« AnteriorContinuar »
m, found as before, and then m, must be transferred to its true position by taking the point n in the same position as regards ob, that m has to ob. Which is done by describing a circle from a as a centre with om, for a radius, and making the arc en equal to dm, of that circle.
2°. Construction of the Tangent. The epicycloid, as has been already observed, lies on the surface of a sphere, having 0,0' for its centre, and oʻb for its radius, consequently the tangent to the point m, m' of this curve is in the tangent plane to this sphere. It has also been proved to lie in the tangent plane to a sphere, the centre of which would be in b, and the radius of which would be bm; the intersection of these two tangent planes must therefore be the tangent in question.
The base of the moving cone, which is turned down in the circle bmc, belongs to the first of these spheres; if therefore a tangent mf is drawn to the circle bmc at the point m, it will be a line situated in the tangent plane to the sphere turned down with bmc on the horizontal plane, consequently the point f, in which mf meets bc, is in the horizontal trace of this plane. This trace must therefore be the line d'f drawn from f, perpendicular to the projection om of the line, which passes through the point of contact.
The circle described with the radius bm is one of the second sphere, also turned down on the horizontal plane, if Mg be drawn a tangent to it, meeting bc in g, and gß be drawn perpendicular to bm, the line gß will be the horizontal trace of the tangent plane to the second sphere. The traces d'f, Bg meet in r, which must belong to the intersection of the two tangent planes, the line mr, drawn through this point r and through m, must consequently be the projection of the tangent to the epicycloid, or, which is the same thing, a tangent to the projection of that curve.
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