Chapter 9

Curvature, Acceleration, and Motion

Speed tells us how quickly a point moves along a curve, but it does not tell us how sharply the curve is bending. A car can speed up on a straight road or travel at constant speed around a tight corner. To distinguish these motions, we need to measure change in direction separately from change in speed.

9.1Curvature

Besides the length of a curve, one of the most powerful things calculus allows us to do is rigorously study its curvature. How can we quantify the fact that some curves bend gently and others turn sharply in space? One approach is to examine the tangent vectors to the curve and determine how quickly they are changing.

Of course, there's a complication to this: a tangent vector can change in length without changing in direction. This doesn't mean that a curve is curving, but rather that the particle tracing it out is accelerating.

To remove this worry, we define the unit tangent vector to a curve. Just divide the derivative by its magnitude!

Definition 9.1 (Unit Tangent Vector). At points where $⃗ 𝑟' (𝑡) \neq ⃗ 0$ , the unit tangent vector to the curve $⃗ 𝑟 (𝑡)$ is the vector of length $1$ parallel to $⃗ 𝑟' (𝑡)$ :

⃗ 𝑇 (𝑡) = ⃗ 𝑟' (𝑡) ‖ ⃗ 𝑟' (𝑡) ‖

This allows a clean definition of curvature: it is how much the unit tangent vector turns per arc length.

Definition 9.2 (Curvature of a Curve). The curvature of a curve is

𝜅 = ∥ 𝑑 ⃗ 𝑇 𝑑 𝑠 ∥ = ‖ ⃗ 𝑇' (𝑡) ‖ ‖ ⃗ 𝑟' (𝑡) ‖

The second equality follows from the chain rule:

∥ 𝑑 ⃗ 𝑇 𝑑 𝑠 ∥ = ∥ 𝑑 ⃗ 𝑇 𝑑 𝑡 𝑑 𝑡 𝑑 𝑠 ∥ = ∥ 𝑑 ⃗ 𝑇 𝑑 𝑡 𝑑 𝑠 𝑑 𝑡 ∥ = ∥ ⃗ 𝑇' ∥ ‖ ⃗ 𝑟' ‖

Figure 9.1 The unit tangent has a fixed length, so the only thing it can do is turn—and the graph beside it records the direction it points, measured against distance travelled rather than against time. The steepness of that graph is the curvature. Where the oval bends hard the direction changes quickly per unit of length and the graph is steep; along the flatter stretches it barely climbs. This is the same trade as the previous chapter, one derivative up: there, length was the area under speed; here, curvature is the slope of direction against length.

This formula can be difficult to apply in general because the unit tangent vector $⃗ 𝑇$ might have a complicated formula, so taking its derivative can require a lot of work. Doing some calculus we can get a simpler formula:

Theorem 9.3 (Curvature of a Curve). For a regular curve in $ℝ 3$ , the curvature of $⃗ 𝑟 (𝑡)$ is given by

𝜅 (𝑡) = ‖ ⃗ 𝑟' (𝑡) \times ⃗ 𝑟'' (𝑡) ‖ ‖ ⃗ 𝑟' (𝑡) ‖ 3

This is something that's relatively easy to compute (though perhaps tedious) from any parameterization: you just need to find the first and second derivatives, take a cross product, and then plug into the formula!

But, if we further restrict ourselves to the case that $⃗ 𝑟 (𝑡) = (𝑡, 𝑓 (𝑡))$ traces the graph of a function, we can simplify this calculation even more:

Theorem 9.4 (Curvature of a Graph). If $𝑦 = 𝑓 (𝑥)$ is a function, the curvature of its graph is

𝜅 (𝑥) = | 𝑓'' (𝑥) | (1 + (𝑓' (𝑥)) 2) 3 / 2

For a circle of radius $𝑅$ , take

⃗ 𝑟 (𝑡) = ⟨ 𝑅 c o s 𝑡, 𝑅 s i n 𝑡, 0 ⟩ .

