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How to use the 3D graphing calculator
- Type an equation on a line of the expression list. Start with
z = x^2 - y^2. Each line gets its own colour. - Rotate by dragging, zoom with the scroll wheel or a pinch, pan with a right-drag. Hover to read the (x, y, z) under the pointer. The buttons 3D, Top, Front, Side and Rotate move the camera.
- Press View for axis ranges, contour lines, grid lines, colour by height, opacity and detail level.
- Use Share to copy a link that contains the whole graph, or download a PNG.
This runs in the browser on a phone as well as a laptop. There is no app to install and no sign-in. It draws with WebGL, so any current mobile browser works.
What you can type
| Form | Example | Notes |
|---|---|---|
| Explicit surface | z = sin(x) cos(y) | Also y = f(x,z) and x = f(y,z). Planes work too: z = 3 - x - y, x = 2 |
| Implicit surface | x^2 + y^2 + z^2 = 9 | Any equation in x, y and z |
| Space curve | (cos(t), sin(t), t/4) | t runs 0 to 2π. Change it with {0 < t < 6π} |
| Parametric surface | (u cos(v), u sin(v), u) | u runs 0 to 2π, v runs 0 to π. Override both: {0 < u < 3} {0 < v < 2π} |
| Cylindrical | r = z | r = f(θ, z). Type theta or the symbol |
| Spherical | rho = 2cos(phi) | ρ = f(θ, φ). Type rho, theta, phi |
| Point | (1, 2, 3) | Drawn with a label |
| Vector field | <-y, x, z> | Arrows coloured by magnitude |
| Slider | a = 1 | Then use a in other lines |
Setting the box
Everything is drawn inside a box that starts at -5 to 5 on all three axes. Surfaces are cut cleanly at its walls, so a surface that reaches z = 25 shows a flat top at z = 5. Press View and change the x, y and z minimum and maximum to fit the thing you are studying. Keep the three ranges similar when shape matters, or a sphere turns into an egg.
The math behind it: level curves, critical points and coordinates
Level curves. A contour (level curve) of z = f(x, y) is the set of points where f equals a constant c, that is f(x, y) = c. Tick Contour lines under View and the grapher draws them on the surface and flattens a copy onto the floor of the box. Lines packed close together mean a steep slope; widely spaced lines mean a gentle one.
| Contour pattern | What the surface is doing | Example |
|---|---|---|
| Nested closed loops | A bowl or a hill | z = x^2 + y^2: circles of radius √c, so c = 4 gives radius 2 |
| Hyperbolas that swap direction at c = 0 | A saddle | z = x^2 - y^2 |
Critical points. A critical point is where the gradient ∇f = (fx, fy) is zero. The second-derivative test uses D = fxx·fyy - fxy². If D < 0 it is a saddle. If D > 0 and fxx > 0 it is a local minimum; if D > 0 and fxx < 0 it is a local maximum. If D = 0 the test says nothing, so look at the graph.
Coordinate systems. Cylindrical coordinates use (r, θ, z) with x = r cos θ and y = r sin θ. Spherical coordinates use (ρ, θ, φ) with x = ρ sin φ cos θ, y = ρ sin φ sin θ and z = ρ cos φ. Here θ is the angle around the z-axis (0 to 2π) and φ is the angle down from the positive z-axis (0 to π). Some textbooks swap those two letters, so check your course's convention before you copy a formula.
Three worked examples
1. The saddle z = x² - y² and its contour lines
Set the box to x and y from -3 to 3 and z from -9 to 9. Then f(3, 0) = 9, f(0, 3) = -9 and f(0, 0) = 0.
- Gradient: ∇f = (2x, -2y), which is zero only at the origin.
- Second derivatives: fxx = 2, fyy = -2, fxy = 0, so D = (2)(-2) - 0 = -4. Negative D means a saddle.
With contour lines on, the lowest and highest heights in this box are -9 and 9. The lines fall at every 1.5 from -7.5 to 7.5. The z = 0 line is x² = y², the two straight lines y = x and y = -x crossing at the origin. At z = 4 the contour is x² - y² = 4, a hyperbola with its vertex at (2, 0). At y = 1 it sits at x = √5 ≈ 2.236. At z = -4 the hyperbola opens along y instead: at x = 1 it sits at y = √5.
