13.12 13.4: SUMMARY

• A parametrization r(t) is called regular if r′(t) ≠ 0 for all t. If r(t) is regular, we define the unit tangent vector .

• Curvature is defined by , where r(s) is an arc length parametrization.

• In practice, we compute curvature using the following formula, which is valid for arbitrary regular parametrizations:

  

• The curvature at a point on a graph y = f(x) in the plane is

  

• If ∥T′(t)∥ ≠ 0, we define the unit normal vector .

• T′(t) = κ(t)υ(t)N(t)

• The osculating plane at a point P on a curve is the plane through P determined by the vectors T_P and N_P. It is defined only if the curvature κ_P at P is nonzero.

• The osculating circle Osc_P is the circle in the osculating plane through P of radius R = 1/κ_P whose center Q lies in the normal direction N_P:

  

  The center of Osc_P is called the center of curvature and R is called the radius of curvature.