Then

⃗ 𝑟' (𝑡) = ⟨ - 𝑅 s i n 𝑡, 𝑅 c o s 𝑡, 0 ⟩, ⃗ 𝑟 ″ (𝑡) = ⟨ - 𝑅 c o s 𝑡, - 𝑅 s i n 𝑡, 0 ⟩ .

The speed is $‖ ⃗ 𝑟' (𝑡) ‖ = 𝑅$ , while

‖ ⃗ 𝑟' (𝑡) \times ⃗ 𝑟 ″ (𝑡) ‖ = 𝑅 2 .

Therefore

𝜅 = 𝑅 2 𝑅 3 = 1 𝑅 .

This agrees with our geometric intuition: small circles bend sharply and have large curvature, while large circles bend gently and have small curvature.

At a point where $𝜅 \neq 0$ , the radius of curvature is $1 / 𝜅$ . The circle with this radius which best matches the curve near that point is called the osculating circle.

Figure 9.2 At a point with nonzero curvature, one circle matches the curve's bending, and its radius is

1 / 𝜅

. Follow it around an oval and it is small and tight where the curve turns hardest, wide and lazy where it does not. On a circle it never changes at all, which is the calculation

𝜅 = 1 / 𝑅

made visible. At an inflection the curve is momentarily straight,

𝜅

passes through zero, and the circle escapes off the page—leaving the tangent line as the limiting object of infinite radius.

Curvature belongs to the geometric curve, not to the speed at which it is traversed. An admissible reparameterization which traces the curve once may change $⃗ 𝑟' (𝑡)$ and $⃗ 𝑟 ″ (𝑡)$ , but it does not change the amount that the unit tangent turns per unit of arc length.

9.2The Frenet Frame

The unit tangent vector provides us with a very useful "pointer" - always oriented directly along a curve. But in any serious application of parametric curves, we need more information: we would like a whole $𝑥, 𝑦, 𝑧$ coordinate frame at each point of the curve.

To start, we'll look for one vector which is orthogonal to, or normal to our curve. How can we find one? The unit tangent has constant length (it is the unit tangent, after all). By the constant-length theorem from Chapter 7, its derivative $⃗ 𝑇' (𝑡)$ is orthogonal to $⃗ 𝑇 (𝑡)$ . To find a unit vector orthogonal to $⃗ 𝑇$ , we normalize $⃗ 𝑇'$ .

Definition 9.5 (Normal Vector). At points where $⃗ 𝑇' (𝑡) \neq ⃗ 0$ ,

⃗ 𝑁 (𝑡) = ⃗ 𝑇' (𝑡) ‖ ⃗ 𝑇' (𝑡) ‖ .

Figure 9.3 Two unit vectors, carried along by the curve.

⃗ 𝑇

points the way you are going;

⃗ 𝑁

points the way you are turning, which is always toward the inside of the bend. They stay perpendicular to each other everywhere they are both defined—

⃗ 𝑇

has constant length, so

⃗ 𝑇'

is perpendicular to it for the reason established in Chapter 7, and

⃗ 𝑁

is just

⃗ 𝑇'

normalized. At the exact inflection

⃗ 𝑇' = ⃗ 0

, so this formula does not define

⃗ 𝑁

; across the inflection, the bend changes sides and the normal turns over.

Given these two vectors, it is easy to find a third unit vector: take the cross product of $⃗ 𝑇$ and $⃗ 𝑁$ . The result is called the binormal because it is a second normal vector to the curve.

Definition 9.6 (Binormal Vector).

⃗ 𝐵 (𝑡) = ⃗ 𝑇 (𝑡) \times ⃗ 𝑁 (𝑡)

Together these three vectors provide a coordinate system at each point along the curve: $⃗ 𝑇$ points in the tangent direction, $⃗ 𝑁$ points in the direction the curve is bending, and $⃗ 𝐵$ is orthogonal to both. This collection of vectors is called the Frenet Frame and is heavily used in computations in physics, engineering, and computer graphics.

Definition 9.7 (Frenet Frame).

