26 - PROBLEM 11. To. pass a plane through a given point, parallel to two given right lines. Let M, Fig. 26, be the point, and NO, and PQ, the two given lines. Analysis. Through the given point, draw a line parallel to each of the given lines. The plane of these two lines will be the required plane, since it contains a line parallel to each of the given lines. .Construction. Through m draw ms, parallel to no, and through m', m's
Page 35 - 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 82 - ... 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 25 - ... to determine the projection of one point on the plane, and through this to draw a line parallel to the 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.
Page 50 - Let the construction be made upon either of the figures, 43, 44, or 45. 89. If any two rectilinear elements of an hyperbolic paraboloid be taken as directrices, with a plane directer parallel to the first directrices, and a surface be thus generated, it will be identical with the first surface. To prove this we have only to prove that any point of an element of the second generation is also a point of the first. Thus, let MN and OP, Fig. 46, be the directrices of the first generation, and NO and...
Page 34 - An ellipse is a curve which is the locus of a point that moves in a plane so that the sum of its distances from two fixed points in the plane is constant.