Velocity, Speed, and Regularity
A parametrized curve describes a moving point. If its position at parameter \(t\) is \(\mathbf{r}(t)=(x(t),y(t))\), then its velocity is \(\mathbf{r}'(t)=(x'(t),y'(t))\). The speed is the magnitude of that velocity. Regularity concerns whether the velocity vanishes; it does not, by itself, tell us the length of the path.
For a planar curve, the speed is \(v(t)=\sqrt{(x'(t))^2+(y'(t))^2}\). If \(\mathbf{r}\) is continuously differentiable on \([a,b]\), its arc length is
This formula integrates speed, which is always nonnegative. A zero-speed parameter can occur without making the integral invalid: if the velocity is continuous, the speed is continuous there too. However, zero velocity means the usual velocity vector cannot specify a tangent direction at that parameter. It is therefore useful to examine such values carefully, especially when the curve has a cusp.
What a Cusp Means
A cusp is a pointed singularity of the geometric curve. A common parametrization of a cusp has zero velocity at the point, and the curve approaches that point along a common tangent line while turning back along the line’s direction. The exact terminology used for singular curve points can vary, so the important distinction here is between the geometric shape and the parametrization: not every parameter with zero velocity is a cusp.
Consider the semicubical cusp \(\mathbf{r}(t)=(t^2,t^3)\). At \(t=0\), the point is the origin and the velocity is zero. The two nearby branches satisfy \(x=t^2\ge 0\) and \(y=\pm x^{3/2}\), so they meet in a pointed shape on the same side of the tangent line \(y=0\). By contrast, a curve can pass smoothly through a point while its parameter temporarily slows to zero. The examples below distinguish these situations.
Regularity Does Not Determine Length
At a regular parameter, the velocity gives a nonzero direction of motion. But regularity is not a length formula: the length still depends on the speed over the whole parameter interval. Conversely, nonregularity does not mean the curve has zero length, infinite length, or a cusp. The integral of speed, with the actual parameter interval and traversal, determines the length.
Worked Example: A Regular Parametrization of a Quarter-Circle
Let \(\mathbf{r}(t)=(\cos t,\sin t)\) for \(0\le t\le \pi/2\). Differentiate each component:
The speed is
It is nonzero for every \(t\) in the interval, so the parametrization is regular throughout. The length is
As a check, the derivative of the antiderivative \(t\) is \(1\), the speed being integrated. The parametrized endpoints are \(\mathbf{r}(0)=(1,0)\) and \(\mathbf{r}(\pi/2)=(0,1)\), consistent with one quarter of the unit circle.
Worked Example: Zero Speed Without a Cusp
Take \(\mathbf{r}(t)=(t^3,0)\) for \(-1\le t\le 1\). Its velocity is
The velocity vanishes at \(t=0\), so the parametrization is nonregular there. The speed and length are
The antiderivative checks because \(\frac{d}{dt}(t^3)=3t^2\). Also, \(x=t^3\) increases from \(-1\) to \(1\) on the entire interval; the point slows to zero at the origin but continues along the same straight line. There is no pointed cusp. This example shows that a zero-speed parameter is compatible with a positive, finite length.
Why Length Integrals Must Be Split at a Cusp
The length integral remains valid across an isolated zero of speed when the parametrization is continuously differentiable. Splitting at that parameter is often essential in the calculation, though, because simplifying a square root can introduce an absolute value. On each side of the cusp, the expression inside that absolute value may have a different sign.
The justification is additivity of the definite integral: since the speed is continuous, it is integrable on the interval, and splitting the interval at \(c\) does not change the total integral. The cusp point itself contributes no separate length. Splitting is a way to handle the speed correctly on each side, not a way of adding or subtracting distance at the point.
Worked Example: Length of a Semicubical Cusp
For \(\mathbf{r}(t)=(t^2,t^3)\) on \([-1,1]\), the velocity and speed are
The absolute value is necessary because \(\sqrt{t^2}=|t|\), not \(t\) for every real \(t\). Split at the cusp parameter \(t=0\): on \([-1,0]\), \(|t|=-t\); on \([0,1]\), \(|t|=t\). Thus
An antiderivative of \(t\sqrt{4+9t^2}\) is \(F(t)=\frac{(4+9t^2)^{3/2}}{27}\), since
The right-hand integral is \(F(1)-F(0)\). The left-hand integral has integrand \(-t\sqrt{4+9t^2}\), whose antiderivative is \(-F(t)\); it is \(-F(0)+F(-1)\). Since \(F(-1)=F(1)\), both pieces equal \(\frac{13\sqrt{13}-8}{27}\). Therefore
Each integral is nonnegative, as required for length. Differentiating \(F\) and \(-F\) verifies the antiderivatives on their respective intervals. The split also makes clear why the cusp does not cancel the length of one branch against the other.
A Common Mistake: Dropping the Absolute Value
For the cusp example, an incorrect simplification would be \(\sqrt{4t^2+9t^4}=t\sqrt{4+9t^2}\) throughout \([-1,1]\). That would give an odd integrand and the false result
This calculation is wrong because the square root of \(t^2\) is \(|t|\). On the negative half of the interval, the speed is \(-t\sqrt{4+9t^2}\), which is positive, not negative. The correction is to keep \(|t|\) and either integrate it directly as a piecewise expression or split the interval at zero.
A different pitfall is to treat the geometric trace alone as sufficient to determine the length of a parametrized traversal. For example, \(\mathbf{r}(t)=(t^2,0)\) on \([-1,1]\) traces the segment from \(0\) to \(1\), then retraces it back to \(0\). Its speed is \(2|t|\), so its traveled length is \(\int_{-1}^1 2|t|\,dt=2\), not \(1\). The parameter value \(t=0\) has zero speed, and the retracing matters. Length measures the distance traveled by the parametrization, including repeated traversal.
A Practical Check at Nonregular Parameters
When the velocity vanishes, do not conclude immediately that the length formula fails or that the curve has a cusp. Compute the velocity, identify every parameter where it is zero, and inspect the local curve or the velocity directions on either side. Then form the speed as a nonnegative quantity and split the integral at any value where its simplified form changes.
Solve \(\mathbf{r}'(t)=\mathbf{0}\) on the given parameter interval.
Check whether the point is a cusp or another kind of stationary point; zero velocity alone does not decide this.
Use \(\|\mathbf{r}'(t)\|\), and retain absolute values when simplifying square roots.
Partition the interval at cusp or other critical parameter values when the speed expression changes form, then add the nonnegative lengths.
Regularity helps describe how a parametrization moves locally. Cusps explain why a tangent direction may fail at a particular point. Arc length, however, comes from integrating speed over the full interval. Keeping those ideas separate prevents both geometric misclassification and incorrect length calculations.
Check Your Understanding
Use the definitions and methods from this tutorial to answer the following questions.
- What condition on \(\mathbf{r}'(t_0)\) makes a parametrization regular at \(t_0\)?
- Why does a zero-speed parameter not, by itself, prove that the curve has a cusp?
- For \(\mathbf{r}(t)=(t^2,t^3)\), why must the speed include \(|t|\)?
- Explain why splitting a speed integral at an isolated cusp does not change the total length.
- What does the parametrization \(\mathbf{r}(t)=(t^2,0)\), \(-1\le t\le1\), show about repeated traversal and length?
