The vector et is the local unit tangent vector to the curve which changes from point to point. Consequently, the time derivative of et w al. be nonzero The time derivative of et can be written as det det ds det dt ds dt d (4) In order to calculate the derivative of et, we note that, since the magnitude of et is constant and equal to one, the only changes that et can have are due to rotation, or swinging O et p +dev d > et+det When we move from s to s +ds, the tangent vector changes from et to et det. The change in direction can be related to the angle dB The direction of det, which is perpendicular to et, is called the normal direction. On the other hand, the magnitude of det will be equal to the length of et(which is one), times dB. Thus, if en is a unit normal vector in the direction of det, we can write det= dBe, dB (6) Here iB/ds is aa local property of the curve, called the curvature, and p=1/k is called the radius of Note that in the picture, the sizes of det, ds, and dB are exaggerated for illustration purposes and actually represent the changes in the limit as ds(and also dt)approach zero Note Curvature and radius of curvature We here two tangent vectors et and e +det, separated by a small ds and having an angle between them If we draw perpendiculars to these two vectors, they will intersect at a point, say, o. Because the two lines meeting at O are perpendicular to each of the tangent vectors, the angle between them will be the same as the angle between et and e+det, dB. The point O is called the center of curvature, and the istance, P, between O and A is the radius of curvature. Thus, from the sketch, we have that ds= pdB, or 1/pThe vector et is the local unit tangent vector to the curve which changes from point to point. Consequently, the time derivative of et will, in general, be nonzero. The time derivative of et can be written as, det dt = det ds ds dt = det ds v . (4) In order to calculate the derivative of et, we note that, since the magnitude of et is constant and equal to one, the only changes that et can have are due to rotation, or swinging. When we move from s to s + ds, the tangent vector changes from et to et + det. The change in direction can be related to the angle dβ. The direction of det, which is perpendicular to et, is called the normal direction. On the other hand, the magnitude of det will be equal to the length of et (which is one), times dβ. Thus, if en is a unit normal vector in the direction of det, we can write det = dβen . (5) Dividing by ds yields, det ds = dβ ds en = κen = 1 ρ en . (6) Here, κ = dβ/ds is a a local property of the curve, called the curvature, and ρ = 1/κ is called the radius of curvature. Note that in the picture, the sizes of det, ds, and dβ are exaggerated for illustration purposes and actually represent the changes in the limit as ds (and also dt) approach zero. Note Curvature and radius of curvature We consider here two tangent vectors et and e + det, separated by a small ds and having an angle between them of dβ. If we draw perpendiculars to these two vectors, they will intersect at a point, say, O′ . Because the two lines meeting at O′ are perpendicular to each of the tangent vectors, the angle between them will be the same as the angle between et and e + det, dβ. The point O′ is called the center of curvature, and the distance, ρ, between O′ and A is the radius of curvature. Thus, from the sketch, we have that ds = ρ dβ, or dβ/ds = κ = 1/ρ. 2