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Transcript
ARC LENGTH and CURVATURE Arc Length : given a space curve r(t) = < f (t), g(t), h(t) >; • the arc length from a to b is L= b 0 a |r (t)|dt Z = b a Z r f 0(t)2 + g 0(t)2 + h0(t)2dt; • arc length function is s(t) = Rt a |r 0 (u)|du, so ds = |r0(t)|; dt • parameterization with respect to arc length is r(t(s)). Curvature : given the unit tangent T(t) = • the curvature κ=| r0 (t) |r0 (t)| ; dT |; ds • chain rule gives |T0(t)| ; κ(t) = 0 |r (t)| • Curvature Theorem |r0(t) × r00(t)| κ(t) = ; 0 3 |r (t)| • for planes curves y = f (x) |f 00(x)| κ(x) = . (1 + (f 0(x))2)3/2 MOTION in SPACE: VELOCITY and ACCELERATION Velocity and Acceleration: assuming r(t) is position of an object in space; • the velocity of the object is v(t) = r0(t); • the speed of the object is v(t) = |v(t)| = |r0(t)|; • the acceleration of the object is a(t) = v0(t) = r00(t); • Newton’s Second Law of Motion: F = ma = mr00(t), for a force F acting on an object with mass m, so given F, two integrations are needed to find r(t). 2