By Luther Pfahler Eisenhart
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Additional resources for A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions)
Hence, with reference F the coordinates of (7 denoted by Q cipal , X =x + Y rl, where the absolute value of y M when l X , -f rm, , to Z Q, C is on the prin any fixed axes in space, are of the form Z^=z + rn, r is the radius of the osculating circle. In order to find the value of of the circle, . is we r, does not have return to the consideration limiting position, and its we let X, F, Z\ Zj, m^ n^ r^ denote respectively coordinates of the cen ter of the circle, the direction-cosines of the diameter through and the radius.
Express the coordinates in terms of the arc and find the radii of first and second curvature. , ; 6. curve Show that make with if 6 - denote the angles which the tangent and binormal to a sin 6 dd = -r and a fixed line in space, then sin </> 7. first When two d<p - p curves are symmetric with respect to the origin, their radii of curvature are equal and their radii of torsion differ only in sign. 8. The osculating circle at an ordinary point of a curve has contact of the sec ond order with the latter and all other circles which lie in the osculating plane and are tangent to the curve at the point have contact of the first order.
1 is is : The osculating sphere to a curve at a point has contact with the curve of the third order ; oilier spheres with their centers on the polar line, and tangent to the curve, have contact with the curve of the second order ; all other spheres tangent to the curve at a point have contact of the first order. The radius of the osculating sphere JS* (93) =,! is given by + TV, and the coordinates of the center, referred to fixed axes (94) xl = x + pi f p T\, y^ = y + pm p rfji, zl in space, are = z + pn p rv.
A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions) by Luther Pfahler Eisenhart