Level Curve Calculator

Calculator Inputs

Choose a preset or enter a custom surface in terms of x and y.

Supports preset and custom functions

Function mode

Preset surface

Custom expression f(x, y)

Allowed functions: sin, cos, tan, sqrt, log, exp, abs, pow.

Point x₀

Point y₀

Target level k

Membership tolerance

Nearby table step

Window sample grid

Window x min

Window x max

Window y min

Window y max

Example Data Table

These examples show typical level values and curve interpretations for common surfaces.

Surface	Equation	Level k	Curve Type	Interpretation
Circular paraboloid	x² + y²	9	Circle	Radius equals 3.
Hyperbolic paraboloid	x² - y²	4	Hyperbola	Opens along the x-direction.
Plane	2x + 3y	12	Line	Every level set is linear.
Gaussian surface	exp(-(x² + y²))	0.5	Circle	Radius is √(-ln 0.5).

Formula Used

A level curve satisfies f(x, y) = k, where k is a constant level.
The gradient is ∇f = (∂f/∂x, ∂f/∂y). This vector is normal to the level curve.
The tangent slope near a regular point is dy/dx = -(fₓ / fᵧ), provided fᵧ ≠ 0.
For a local approximation, the calculator linearizes the curve as fₓ(x₀,y₀)(x-x₀)+fᵧ(x₀,y₀)(y-y₀)=k-f(x₀,y₀).
Partial derivatives are estimated numerically with a central-difference method, which is accurate for smooth surfaces near the point.
The window scan samples many points between the selected limits and estimates whether the chosen level likely appears there.

How to Use This Calculator

Select a preset surface or choose custom mode for your own function.
Enter the point (x₀, y₀) where you want to inspect the surface.
Type the target level k that defines the contour set f(x, y) = k.
Choose a tolerance. This checks whether the point approximately lies on the selected level curve.
Set a nearby table step and a window range for broader contour sampling.
Press Calculate Level Curve to show results immediately above the form.
Use the export buttons to download a CSV summary or a compact PDF report.

Frequently Asked Questions

1. What does a level curve represent?

A level curve is the set of all points where a two-variable function has the same value. It is also called a contour line.

2. Can I enter my own equation?

Yes. Choose custom mode and enter a function using x and y. Standard operations and common mathematical functions are supported.

3. Why does the tangent slope become undefined?

If the partial derivative with respect to y is near zero, the contour can have a vertical tangent locally. In that case, dy/dx is undefined.

4. What is the role of the gradient?

The gradient points in the direction of steepest increase. It is perpendicular to the level curve at regular points.

5. How is point membership tested?

The calculator computes f(x₀, y₀) and compares it with k. If the absolute difference is within your tolerance, the point is treated as lying on the curve.

6. What does the window scan tell me?

It samples the selected x and y range, estimates the function range there, and checks whether your chosen level likely occurs inside that window.

7. Are the derivatives exact?

For custom expressions and preset functions, derivatives are estimated numerically. They are usually very accurate near smooth points, but small numerical error is possible.

8. When should I export CSV or PDF?

Use CSV for spreadsheets and repeated analysis. Use PDF when you want a neat report for class notes, homework, or printed records.