Arc Length of a Parametric Curve
A parametric curve gives both coordinates as functions of a parameter, (x(t), y(t)), the natural description of anything moving: t is time, and the curve is the trajectory. Its length from t = a to t = b is:
L = ∫ab √( [x'(t)]² + [y'(t)]² ) dt
The integrand has a beautiful physical name: it's the speed. The velocity vector at time t is (x'(t), y'(t)); its magnitude √(x'² + y'²) is how fast the point is moving; and integrating speed over time gives distance traveled, the formula literally says length = ∫ speed dt the odometer principle promoted to curved paths in the plane. No other arc length formula states its meaning so plainly.
This is also the master version of the arc length story: the rectangular formula ∫√(1 + f'²)dx is this one with the trivial parameterization x = t, y = f(t), and the polar formula √(r² + r'²) is this one with x = r cosθ, y = r sinθ pre-simplified.
How to Use This Calculator
Enter x(t) and y(t), implicit multiplication like 3cos(t) works, and the parameter window. The calculator differentiates both coordinates symbolically, shows x' and y', integrates the speed numerically, and draws the actual path. The chips include two of the subject's celebrity results: the unit circle (length 2π, the definition of π in disguise) and the cycloid arch, whose length is exactly 8, a result so surprising in the 1600s that it helped launch the calculus itself.
Worked Example: the Cycloid's Exact 8
The cycloid, the path of a point on a rolling wheel's rim, is x = t − sin t, y = 1 − cos t. Derivatives: x' = 1 − cos t, y' = sin t, so
x'² + y'² = (1 − cos t)² + sin²t = 2 − 2cos t = 4sin²(t/2)
The speed is 2|sin(t/2)|, and one arch (t from 0 to 2π) has length ∫₀^{2π} 2 sin(t/2) dt = 8 exactly four times the wheel's diameter, with no π anywhere. A wheel of radius 1 rolls forward 2π ≈ 6.283 per revolution while its rim point travels 8: the point moves farther than the wheel.
Common Mistakes to Avoid
- Adding the derivatives instead of their squares. The integrand is √(x'² + y'²), a vector magnitude, not x' + y' and not √(x' + y')². Componentwise Pythagoras or nothing.
- Retracing the path. If the parameterization loops (the circle beyond 2π), the integral honestly counts every lap, length of the journey not the tire track. Bound t to one traversal for geometric length.
- Assuming the parameterization is unique. (cos 2t, sin 2t) over [0, π] traces the same circle at double speed, different integrand, same length. Arc length is parameterization-independent; speed is not.
- Expecting closed forms. The ellipse chip is the famous cautionary tale: its perimeter has no elementary formula (elliptic integrals were invented for it). The calculator's numeric ≈ 15.865 for the 3×2 ellipse is the professional answer.
Real-World Applications
Parametric arc length is the path-length computation of the moving world. GPS devices and fitness watches integrate speed over time, exactly this formula, to report distance run along a curving road; flight management systems do it along great-circle-plus-wind trajectories; and delivery routing sums it over road polylines. In robotics and CNC machining, tool paths are parametric splines, and their arc lengths set cycle times and feed rates, CAM software evaluates this integral for every contour it cuts.
Computer graphics leans on it constantly: animating a character along a Bézier path at constant speed requires reparameterizing by arc length (otherwise motion crawls where control points crowd), and font rendering measures curve lengths to place dashes evenly. The cycloid's starring role is historical and physical at once, it's the brachistochrone, the fastest-descent ramp between two points, and the curve of Huygens' isochronous pendulum clock; its tidy length 8 was one of the first triumphs of the very integral this page computes.
Frequently Asked Questions
Why is the integrand called the speed?
(x'(t), y'(t)) is the velocity vector, instantaneous direction and rate of the moving point. Its magnitude is speed, and distance is speed integrated over time. The formula is kinematics stated as geometry.
Does a different parameterization change the answer?
No, traversing the same path faster shrinks the time window exactly as it grows the speed, and the integral is invariant. This parameterization-independence is what makes "the length of the curve" a well-defined geometric quantity.
How does this extend to 3D curves?
Add a term: L = ∫√(x'² + y'² + z'²)dt. A helix (cos t, sin t, ct), a coiled spring or DNA strand, has constant speed √(1 + c²), making spring wire length a one-line computation. The pattern continues into any dimension.
Why does the ellipse have no perimeter formula when the circle does?
The circle's constant speed makes its integral trivial; the ellipse's speed √(9sin²t + 4cos²t) varies, and its antiderivative provably escapes elementary functions. The "elliptic integrals" born from this problem became a rich field of their own, and a standard example that most arc lengths are numeric affairs.
What's the connection to dy/dx for parametric curves?
Same derivatives, different combination: slope is the ratio y'/x' while length integrates the magnitude √(x'² + y'²). The Parametric Derivative Calculator handles the ratio, together the two tools read a trajectory's direction and its distance from one pair of derivatives.
What is arc length parameterization and why do graphics people want it?
It's re-describing the curve so the parameter is distance along it, s instead of t, making unit parameter steps advance unit distances. Animations then move at visually constant speed, dashes space evenly, and speedometers read true. Computing it requires inverting s(t) = ∫√(x'²+y'²)dt, usually numerically, this tool's integral is the s(t) being inverted.
How is the chord length related to the arc length?
The straight segment between the endpoints is always the shorter route: chord ≤ arc, with equality only for straight paths. Computing the chord √((x(b)−x(a))² + (y(b)−y(a))²) gives an instant lower-bound sanity check on any arc length result, a habit that catches setup errors quickly.