Download Differential geometry : a first course by D. Somasundaram PDF

By D. Somasundaram

Differential Geometry: a primary path is an creation to the classical thought of area curves and surfaces provided on the Graduate and put up- Graduate classes in arithmetic. in response to Serret-Frenet formulae, the speculation of area curves is built and concluded with a close dialogue on primary lifestyles theorem. the idea of surfaces comprises the 1st basic shape with neighborhood intrinsic homes, geodesics on surfaces, the second one basic shape with neighborhood non-intrinsic houses and the elemental equations of the skin concept with numerous purposes.

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Theorem 2. If r = r(s) is the given curve y, then the centre C and radius R of spherical curvature at a point P on /are given by C = r + pn + op'b, R = jp2 + o2p'2 Proof. If C is the centre and R is the radius of the osculating sphere, then its equation is (C - R)2 = R2 where R is the position in vector of any point on the sphere. The points of intersection of the curve and the sphere are given by F(s) = (C - r)2 - R2 = 0. Since the sphere has four point contact with yat P, the conditions of four point contact are F(s) = F'(s) = F"(s) = F'"(s) = 0 which give rise to the following equations.

Corresponding to the pointw = w0, we have "o = 0('o) a t ' = 'oNow F(u) = F(0(O) =/(0 where/is a function of f only. (2) F'(w) = 0, then /(/) = 0 as 0(0 * 0 If F'(u) = 0 and F"(w) * 0, then from (1) and (2) we get 7 (0 = 0 and / (0 * 0, since 0(0, 0(0*0. (3) If F\u) = 0, F"(u) = )0 and F'"(w) * 0, then from (3) /(*) = 0, / ( 0 = 0 and / (0 * 0 as 0 (0 is regular. Hence the surface S given by/(0 has three point contact with the curve yat r[0(ro)]. Proceeding like this, if F\u) = F"(w) = ... = F{"-l\u) = 0 and F{n\u) * 0 at u = w0> then 7(0 = 7(0 = ...

Hence C is a o ds ds constant vector which means that the centre of the osculating sphere is a fixed point Therefore the given curve must lie on a sphere. Hence the condition is sufficient. 12 LOCUS OF CENTRES OF SPHERICAL CURVATURE Unless the curve lies on a sphere, the centres of spherical curvature change from point to point as the point moves on the curve. Hence it is but natural to study the locus of the centres of spherical curvature of the given curve. Let C be the given curve and C{ be the locus of centres of spherical curvature.

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