⃗ 𝑇 (𝑡) = ⃗ 𝑟' (𝑡) ‖ ⃗ 𝑟' (𝑡) ‖ ⃗ 𝑁 (𝑡) = ⃗ 𝑇' (𝑡) ‖ ⃗ 𝑇' (𝑡) ‖ ⃗ 𝐵 (𝑡) = ⃗ 𝑇 (𝑡) \times ⃗ 𝑁 (𝑡)

The conditions in these formulas matter: $⃗ 𝑟' (𝑡) \neq ⃗ 0$ is needed to define $⃗ 𝑇$ , and $⃗ 𝑇' (𝑡) \neq ⃗ 0$ is needed to define $⃗ 𝑁$ by normalization.

Figure 9.4 The same frame, one dimension up. A curve in space still has its

⃗ 𝑇

and its

⃗ 𝑁

, and they still span the plane it is momentarily bending in—the shaded square. What is new is that this plane can now tilt as you move, so it takes a third vector to say which way it is facing:

⃗ 𝐵 = ⃗ 𝑇 \times ⃗ 𝑁

, perpendicular to both. On the helix the frame turns steadily; on the trefoil it tumbles.

9.3Acceleration in the Frenet Frame

Acceleration records every change in velocity. Since velocity has both a magnitude and a direction, acceleration can come from changing either one. The Frenet frame separates these two effects.

Let

𝑣 (𝑡) = ‖ ⃗ 𝑣 (𝑡) ‖

denote scalar speed. Since $⃗ 𝑇$ points in the direction of motion, velocity can be written

⃗ 𝑣 = 𝑣 ⃗ 𝑇 .

Differentiating with the product rule gives

⃗ 𝑎 = 𝑑 ⃗ 𝑣 𝑑 𝑡 = 𝑑 𝑣 𝑑 𝑡 ⃗ 𝑇 + 𝑣 𝑑 ⃗ 𝑇 𝑑 𝑡 .

Now use the chain rule and the definitions of speed and curvature:

𝑑 ⃗ 𝑇 𝑑 𝑡 = 𝑑 ⃗ 𝑇 𝑑 𝑠 𝑑 𝑠 𝑑 𝑡 = 𝜅 ⃗ 𝑁 𝑣 .

Substituting this into the acceleration formula gives

⃗ 𝑎 = 𝑑 𝑣 𝑑 𝑡 ⃗ 𝑇 + 𝜅 𝑣 2 ⃗ 𝑁 .

The coefficient

𝑎 𝑇 = 𝑑 𝑣 𝑑 𝑡

is the tangential acceleration: it measures change in speed. The coefficient

𝑎 𝑁 = 𝜅 𝑣 2

is the normal acceleration: it measures change in direction. At a point where $𝜅 = 0$ , the normal component is zero even though the normalized vector $⃗ 𝑁$ may not be defined there.

Figure 9.5 Acceleration answers two different questions at once, and the frame separates them. The part along

⃗ 𝑇

𝑑 𝑣 / 𝑑 𝑡

: it is how fast you are speeding up. The part along

⃗ 𝑁

𝜅 𝑣 2

: it is how hard you are turning. Speeding up in a straight line puts everything into the first and nothing into the second; holding a steady speed around a bend does the reverse. Neither is more real than the other, and a lap uses both.

Curvature and acceleration therefore describe different things. Curvature is a geometric property of the path and does not change when we traverse the same path at a different speed. Acceleration belongs to the particular motion along that path.

Figure 9.6 One bend, taken at whatever speed you choose. The curvature never moves—it belongs to the road, not to the driver. What moves is

𝑎 𝑁 = 𝜅 𝑣 2

, and it moves as the square: take the same corner twice as fast and you need four times the sideways acceleration to stay on it. That is the whole difference between a geometric property of a curve and a property of how you happen to traverse it.

This completes our study of parametric curves. We learned how to construct a curve, differentiate and integrate its coordinate functions, measure its length and curvature, and separate the ways a motion changes speed and direction. In Part III, we turn from functions whose outputs are vectors to scalar functions whose inputs are points in a higher-dimensional space.