Level Curve or Surface Plotter

Math Calculus • Multivariable Calculus

Written by STEM Calculators Team Published December 30, 2025 Updated February 24, 2026

7. Level Curve Or Surface Plotter

Plot level curves \(f(x,y)=c\) or trace curves of level surfaces \(f(x,y,z)=c\) on a chosen plane. Use the slider to vary \(c\).

Mode: Function f: Slice plane (3D):

Slice value (3D)

z₀:

In 3D mode, we plot the trace curve where the level surface intersects the chosen plane through this value.

Level value

c (slider):

c = 1

c (exact): c min: c max: Extra levels (optional):

Input supports pi, e, sqrt(2), trig (sin, cos), etc. Drag to pan, mouse wheel to zoom, double-click to reset. Hover to probe \(f\) and \(f-c\).

Evaluation point (for gradient / normal)

x₀: y₀:

The arrow drawn at the point shows a normal direction (gradient) to the level curve (2D), or the in-plane normal to the trace curve (3D slice).

Plot window (displayed axes)

x min: x max: y min: y max:

Graph settings (optional)

Resolution: Show grid: Fill by f: Show gradient normal: Slider animation:

Ready

Enter \(f\) and click “Plot”.

Graph

Contours with axes + tick values

Axes units are in your input units. Main level \(c\) is highlighted; optional extra levels are also drawn.

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Theory

Level curves (2D)

For a scalar field \(f(x,y)\), a level curve at value \(c\) is the set of points satisfying

\[ f(x,y)=c. \]

Changing \(c\) produces a family of curves. The slider in the calculator lets you sweep through these values.

Level surfaces (3D) and trace curves

For a scalar field \(f(x,y,z)\), a level surface is defined by

\[ f(x,y,z)=c. \]

Visualizing an entire implicit surface is harder on a 2D screen, so the calculator shows trace curves: intersections of the level surface with a plane (e.g., \(z=z_0\), \(y=y_0\), or \(x=x_0\)).

Example (plane \(z=z_0\)):

\[ f(x,y,z_0)=c \quad\text{(a 2D level curve in the \(x\!-\!y\) plane).} \]

Gradient and normals

The gradient points in the direction of steepest increase and is normal to level sets:

\[ \nabla f=\left\langle \frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right\rangle \quad\text{(2D)},\qquad \nabla f=\left\langle \frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right\rangle \quad\text{(3D)}. \]

If a point lies on the level curve \(f(x,y)=c\), then \(\nabla f\) at that point is perpendicular to the curve. In 3D, \(\nabla f\) is perpendicular to the level surface, and its projection onto a slice plane gives an in-plane normal to the trace curve.

Worked example: \(f(x,y)=x^2+y^2\)

Level curves satisfy \(x^2+y^2=c\). For \(c>0\), these are circles of radius \(\sqrt{c}\). The gradient is

\[ \nabla f=\langle 2x,2y\rangle, \]

which points radially outward, perpendicular to each circle.

Frequently Asked Questions

What is a level curve of a function f(x,y)?

A level curve is the set of points (x,y) where the function has a constant value c, written as f(x,y)=c. Changing c produces a family of contour curves on the same axes.

How is a 3D level surface shown on a 2D graph in this tool?

A full level surface f(x,y,z)=c is visualized as a trace curve by intersecting the surface with a plane such as z=z0, y=y0, or x=x0. The tool plots that intersection curve in the chosen slice plane.

Why is the gradient perpendicular to a level curve?

Along a level curve f(x,y)=c, the function value does not change, so the direction of greatest increase must be orthogonal to the curve. The gradient vector points in the direction of steepest increase and is normal to level sets.

What does the probe value f-c mean when hovering on the graph?

The quantity f-c measures how far the point is from satisfying the level-set equation f=c. Points on the plotted level curve have f-c=0.