11.5:  Arc Length and Curvature

A.	Theorem 11.6:  Arc Length of a Space Curve

If C is a smooth curve given by r(t) = f(t)i + g(t)j + h(t)k , on an interval [a,b], then the arc length of C on the interval is

B.	Definition of Arc Length Function

Let C be a smooth curve given by r(t) defined on the closed interval [a,b].  For a ≤ t ≤ b, the arc length function is given by

The arc length s is called the arc length parameter.
Note:  It follows  

C.	Theorem 11.7:  Arc Length Parameter

If C is a smooth curve given by

r(s) = x(s)i + y(s)j       OR       r(s) = x(s)i + y(s)j + z(s)k  

where s is the arc length parameter, then

Moreover, if t is any parameter of the vector-valued function r such that , then t must be the arc length parameter.

D.	An important use of the arc length parameter is to find curvature – the measure of how sharply a curve bends.  If a curve bends more sharply at P than at Q, we say the curvature is greater at P than at Q.  

Definition of Curvature

Let C be a smooth curve (in the plane or in space) given by r(s), where s is the arc length parameter.  The curvature at s is given by 

Note:  For a circle, the curvature and the radius are inversely related.  As the radius increases, the curvature decreases.
E.	Theorem 11.8:  Formulas for Curvature

F.	Theorem 11.9:  Curvature in Rectangular Coordinates

If C is the graph of a twice differentiable function given by y = f(x), then the curvature at the point (x, y) is given by

G.	Let C be a curve with curvature K at point P.  The circle passing through point P with radius     r = 1/K is called the circle of curvature if the circle lies on the concave side of the curve and shares a common tangent line with the curve at point P.  the radius is called the radius of curvature at P, and the center of the circle is called the center of curvature.  

H.	Arc length and curvature are closely related to the tangential and normal components of acceleration.  The tangential component of acceleration is the rate of change of the speed, which in turn is the rate of change of the arc length.  The tangential component is a function of the arc length and is independent of the curvature.

On the other hand, the normal component of acceleration is a function of both speed and curvature.  This component measures the acceleration action perpendicular to the direction of motion.  Imagine you driving a car around a turn.  If your speed is high and the turn is sharp, you feel yourself thrown against the car door.  By lowering your speed or taking a wider turn, you are able to lesson this sideways thrust.  

I.	Theorem 11.10:  Acceleration, Speed, and Curvature
If r(t) is the position vector for a smooth curve C, then the acceleration vector is given by

where K is the curvature of C and ds/dt is the speed.

J.	Suppose a moving object was mass m is in contact with a stationary object.  The total force required to produce an acceleration a along a given path is

The portion of this total force that is supplied by the stationary object is called the force of friction.  If a car is moving with constant speed is rounding a turn, the roadway exerts a frictional force that keeps the care from sliding off the road.  If the car is not sliding, the frictional force is perpendicular to the direction of motion and had magnitude equal to the normal component of acceleration.