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3D Graphing Calculator
Type z = f(x, y) to draw a 3D surface, or x(t), y(t), z(t) for a space curve. Drag to rotate, scroll to zoom, switch on contour lines and a contour map, plot several functions together and download the view as a PNG image.
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Examples:
Functions
Plot range & display
Your browser has WebGL turned off or unavailable, so the 3D view cannot be drawn. Try another browser or enable hardware acceleration.
Drag to rotate · scroll or pinch to zoom · right-drag or Shift-drag to pan
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How to use the 3D Graphing Calculator
- Type a surface such as sin(x)*cos(y), x^2 - y^2 or e^(-(x^2+y^2)) in the z = box, or tap an example.
- Add more with + Surface or + Parametric curve (enter x(t), y(t), z(t) and the t range).
- Set the plot range for x and y. Leave z min and z max empty for automatic scaling.
- Rotate and zoom: drag the graph, scroll or pinch to zoom, Shift-drag to pan, or use the Top, Front and Side buttons.
- Turn on contours, mesh lines or color by height, then press Download PNG to save the picture.
Surfaces worth trying
| Name | z = f(x, y) | Suggested range | What to look for |
|---|---|---|---|
| Paraboloid | x^2 + y^2 | −3 to 3 | Circular contours, minimum at the origin |
| Saddle (hyperbolic paraboloid) | x^2 - y^2 | −3 to 3 | Up in one direction, down in the other; contours are hyperbolas |
| Egg crate | sin(x)*cos(y) | −5 to 5 | A grid of peaks and pits |
| Ripple | sin(sqrt(x^2 + y^2)) | −10 to 10 | Concentric waves |
| Gaussian bump | e^(-(x^2 + y^2)) | −3 to 3 | Bell shape; its integral over the plane is π |
| Monkey saddle | x^3 - 3x*y^2 | −2 to 2 | Three valleys around a flat critical point |
| Cone | sqrt(x^2 + y^2) | −3 to 3 | Straight sides, a corner at the origin |
| Plane | 2x - y + 1 | −5 to 5 | Evenly spaced straight contours |
Reading a 3D graph
- Height and color: with Color by height on, low values are dark and high values are light, so the top view works as a heat map.
- Contours and the gradient: the gradient ∇f = ⟨∂f/∂x, ∂f/∂y⟩ always points across the contour lines, straight uphill. Where contours bunch together the gradient is large.
- Critical points: peaks and pits are surrounded by closed contour loops; at a saddle, contours cross in an X. Confirm them with the Partial Derivative Calculator by checking where both first partials are zero.
- Traces: the Front view (looking along the y-axis) shows the surface's profile in the x–z plane, and the Side view shows the y–z profile.
Parametric curve ideas
| Curve | x(t) | y(t) | z(t) | t range |
|---|---|---|---|---|
| Helix | cos(t) | sin(t) | t/(2pi) | 0 to 6pi |
| Conical spiral | t*cos(4t) | t*sin(4t) | t | 0 to 2pi |
| Trefoil knot | sin(t) + 2sin(2t) | cos(t) - 2cos(2t) | -sin(3t) | 0 to 2pi |
| Viviani's curve | 1 + cos(t) | sin(t) | 2sin(t/2) | 0 to 4pi |
| Line through a point | 1 + 2t | -1 + t | 3 - t | -2 to 2 |
Frequently asked questions
How do I graph a function of two variables in 3D?
Type the formula for z in terms of x and y, for example z = x^2 + y^2. The calculator evaluates it on a grid over your x and y range and connects the points into a lit surface. Use ^ for powers, sqrt() for roots and functions like sin, cos, ln and e^.
How do I plot a parametric curve in 3D?
Press + Parametric curve and enter x(t), y(t) and z(t) together with the start and end of t. A helix, for example, is x = cos(t), y = sin(t), z = t/(2pi) for t from 0 to 6pi. The curve is drawn as a tube so it stays visible from every angle.
What are contour lines?
A contour line joins all points where the surface has the same height, like the lines on a topographic map. The calculator draws them on the surface and can project them onto the floor as a contour map. Closely spaced contours mean the surface is steep there.
Why is part of my surface missing?
Points where the function is undefined (for example sqrt of a negative number, ln of zero, or division by zero) are left out. Parts that are outside the z range are clipped at the top or bottom of the box. Clear z min and z max to let the calculator choose the range automatically.
Can I plot two surfaces to see where they intersect?
Yes. Add a second surface and both are drawn in the same box with different color maps. Where one passes through the other you can see the intersection curve; the Two surfaces example shows a bowl meeting an upside-down bowl along a circle.
Does it need a fast computer?
No. It uses WebGL, which is built into every modern browser, and the default grid of 70 × 70 cells draws instantly on phones. Choose Fine or Very fine resolution for smoother surfaces on bigger screens. Everything runs locally; nothing is uploaded.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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