Hover at (2, 1) and the readout should say z = 3. The gradient there is (4, -2), and the contour through that point has tangent direction (1, 2). Their dot product is 4 - 4 = 0: the gradient is perpendicular to the level curve.
2. Where a sphere meets a plane
Take the sphere x² + y² + z² = 25 (radius 5) and the plane z = 3. On the plane, x² + y² = 25 - 9 = 16, so the intersection is a circle of radius √16 = 4 centred at (0, 0, 3). In general, a plane at distance d from the centre cuts a sphere of radius R in a circle of radius √(R² - d²).
The third line in the loaded example, (4cos(t), 4sin(t), 3), draws that circle as a space curve. Check a point: at t = 1 it is (2.161, 3.366, 3), and 2.161² + 3.366² + 3² = 25. Every point on the curve passes the same test.
Tilt the plane to z = 3 - x - y, which is x + y + z = 3. Its distance from the origin is 3/√3 = √3 ≈ 1.732, so the circle has radius √(25 - 3) = √22 ≈ 4.690.
3. A helix on a cylinder
The curve (2cos t, 2sin t, t/4) for t from 0 to 6π winds three times around a cylinder of radius 2. Every point satisfies x² + y² = 4 cos²t + 4 sin²t = 4, so it stays on that cylinder. The cylinder itself is just r = 2 in cylindrical form.
- Pitch (the rise per full turn): z grows by 2π/4 = π/2 ≈ 1.571 each turn, so it reaches 6π/4 ≈ 4.712 after three.
- Steepness: tan α = 1.571 / (2π × 2) = 0.125, so the helix climbs at about 7.13° from horizontal.
Reading the graph and common mistakes
- Cone in cylindrical form.
r = z {0 < z < 4}is a cone. Its cross-section at height z is a circle of radius z, which is the same as x² + y² = z². Without the brace the grapher uses the whole z range of the box, and negative z draws the mirror half to give a double cone. - A sphere from spherical coordinates.
rho = 2cos(phi)is not centred on the origin. Multiply by ρ: ρ² = 2ρ cos φ, so x² + y² + z² = 2z, which is x² + y² + (z - 1)² = 1. That is a sphere of radius 1 centred at (0, 0, 1). - Vector fields.
<-y, x, z>spins around the z-axis while drifting upward. The arrows sit on a 7 by 7 by 7 lattice inside the box, longer and brighter where the field is stronger. At (1, 2, 3) the magnitude is √(4 + 1 + 9) = √14 ≈ 3.742. The horizontal part is perpendicular to the radius, since (-y, x)·(x, y) = 0. - Half a surface. A parametric surface runs v from 0 to π unless you say otherwise. To revolve a profile fully, add
{0 < v < 2π}. - Radians. RAD is selected by default, so
sin(90)is not 1. Switch the RAD | DEG control to DEG to work in degrees, or read the radians versus degrees guide.
Frequently asked questions
Is the 3D graphing calculator free, and does it work on a phone?
Yes to both. It runs in your phone or laptop browser with no app and no sign-in. Share copies a link to your graph, and you can download a PNG.
How do I graph in cylindrical or spherical coordinates?
Type r = f(theta, z) for cylindrical, for example r = z, or rho = f(theta, phi) for spherical, for example rho = 3. You can type the words or the symbols. Add braces such as {0 < z < 4} to limit a range.
How do I plot parametric curves and surfaces in 3D?
Write three expressions in parentheses. With t it is a curve: (cos(t), sin(t), t/4). With u and v it is a surface: (u cos(v), u sin(v), u). To revolve y = x² about the x-axis, try (u, u^2 cos(v), u^2 sin(v)) {0 < u < 2} {0 < v < 2π}.
Can it draw vector fields?
Yes. Type <P, Q, R> using x, y and z, such as <-y, x, z>. Arrows are coloured and sized by magnitude.
How do I see contour lines for z = f(x, y)?
Press View and tick Contour lines. They appear on the surface and as a projection on the floor of the box. The levels are evenly spaced between the lowest and highest height inside your box, so changing the z range changes them.