Calc 3 Lecture Notes Section 11.4 Page 8 of 8

### Section 11.4: Curvature

Big idea: Curvature is a quantity that describes what it sounds like: the curviness of a curve at any given point on the curve…, or how sharply a curve is changing direction.

Big skill: You should be able to compute the unit tangent vector and curvature of any curve traced by a vector-valued function r(t) or described by the graph of a scalar function y = f(x).

In this section, we want to capture the notion that a curve described by 90° of a quarter-mile radius circle is not nearly as “sharp” a curve described by 90° of a 30 foot radius circle. In the first case, we curve through 90° over a distance of for a ratio of , while in the second case, we have a ratio of . Thus, we rotate through 44 times more of an angle for every foot traveled in the second curve than in the first curve.

Also, notice what happens when we compute the curvature in terms of radians per foot:

and

As it turns out, it is (theoretically) pretty convenient to define curvature in terms of angle turned through per arc length of curve. So, we need to look at how to describe a curve in terms of its arc length instead of its independent parameter t.

Parameterizing a curve in terms of arc length:

Recall that there are infinitely many ways of parameterizing a curve. For example, the parabola
y = x2 can be parameterized as x = t and y = t2, or as x = 5t and y = 25t2, etc.

When possible, it’s often useful to parameterize a curve in terms of arc length so that each 1-unit increment in the parameter traces out exactly one unit of arc length on the curve.

To do this, recall that for the curve given by the vector function , we can define an “arc length function” s(t) by:

The above integral relates arc length s to the given parameter t.

The idea is to evaluate above integral to find s(t), invert the equation to obtain t(s), and then substitute t(s) into the original parameterization to obtain an arc length parameterization.

Practice:

1. Find an arc length parameterization for a circle of radius a traced out by the vector-valued function for 0 £ t £ 2p.
1. Find an arc length parameterization for a helix of radius R that passes through the point (R, 0, 0), and completes one full turn over a vertical distance of H (i.e., (R, 0, H) is another point on the curve).

Unit Tangent Vector

We will now define the unit tangent vector, which will be a vector tangent to a curve that always has a magnitude of one. So, any change in a unit tangent vector will always be a change in its direction, since its magnitude (The only other quantity that could change) is always a constant. Thus, a change in the unit tangent vector will capture a change in angle along a curve.

Recall that for a vector function r(t), the derivative r¢(t) is tangent to the curve for all t, and that to create a unit vector, we just divide a vector by its magnitude. Thus, the unit tangent vector T(t) for any value of t is defined by:

T(t) will always points in the direction of the orientation of the curve.

Practice:

1. Find the unit tangent vector T(t) for a circle of radius a traced out by the vector-valued function .
1. Find the unit tangent vector T(t) for the parabola traced out by the vector-value function .
1. Find the unit tangent vector T(t) for the helix in practice #2.

Curvature

We intuitively think of curvature as a measure of how sharply a curve “bends” at some point. But how can we quantify this?

Idea: Suppose the curve has been parameterized in terms of arc length; then the amount the unit tangent vector turns as we take a small step along the curve will be a measure of curvature, since a sharp curve will rotate the tangent vector by a lot over a short distance.

Practice:

1. Find an approximation of the curvature of at t = 0.

Definition 4.1: Curvature

The curvature k of a curve at any point is defined as the scalar quantity

.

The above definition requires that the curve be parameterized in terms of its arc length, which is not usually possible to do. Using the Chain Rule results in a more useful form for curvature:

Recall that since , we can compute s¢(t) easily using the fundamental theorem of calculus:

Alternative Formula for Curvature:

Practice:

1. Find the curvature of a circle of radius a using both the definition and the alternative formula.
1. Find the curvature for the helix in practice #2 using both the definition and the alternative formula.
1. Find the curvature for any straight line.
1. Find the curvature of the parabola traced out by .

Theorem 4.1: (Another) Alternative Formula for the curvature

The curvature of a smooth curve traced out by the vector-valued function r(t) can also be calculated from:

(proof omitted because it is long and painful)

Practice:

1. Use Theorem 4.1 to find the curvature of a circle of radius a.
1. Use Theorem 4.1 to find the curvature of a helix.
1. Use Theorem 4.1 to find the curvature of a straight line.
1. Use Theorem 4.1 to find the curvature of the parabola traced out by .

(Another)2 Alternative Formula for Curvature Given y = f(x):

For a 2D plane curve defined by the Cartesian function y = f(x), we can form a vector-valued function by taking . Thus, , and . Plugging this into our equation:

(Another)3 Alternative Formula for Curvature Given r = f(q):

For a 2D plane curve defined by the polar function r = f(q),

Practice:

1. Use this new formula to find the curvature of the general parabola y = ax2 + bx + c. Notice that the largest curvature is at the vertex, where x = -b/2a.