An elementary treatise on coordinate geometry of three dimensions . curvatures of the curves of inter-section of the ellipsoid —+4 + -- = 1 and its onfocala whose para-meters are A and /z, are ajbjCi «2^2C2 Shew how this result may be deduced from that of Ex. 18, £ 232. 253. Geodesic torsion. If ot, (fig. 70), is the tangent at O to a curve drawn on a surface, and the osculating plane of the curve makes an angle w with the normal section through OT, then to is the angle between the principal normal to the curve and the normal to the surface, and therefore _w -qm,+ cos (0 = J. - * ^=—- (II
An elementary treatise on coordinate geometry of three dimensions . curvatures of the curves of inter-section of the ellipsoid —+4 + -- = 1 and its onfocala whose para-meters are A and /z, are ajbjCi «2^2C2 Shew how this result may be deduced from that of Ex. 18, £ 232. 253. Geodesic torsion. If ot, (fig. 70), is the tangent at O to a curve drawn on a surface, and the osculating plane of the curve makes an angle w with the normal section through OT, then to is the angle between the principal normal to the curve and the normal to the surface, and therefore _w -qm,+ cos (0 = J. - * ^=—- (II The binomial makes an angle 90° + ^ with the normal tothe surface. Let us take as the positive direction of thebinomial that which makes an angle 90° — w with the normalto the surface, and then choose the positive direction of the 374 COORDINATE GEOMETRY [CH. XVII. tangent to the curve, so that the tangent, principal normaland binormal can be brought into coincidence with OX,OY, OZ respectively. Then — pL — qmo + 7iosin co - Li „ — - .Jl+pi + qz. Fig. 70. Differentiate with respect to st the arc of the curve, andwe have, by (1), dco cos to thirty + smx) + mz{slx + tmx) cos to ds s/l-\-p2 + q2 d -* •(2) Now take O as the origin, and let the equation to thesurface be 2: :- + £+.... Pi P-2 Then at the origin (2) becomes doe cos to rti^ni, cosco-7- = — a.—t as cr px Let OT make an angle 6 with OX. Then lx = cos 0, mx = sin 6, nx = 0; and since n2 — cos to, l3 = m^o = sin 6 cos w, ms — — l{n2 = — cos 6 cos w. •(3) §263] EXAMPLES XIV. Therefore (.3) becomes 1 ,/ and hence, by §251j the value of — , is the torsion oi the geodesic that touches the curve at o. It is called the geodesic torsion of the curve, and is evidently the -;nn>- forall curves which touch OT at O. Cor. 1. If a curve touches a line of curvature at O itsgeodesic torsion at O is zero. Cor. 2. The torsion of a curve drawn on a developable issin 6 cos 6 dwp +dswhere 6 is
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
