An elementary treatise on coordinate geometry of three dimensions . re tan </.= A tan /?. 24. If the generator of a skew surface make with the tangent and principal normal of the line of striction angles whos«- cosines are J and //, prove that -— = -, where p is the radius of curvature of theline of striction. s P 25. Prove that the line of striction on the skew surface generated by the line x — a cos 6_y — « sin 6_ cos 0 cos - sin 0 cos sin -2 2 2 is an ellipse in the plane 2y + z = 0, whose seniiaxcs are . —-, and 5 whose centre is ( -, 0, 0 ). [CH. XVI. CHAPTER XVI. CURVATURE OF SURFACES
An elementary treatise on coordinate geometry of three dimensions . re tan </.= A tan /?. 24. If the generator of a skew surface make with the tangent and principal normal of the line of striction angles whos«- cosines are J and //, prove that -— = -, where p is the radius of curvature of theline of striction. s P 25. Prove that the line of striction on the skew surface generated by the line x — a cos 6_y — « sin 6_ cos 0 cos - sin 0 cos sin -2 2 2 is an ellipse in the plane 2y + z = 0, whose seniiaxcs are . —-, and 5 whose centre is ( -, 0, 0 ). [CH. XVI. CHAPTER XVI. CURVATURE OF SURFACES. 219. We now proceed to investigate the curvature at apoint on a given surface of the plane sections of the surfacewhich pass through the point. In our investigation weshall make use of the properties of the indicatrix definedin § 184. If the point is taken as origin, the tangent plane at theorigin as aJ2/-plane, and the normal as 0-axis, the equationsto the surface and indicatrix are 2z = rx2 + 2sxy + ty2-\,2 — h, 2h = rx2 + 2sxy + Fig. 60. 220. Curvature of normal sections through an ellipticpoint. If rt — s2>0 the indicatrix is an ellipse, (fig. 60).Let C be its centre, CA and CB its axes, and let CP be any §§219-221] CURVATURE OF NORMAL SECTIONS semidiameter. Then, if p is the radius of curvature of the CP normal section ocp, p = hb „--> and therefore the radii of curvature of normal sections are proportional to the 3quof the semidiameters of the indicatrix. The sections OCB,OCA, which have the greatest and least curvature, arccalled the principal sections at O and their radii of curvatureare the principal radii. If px, p2 are the principal radii, andCA = a, CB = 6, a2 12 If the axes OX and OY are turned in the plane XOYuntil they lie in the principal sections OCA, OCB respectively,the equations to the indicatrix become Z = -h, x2 y2 = 1, or and the equation Pi Pz to the surface is :2k, 2Z X2 Pi Pi If CP = r, and the normal section OCP ma
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
