Elements of Descriptive Geometry: With Their Application to Spherical Trigonometry, Spherical Projections, and Warped Surfaces

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Barnes, 1854 - Geometry, Descriptive - 174 pages
 

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Page 160 - From 1 to 2 From 2 to 3 From 3 to 4 From 4 to 5 From 5 to 6 From 6 to 7 From 7 to...
Page 41 - A cylinder is conceived to be generated by the revolution of a rectangle about one of its sides as an axis.
Page 110 - The axis of a circle is a line passing through its centre, perpendicular to its plane : the points in which this line meets the surface of the sphere are called the poles of the circle. Either pole is at the same distance from every point of the circumference of the circle ; since, if a line be perpendicular to a plane all points of the plane equidistant from the foot of the perpendicular are equidistant from any po'nt of the line.
Page 26 - It may be shown, in a similar manner, that the vertical projection of the line is perpendicular to the vertical trace of the oblique plane. § 50. The converse of this proposition is also true, that is, if the projections of a line are respectively perpendicular to the traces of a plane, the line in space is perpendicular to the plane. For, the projecting planes of the line will be respectively perpendicular to the traces of the oblique plane, and therefore perpendicular to the oblique plane ; hence,...
Page 48 - I'H'), and tangent to the surface of the cone. If we draw through E the tangent EP, it will be the horizontal trace of the second plane which is parallel to the given line (IH, I'H'), and tangent to the surface of the cone. The vertical trace of this plane is easily constructed. § 96. This problem becomes impossible when the line •(AE, A'E'), which is drawn through the vertex of the cone and parallel to the given line, passes within the surface : in this case, it would pierce the base of the cone...
Page 124 - THEOREM The angle formed within a circle by the intersection of two chords is measured by half the sum of the two intercepted arcs.
Page 14 - ... the projections of cd, PI. I., Fig. 2. 9. Remark. A general principle, which it is important to be perfectly familiar with, is embodied in several of the preceding examples; viz. When any line is parallel to either plane of projection, its projection on that plane is equal and parallel to itself, and its projection on the other plane is parallel to the ground line. 10. The preceding remark serves to show how to find the true length of a line, when its projections are given. When the line, as...
Page 109 - 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.
Page 132 - ... 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...
Page 121 - ... the point of sight. NM is the polar distance and m the stereographic projection of M. Cm is the tangent of the arc Co, computed to the radius CS =CE, and Co is one half of NM. That is, the stereographic projection of any point of the surface of the sphere is at a distance from the centre of the primitive circle equal to the tangent of one half its polar distance. In this projection, it should be observed that the polar distance of a point is always its distance from the pole opposite the point...

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