Intrinsic Geometry

Abstract

The Fundamental Form of a Surface Properties of a curve or surface which depend on the coordinate space that curve or surface is embedded in are called extrinsic properties of the curve. For example, the slope of a tangent line is an extrinsic property since it depends on the coordinate system in which rises and runs are measured. In contrast, intrinsic properties of surfaces are properties that can be measured within the surface itself without any reference to a larger space. For example, the length of a curve is an intrinsic property of the curve, and thus, the length of a curve (t) = r (u (t) ; v (t)) ; t in [a; b] ; on a surface r (u; v) is an intrinsic property of both the curve itself and the surface that contains it. As we saw in the last section, the square of the speed of (t) is ds dt 2 = g 11 du dt 2 + 2g 12 du dt dv dt + g 22 dv dt 2 in terms of the metric coe¢ cients g 11 = r u r u ; g 12 = r v r u ; and g 22 = r v r v Thus, very short distances ds on the surface can be approximated by (ds) 2 = g 11 (du) 2 + 2g 12 dudv + g 22 (dv) 2 (1) That is, if du and dv are su¢ ciently small, then ds is the length of an in…nites-1 imally short curve on the surface itself. Equation (1) is the fundamental form of the surface, which intrinsic to a surface because it is related to distances on the surface itself. Moreover, any properties which can be derived solely from a surface's fundamental form are also intrinsic to the surface. EXAMPLE 1 Find the fundamental form of the right circular cylin-der of radius R; which can be parameterized by r (u; v) = hR cos (u) ; R sin (u) ; vi Solution: Since r u = hhR sin (u) ; R cos (u) ; 0i and r v = h0; 0; 1i ; the metric coe¢ cients are g 11 = r u r u = R 2 sin 2 (u) + R 2 cos 2 (u) + 0 2 = R 2 g 12 = r u r v = 0 + 0 + 0 = 0 g 22 = r v r v = 0 2 + 0 2 + 1 2 = 1 Thus, ds 2 = R 2 du 2 + dv 2 : If the parameterization is orthogonal, then g 12 = r u r v = 0; so that ds 2 = g 11 du 2 + g 22 dv 2 For example, the xy-plane is parameterized by r (u; v) = hu; v; 0i ;which implies that r u = i and r v = j and that g 11 = g 22 = 1; g 12 = 0: The fundamental form for the plane is ds 2 = du 2 + dv 2