Contour Plot Generator

Visualize 3D functions with contour plots and level curves. Explore mathematical surfaces with interactive controls.

Contour Plot Definition: A contour plot shows level curves of a function f(x,y) where f(x,y) = constant. Each curve represents points with equal function value.

Paraboloid
x² + y²
Saddle
x² - y²
Gaussian
e⁻⁽ˣ²⁺ʸ²⁾/²
Waves
sin(x)·cos(y)
Monkey Saddle
x³ - 3xy²
Ripple
sin(√(x²+y²))
f(x,y) =
Enter a function of x and y. Use standard math notation: ^ for exponent, * for multiplication, sin(), cos(), exp(), sqrt(), etc.
x² + y²
sin(x) + cos(y)
x² - y²
e⁻⁽ˣ²⁺ʸ²⁾
x·y
sin(x)·cos(y)
√(x² + y²)
1/(1+x²+y²)
x³ - 3xy²
cos(x)·sin(y)
e⁻|xy|
sin(√(x²+y²))
Range of x-values for the plot
Range of y-values for the plot
5 15 levels 30
More levels show finer detail but take longer to compute
Higher resolution gives smoother plots but slower computation
Initializing... 0%
Generating contour plot...

Understanding Contour Plots

A contour plot (also called a level plot) is a graphical technique for representing a 3-dimensional surface on a 2-dimensional plane. It plots contour lines (also called isolines) of a function f(x,y), where each line corresponds to constant values of the function.

Mathematical Definition:

For a function f(x,y), a contour line (level curve) at level c is the set of points (x,y) satisfying:

f(x, y) = c

where c is a constant. Different values of c produce different contour lines.

Interpretation of Contour Plots

1

Steepness: Closely spaced contour lines indicate a steep slope, while widely spaced lines indicate a gentle slope.

2

Peaks and Valleys: Closed contour lines typically represent peaks (if values increase toward the center) or valleys/depressions (if values decrease toward the center).

3

Ridges and Troughs: Elongated contour patterns with higher values on both sides indicate ridges, while patterns with lower values on both sides indicate troughs.

Common Functions and Their Contour Patterns

Function Contour Pattern Description
f(x,y) = x² + y² Concentric circles Paraboloid (circular symmetry)
f(x,y) = x² - y² Hyperbolas Saddle surface
f(x,y) = sin(x) + cos(y) Wave pattern Periodic surface
f(x,y) = exp(-(x²+y²)) Concentric circles Gaussian bell curve
f(x,y) = xy Rectangular hyperbolas Saddle oriented at 45°
f(x,y) = x³ - 3xy² Complex symmetric pattern Monkey saddle (3-fold symmetry)

Applications of Contour Plots

Topography: Contour maps show elevation with contour lines representing equal height above sea level.

Meteorology: Weather maps use isobars (pressure contours), isotherms (temperature contours), and isohyets (precipitation contours).

Engineering: Stress and strain distributions, temperature gradients, and fluid flow patterns are often visualized with contour plots.

Statistics: Contour plots visualize probability density functions and likelihood surfaces in statistical modeling.

How the Calculator Works

1

Function Evaluation: The calculator evaluates your function on a grid of (x,y) points within the specified range using the nerdamer.js symbolic math library.

2

Contour Detection: The algorithm finds level curves by detecting where the function crosses constant values and connecting these points into smooth curves.

3

Visualization: The plotly.js library renders the contour plot with appropriate coloring, labels, and interactive features.

Calculator Features:

  • Supports a wide variety of mathematical functions and expressions
  • Interactive 3D surface view in addition to contour plot
  • Adjustable color schemes and visualization parameters
  • Computes and displays key properties of the surface
  • Allows downloading generated plots as images

Frequently Asked Questions

A contour plot shows explicit contour lines connecting points of equal value, while a heatmap uses color gradients to represent values without explicit lines. Contour plots are better for identifying specific level values, while heatmaps provide a more immediate visual impression of value distribution.

Contour lines can be disconnected when the level set f(x,y)=c consists of multiple separate components. This often happens with functions that have multiple peaks or valleys at the same height, or with functions that have discontinuities or singularities.

The spacing between contour lines indicates the gradient magnitude. Closely spaced contours mean the function is changing rapidly (steep slope), while widely spaced contours indicate gradual change. Equally spaced contours indicate a constant gradient, while varying spacing indicates changing steepness.

The calculator handles many functions with discontinuities, but extreme singularities (like division by zero) may cause issues. For functions with singularities, it's best to choose ranges that avoid the problematic regions, or use the resolution control to reduce computation near singularities.

Color schemes map function values to colors. Sequential schemes (like Viridis and Plasma) are good for showing ordered data from low to high. Diverging schemes (like Cool) are useful when there's a meaningful middle value. Rainbow schemes show clear distinctions but can create artificial boundaries. Choose based on what aspect of your data you want to emphasize.