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General problems from the orthographic projections of descriptive geometry; with their applications to oblique--including isometrical--projections, ... projection ("One plane descriptive"), and - Softcover
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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. 1860 Excerpt: ...intersections with the paraboloid. The point in which any element of the ruled surface meets the curve cut from the paraboloid by the plane passed through that element, will be a point of the intersection of the two surfaces. The construction may be simplified by passing the auxiliary planes which contain rectilinear elements of the ruled surface, parallel to one of the plane directers of the hyperbolic paraboloid; in which case these planes will cut the paraboloid in rectilinear elements. Graphical Constructions. After the preceding explanations, it is sufficient to give the constructions for the first and third special forms of the third general case. C.(a). PL XV., Fig. 80, 6hows the construction of the intersection of a given line, MN--M'N', with the hyperboloid whose gorge is FXY, and whose generatrix is Gil--G'A". EXY is the gorge, and gh--G'lF the generatrix of the auxiliary hyperboloid, formed by revolving the hyperbola cut from the given hyperboloid by a vertical plane WW about a vertical axis at a. X'Y' is the vertical trace of the plane of the gorges. If MN--WW be revolved about the vertical axis at a, it will generate a cone whose base is cVS, and which will intersect the hyperboloid EXY in horizontal circles, points of which, found as in Prob. LXX., are at the intersections of any elements of this cone with the generatrix gh--G'H'. E'C'--aC is the parallel to this generatrix, and through the vertex of the cone, hence gc is the trace of the plane which contains gh--G'H' and cuts the elements ba and ca from the cone. These elements meet the generatrix gh--G'H' at AH' and kk'. But the given line, MN--M'N', meets the hyperholoid in horizontal circles through these points, that is at n'n and M'M, which are the required points. C. (c). PI. XV., ...
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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. 1860 Excerpt: ...intersections with the paraboloid. The point in which any element of the ruled surface meets the curve cut from the paraboloid by the plane passed through that element, will be a point of the intersection of the two surfaces. The construction may be simplified by passing the auxiliary planes which contain rectilinear elements of the ruled surface, parallel to one of the plane directers of the hyperbolic paraboloid; in which case these planes will cut the paraboloid in rectilinear elements. Graphical Constructions. After the preceding explanations, it is sufficient to give the constructions for the first and third special forms of the third general case. C.(a). PL XV., Fig. 80, 6hows the construction of the intersection of a given line, MN--M'N', with the hyperboloid whose gorge is FXY, and whose generatrix is Gil--G'A". EXY is the gorge, and gh--G'lF the generatrix of the auxiliary hyperboloid, formed by revolving the hyperbola cut from the given hyperboloid by a vertical plane WW about a vertical axis at a. X'Y' is the vertical trace of the plane of the gorges. If MN--WW be revolved about the vertical axis at a, it will generate a cone whose base is cVS, and which will intersect the hyperboloid EXY in horizontal circles, points of which, found as in Prob. LXX., are at the intersections of any elements of this cone with the generatrix gh--G'H'. E'C'--aC is the parallel to this generatrix, and through the vertex of the cone, hence gc is the trace of the plane which contains gh--G'H' and cuts the elements ba and ca from the cone. These elements meet the generatrix gh--G'H' at AH' and kk'. But the given line, MN--M'N', meets the hyperholoid in horizontal circles through these points, that is at n'n and M'M, which are the required points. C. (c). PI. XV., ...
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