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

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A. S. Barnes and Company, 1840 - Geometry, Descriptive - 174 pages
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Page 41 - If the base be a circle, such cone is a right cone with a circular base ; it can be generated by the revolution of a right-angled triangle about one of its legs, and its rectilinear elements make equal angles with the axis.
Page 41 - A cylinder is conceived to be generated by the revolution of a rectangle about one of its sides as an axis.
Page 114 - 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...
Page 48 - 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 within the circle CDG. If the parallel should become an element of the cone, the problem would be possible, but would admit of one solution only. 97. The last three problems...
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 129 - The angle formed by the intersection of two chords is measured by half the sum of the two intercepted arcs. Let the two chords AB, CD intersect each other at the point E ; then will the angle DEB, or its equal, AEC, be measured by half the two arcs DB and A C.
Page 160 - AB, the horizontal projection of one directrix, the parts from 2 to 3, from 3 to 4, from 4 to 5, &c. be each made equal to the part from 1 to 2 ; and on CD...
Page 14 - A'B' are the projections, is the line of intersection of these two planes, and since the planes are determined in position, their intersection is also determined. If the horizontal projection only be given, the line is somewhere in a plane passing through the horizontal projection and perpendicular to the horizontal plane, but its position in this projecting plane is not determined. So, when the vertical projection only is given, the line may have any position in the plane passing through the projection...
Page 88 - ... (137). The line Nfc is the horizontal and N'fc' the vertical projection of the tangent. Drawing a tangent plane along the element (OAG, O'A'G'), GH is its horizontal trace, and the line in which it intersects the tangent plane (of which NH is the trace) is tangent to the upper curve at the point (O,O'). The lines HO and H'O' are the projections of this intersection, which are respectively tangent to the projections of the curve.
Page 80 - A", c", g'\ and a" are determined in the same manner. The line I /A" is the position of the tangent line when the plane of the curve is revolved to coincide with the horizontal plane. 133. These points might also be determined by the general method of finding the hypothenuse of a triangle whose base is the distance from the horizontal projection of the point to the axis, and perpendicular, the altitude of the point above the horizontal plane, and then laying off these several distances from the...

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