2.4 Curvature Kennesaw State University（2.4曲率肯尼索州立大学）.pdfVIP

88 CHAPTER 2. VECTOR FUNCTIONS 2.4 Curvature 2.4.1 DeÖnitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small for curves which bend very little and to be large for curves which bend sharply. If we move along a curve, we see that the direction of the tangent vector will not change as long as the curve is áat. Its direction will change if the curve bends. The more the curve bends, the more the direction of the tangent vector will change. So, it makes sense to study how the tangent vector changes as we move along a curve. But because we are only interested in the direction of the tangent vector, not its magnitude, we will consider the unit tangent vector. Curvature is deÖned as follows: DeÖnition 150 (Curvature) Let C be a smooth curve with position vector ! r (s) where s is the arc length parameter. The curvature of C is deÖned to be: ! dT = (2.11) ds ! where T is the unit tangent vector. Note the letter used to denote the curvature is the greek letter kappa denoted . ! Remark 151 The above formula implies that T be expressed in terms of s, arc length. And therefore, we must have the curve parametrized in terms of arc length. Example 152 Find the curvature of a circle of radius a. We saw earlier that the parametrization of a circle of radius a with respect to arc ! D s sE ! length length was r (s) = acos ;as