Enter parametric inputs
Use t as the parameter. Supported functions include sin, cos, tan, sqrt, log, exp, abs, min, max, and pi.
Example data table
| x(t) | y(t) | z(t) | t value | Start | End | Step | Use case |
|---|---|---|---|---|---|---|---|
| cos(t) | sin(t) | — | 1.5708 | 0 | 6.2832 | 0.5236 | Unit circle |
| 2*cos(t) | 3*sin(t) | — | 0.7854 | 0 | 6.2832 | 0.3927 | Ellipse |
| t*cos(t) | t*sin(t) | — | 2 | 0 | 10 | 0.5 | Spiral |
| cos(t) | sin(t) | t/4 | 3.1416 | 0 | 12.5664 | 0.6283 | Helix |
Formula used
Position vector: r(t) = <x(t), y(t), z(t)> for 3D, or r(t) = <x(t), y(t)> for 2D.
First derivatives: x′(t), y′(t), and optional z′(t) are estimated numerically using the central difference rule.
Central difference: f′(t) ≈ [f(t+h) − f(t−h)] / (2h).
Second derivative: f″(t) ≈ [f(t+h) − 2f(t) + f(t−h)] / h².
Slope in the plane: dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.
Speed: |r′(t)| = √[(dx/dt)² + (dy/dt)² + (dz/dt)²].
2D curvature: κ = |x′y″ − y′x″| / (x′² + y′²)^(3/2).
3D curvature: κ = |r′(t) × r″(t)| / |r′(t)|³.
Arc length approximation: the calculator sums distances between sampled points from the chosen start to end values of t.
How to use this calculator
- Enter formulas for
x(t)andy(t). Addz(t)only when you need a 3D curve. - Choose the parameter value where you want point coordinates, slope, speed, and curvature.
- Set a start value, end value, and sampling step for the interval table and arc length estimate.
- Set the derivative step. Smaller values often improve precision, but very tiny values can increase numerical noise.
- Click Calculate to display the result summary above the form.
- Use Download CSV for tabular results and Download PDF to save the printed output as a PDF.
FAQs
1. What does this calculator solve?
It evaluates parametric equations at a chosen parameter value, then estimates derivatives, slope, speed, curvature, arc length, displacement, and a sample table across an interval.
2. Which functions can I enter?
You can use t, numbers, parentheses, arithmetic operators, and common functions such as sin, cos, tan, sqrt, exp, log, abs, min, max, and pi.
3. Why is dy/dx sometimes undefined?
The planar slope is calculated as (dy/dt)/(dx/dt). When dx/dt is zero or extremely close to zero, the tangent is vertical, so dy/dx is undefined.
4. How is arc length estimated?
The tool samples points from the chosen start to end values of t, then adds the distance between consecutive points. Smaller sampling steps usually improve the estimate.
5. Can I use the calculator for 3D curves?
Yes. Enter z(t) to activate 3D calculations. The preview still plots x(t) against y(t), while speed, displacement, and curvature include the z component.
6. What is the derivative step for?
It controls the central difference approximation used for first and second derivatives. A sensible small value improves accuracy without making rounding error too dominant.
7. Why do some expressions fail to evaluate?
Expressions can fail when they contain unsupported characters, invalid syntax, division by zero, or values that produce non-finite outputs such as infinity or not-a-number.
8. What does the preview chart show?
The chart traces the path formed by x(t) and y(t) over the selected interval. It marks the first sample in green and the last sample in red.