Section 3.3 - Arc Length and Curvature

Section Objectives

  1. Determine the length of a plane or space curve defined by a vector-valued function.
  2. Find the arc-length parameterization for a smooth curve.
  3. Find the curvature of a smooth curve at a point.
  4. Compute the principal unit normal vector for a smooth curve.



Arc length

Suppose that a smooth curve is defined by the vector-valued function for . Then the arc length of is given by



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Arc-length parameter

Suppose defines a smooth curve for . The arc-length parameter is the function


It is easy to see that is an increasing function. In fact, (since the is smooth). When using vector-valued functions of the variable , we usually think about as representing time. The arc-length parameter provides an alternative to the time parameterization. The arc-length parameter is an imprtant theoretical tool.



Examples







Curvature

Describing the "curviness" of a curve is one of the many things for which the arc-length parameter is well-suited. After some thought, you will probably agree that this definition provides a good description of curviness.


Let be the smooth curve defined by , where is the arc-length parameter. The curvature at is given by


Unfortunately the definition of curvature requires the use of the arc-length parameterization, which can be computationally inconvenient. The following theorem provides some alternative formulas that might be easier to work with.


Theorem: If is a smooth curve defined by . Then the curvature at is given by

Alternatively,

If is the graph of the function and both and exist, then the curvature at is given by



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Normal and binormal vectors

Let be a smooth curve defined by on the open interval . If , then the principal unit normal vector is

and the binormal vector is


Notice that points in the direction that is changing. That is to say that points "into the turns." Furthermore notice that is a unit vector orthogonal to both and .


The principal unit normal vector is usually challenging to compute because can be a very complicated function. In section 3.4, we will see alternatives to the formula above.



Examples