Parametric Equation Calculator
Evaluate parametric curves at chosen parameter. Enter x(t) and y(t), pick t, then compute coordinates and slope instantly with details.
Equation Preview
Helping Notes
Only three inputs are required: x(t), y(t), and a numeric t. Use ^ for powers, exp(t) for \(e^t\), ln() for natural log.
If \(dx/dt=0\) at your t, the slope \(dy/dx\) is undefined (vertical tangent). Try a different t if needed.
Results
x(t)
y(t)
Slope dy/dx
What Is a Parametric Equation Calculator?
A Parametric Equation Calculator evaluates curves defined by functions of a parameter, typically in 2D or in 3D. Instead of expressing y directly as a function of x, parametric form lets you model loops, cusps, and paths that fail the vertical line test. The calculator parses your components, computes derivatives and slopes, finds tangent vectors and speed, and can integrate for arc length and enclosed area when appropriate. It also supports eliminating the parameter to recover an implicit Cartesian relation when possible, giving you two consistent representations of the same curve for checking and further analysis.
About the Parametric Equation Calculator
The engine differentiates each component, then forms composite quantities such as (when ) and the second derivative . Speed is the magnitude of the velocity vector, and arc length integrates speed over a parameter interval. For area enclosed by a positively oriented 2D parametric loop, the tool uses (or an equivalent symmetric form). In 3D, it provides the tangent vector and optional curvature via cross‑products. Results appear symbolically when feasible and numerically otherwise, with step‑by‑step intermediate expressions so you can follow the logic.
Velocity & speed: ,
Slope (2D): Second derivative:
Arc length: (2D), or (3D)
Area (2D loop):
Curvature (2D):
How to Use This Parametric Equation Calculator
- Enter , (and optionally ) and set the parameter interval .
- Choose outputs: slope, tangent/speed, arc length, area, curvature, or eliminate the parameter for a Cartesian relation.
- Compute. The tool differentiates, simplifies, and integrates where needed; symbolic results appear with numeric evaluations.
- Inspect domain warnings (e.g., where makes undefined) and adjust intervals if required.
Examples
- Circle: ⇒ , , .
- Polynomial path: ⇒ , .
- Projectile model: ⇒ slope .
- 3D helix: ⇒ , .
Formula Snippets Ready for Rendering
FAQs
What are parametric equations?
They define coordinates as functions of a parameter, enabling curves that are difficult or impossible to express as y = f(x).
How do I find dy/dx from parametric form?
Use dy/dx = (dy/dt)/(dx/dt) where x′(t) ≠ 0. If x′ = 0, the slope is undefined or vertical at that parameter.
Can the calculator handle 3D curves?
Yes. Provide x(t), y(t), z(t). It returns velocity, speed, and optional curvature diagnostics.
How is arc length computed?
Integrate the speed: L = ∫√((x′)² + (y′)²) dt in 2D, or ∫||r′(t)|| dt in 3D over your interval.
How do I compute the area enclosed by a parametric loop?
Use A = ∫ y(t) x′(t) dt with positive orientation; reversing orientation flips the sign of area.
What if the curve self‑intersects?
Area integrals may cancel. Split the interval around crossings or use absolute area if that fits your goal.
Why is my slope undefined at some t?
Because x′(t) = 0 there. The curve may have a vertical tangent or cusp; check nearby parameter values.
Can I eliminate the parameter?
Often. Solve t from one component and substitute into the other, or use algebraic elimination where feasible.
Does the tool support radians and degrees?
Angles inside trig functions are interpreted consistently; convert to radians if needed for calculus operations.