An elementary treatise on coordinate geometry of three dimensions . henormal section of algebraically least curvature through thereal axis of the indicatrix (2). These indicatrices projecton the &2/-plane into conjugate hyperbolas whose commonasymptotes are the inflexional tangents. As in § 220, thesections of greatest and least curvature are the principalsections. If the axes OX and OY lie in the principalsections the equations to the indicatrices are (1) a? 62 i; (2) • — —h x —VI— _ i a2 62 If pv p2 measure the principal radii in magnitude andsign> _.«« _ /-62 = Lt- and therefore the equa
An elementary treatise on coordinate geometry of three dimensions . henormal section of algebraically least curvature through thereal axis of the indicatrix (2). These indicatrices projecton the &2/-plane into conjugate hyperbolas whose commonasymptotes are the inflexional tangents. As in § 220, thesections of greatest and least curvature are the principalsections. If the axes OX and OY lie in the principalsections the equations to the indicatrices are (1) a? 62 i; (2) • — —h x —VI— _ i a2 62 If pv p2 measure the principal radii in magnitude andsign> _.«« _ /-62 = Lt- and therefore the equations to the indicatrices are (1) z = h, - + V- = 2h, (2)z=-K,Pi P% Pi Pi §§221,222] CURVATURE OF NORMAL SECTK and the equal ion to i be surface is /i PaThe radius of curvature of the normal Bection thaimakes an angle d with the KB-plane is given by I __cosJ0 sin-A P Pi A> 222. Curvature of normal sections through a para-bolic point. If rt — s~ = (), the indicatrix consists of twoparallel straight lines, (fig. 62). The inflexional tangents. Fig. 62. eoincidc, and the normal section whieli contains them hasits curvature zero. The normal section at right angles tothe section of zero curvature has maximum two sections are the principal sections. If OX andOY lie in the principal sections, the equations to theindicatrix are z=h x2 = <l2 where (t = CA. The finite principal radius px is given by Pi = Lt^; Hence the equations to the indicatrix and surface arc ,r2 -a z=h, 2h=-; 2z=- +.... Pi Pi If p is the radius of curvature of the section OCP which makes an angle $ with the principal section OCA, 1 cos- 0 Pi 330 COORDINATE GEOMETRY [ch. xvi. 223. Umbilics. If r = t and 8 = 0, the indicatrix is acircle and the principal sections are indeterminate, since allnormal sections have the same curvature. Points at whichthe indicatrix is circular are umbilics. 224. The curvature of an oblique section. The re-lation between the curvatures of a normal sectio
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
