Graph library

3D graph examples

Every example opens a live surface you can rotate, cut, and export. The formula is on the card, so you can check the shape against the equation before you plot anything.

Explicit surfaces: z = f(x, y)

One height for every point on the plane. Each page carries the formula, a teaching note, the live calculator, and a PNG export.

Saddle Surface

z = x^2 - y^2

A saddle surface curves up in one direction and down in the perpendicular direction.

Gaussian Surface

z = exp(-(x^2 + y^2))

A smooth bell-shaped surface centered at the origin.

Elliptic Paraboloid

z = x^2 + y^2

A bowl-shaped surface that opens upward.

Inverted Paraboloid

z = 12 - x^2 - y^2

A dome-shaped surface with a highest point at the center.

Monkey Saddle

z = x^3 - 3*x*y^2

A three-way saddle with three valleys and three ridges.

Sine Wave Surface

z = sin(x) + sin(y)

Two perpendicular sine waves combined into one surface.

Ripple Surface

z = sin(sqrt(x^2 + y^2))

Circular waves radiate outward from the center.

Damped Ripple

z = sin(x^2 + y^2) / (1 + x^2 + y^2)

A ripple surface whose waves fade away from the origin.

Cone Surface

z = sqrt(x^2 + y^2)

A circular cone rising from the origin.

Plane

z = 2*x - y + 1

A flat plane with constant slope.

Tilted Wave Plane

z = 0.35*x + sin(y)

A sloped plane with a wave running across it.

Sine Cosine Product

z = sin(x) * cos(y)

A checkerboard-like wave surface.

Absolute Value Pyramid

z = abs(x) + abs(y)

A four-sided pyramid made from absolute values.

Cubic Surface

z = x^3 - y

A cubic sheet tilted by the y term.

Log Bowl

z = log(1 + x^2 + y^2)

A bowl that rises slowly as it moves outward.

Square Root Dome

z = sqrt(max(0, 16 - x^2 - y^2))

A dome of height 4 inside radius 4, sitting on a flat base where the formula clamps to zero.

Hyperbolic Valley

z = x*y

A saddle-like surface formed by multiplying x and y.

Cosine Hills

z = cos(x) + cos(y)

A regular field of smooth hills and basins.

Egg Crate Surface

z = sin(x) * sin(y)

A repeating grid of peaks and pits.

Radial Decay

z = 1 / (1 + x^2 + y^2)

A smooth mound that decays toward zero.

Inverse Square

z = min(4, 1 / (x^2 + y^2))

A radial surface that blows up at the origin and levels off at a flat top of height 4.

Mexican Hat Surface

z = (1 - x^2 - y^2) * exp(-(x^2 + y^2) / 2)

A central peak surrounded by a circular valley.

Ridge Surface

z = exp(-x^2)

A ridge that is constant in the y direction.

Valley Surface

z = y^2 - 2*x

A parabolic valley tilted along the x-axis.

Spiral Height Surface

z = atan(y / (x + 0.001))

A wrapped height field based on angle.

Checkerboard Surface

z = floor(abs(sin(x) * sin(y)) * 4) / 4

A stepped checkerboard made from quantized waves.

Fourier Demo Surface

z = sin(x) + sin(2*y)/2 + sin(3*x)/3

A layered wave made from several sine terms.

Distance Field

z = min(sqrt((x - 2)^2 + y^2), sqrt((x + 2)^2 + y^2))

A surface showing distance to the nearer of two points.

Elliptic Gaussian

z = exp(-(x^2 / 8 + y^2 / 2))

A stretched bell surface with different widths by axis.

Anisotropic Paraboloid

z = x^2 / 4 + y^2

A bowl that curves more sharply along y than x.

Two Peak Surface

z = exp(-((x - 2)^2 + y^2)) + exp(-((x + 2)^2 + y^2))

Two smooth hills separated along the x-axis.

Basin and Ridge

z = x^2 - cos(3*y)

A parabolic basin crossed by repeated ridges.

Exponential Slope

z = exp(0.25*x) * cos(y)

A wave whose amplitude grows along x.

Ring Valley

z = (sqrt(x^2 + y^2) - 3)^2

A circular valley with a raised center and outside wall.

Smooth Cliff Surface

z = tanh(x) + 0.2*y

A smooth step blended with a gentle tilt.

Parametric surfaces

Shapes described with two parameters u and v. Each page gives the full equations and states which related surface the live preview draws.

Torus Equations

z = sin(sqrt(x^2 + y^2))

Two angles trace a torus: u goes around the main ring, v goes around the tube, and R and r set the two radii.

Möbius Equations

z = atan(y / (x + 0.001))

The half-twist factor (1 + v cos(u/2)) is what turns a rectangle into a strip with one side and one edge.

Helicoid Equations

z = atan(y / (x + 0.001)) + 0.15*sqrt(x^2 + y^2)

A helicoid is a straight line that rotates and climbs: x = u cos v, y = u sin v, z = v.

Wave Sheet

z = sin(x) * cos(y)

A parametric sheet whose height is a wave product.

Ripple Disk

z = sin(sqrt(x^2 + y^2))

A circular ripple described naturally with radius and angle.

Spherical Cap

z = sqrt(max(0, 16 - x^2 - y^2))

A cap from a sphere parameterized by angles.

Spiral Ramp

z = atan(y / (x + 0.001)) + 0.2*sqrt(x^2 + y^2)

A ramp that rises as it rotates.

Saddle Patch

z = x^2 - y^2

A saddle written as a two-parameter patch.

Parametric Paraboloid

z = x^2 + y^2

A paraboloid written with radius and angle.

Cylinder Wave Equations

z = sin(sqrt(x^2 + y^2)) + 0.2*y

One angle wraps around a tube while a second parameter runs along it: x = cos u, y = sin u, z = v + 0.2 sin 4u.

Not sure which one to open?

The guides explain what to look for once a surface is on screen.

Read the guides