Arc Length and Curvature Section 14.3 Arc Length and Curvature

ARC LENGTH Let a space curve C be given by the parametric equations x = f (t) y = g(t) z = h(t) on the interval a ≤ t ≤ b, where f ′, g′, and h′ are continuous. If the curve is traversed exactly once as t increases from a to b, then the length of the curve is

ALTERNATIVE FORMULA FOR ARC LENGTH If , is the vector equation of the curve C, then the arc length formula can be written as

THE ARC LENGTH FUNCTION Suppose that C is a piecewise-smooth curve given by a vector function r(t) = f (t)i + g(t)j + h(t)k, a ≤ t ≤ b, and C is traversed exactly once as t increased from a to b. We define its arc length function s by NOTE: ds/dt = | r′(t) |

PARAMETERIZATION WITH RESPECT TO ARC LENGTH Given the arc length function s(t) of a curve given by r, it is often possible to solve for the parameter t in terms of s. This allows us to parameterize the curve with respect to arc length by writing r as r = r(t(s)). This is useful because arc length arises naturally from the shape of the curve and does not depend on a particular coordinate system.

CURVATURE The curvature of a curve at a given point is a measure of how quickly the curve changes direction at that point. The curvature is define to be where T is the unit tangent vector,

AN ALTERNATE CURVATURE FORMULA

A THEOREM ABOUT CURVATURE The curvature of the curve given by the vector function r is

CURVATURE OF A TWO-DIMENSIONAL PLANE CURVE If y = f (x) is a two-dimensional plane curve, the curvature is given by

THE PRINCIPAL UNIT NORMAL VECTOR Given the smooth space curve r(t). If r′ is also smooth, we define the principal unit normal vector N(t) (or simply unit normal) as where T(t) is the unit tangent vector.

THE BINORMAL VECTOR The vector B(t) = T(t) × N(t) is called the binormal vector. It is perpendicular to both T and N and is also a unit vector.

THE NORMAL AND OSCULATING PLANES The plane determined by the normal and binormal vectors N and B at the point P on a curve C is called the normal plane of C at P. It consists of all lines that are orthogonal to the tangent vector T. The plane determined by the vectors T and N is called the osculating plane of C at P. It is the plane that comes closest to containing the part of the curve near P. (For a plane curve, the osculating plane is simply the plane that contains the curve.)

THE OSCULATING CIRCLE The circle that lies in the osculating plane of C at P, has the same tangent as C at P, lies on the concave side of C (toward which N points), and has radius ρ = 1/κ is called the osculating circle (or circle of curvature) of C at P. It is the circle that best describes how C behaves near P; it shares the same tangent, normal, and curvature at P.