3D Graphing Calculator NZ

How to use the 3D graphing calculator

Type a function of x and y into the equation field on the left. Use x and y as your two input variables; the calculator will plot the resulting z height as a coloured 3D surface. Click Plot / Update to draw the surface. You can enter up to three surfaces simultaneously and they will be overlaid in different colours.

Use the Domain fields to set the x and y range shown. Narrow the range to zoom in on a specific region, or widen it to see the global shape of the function. Higher resolution gives a smoother surface but takes slightly longer to compute.

To navigate the 3D view: click and drag to rotate, scroll to zoom, and right-click and drag to pan. On a touchscreen, use one finger to rotate and two fingers to zoom. Press Reset View at any time to return to the default camera angle.

What you can type

Use x and y as your variables. All the same functions as the 2D calculator are supported: sin, cos, tan, sqrt, abs, log, ln, exp, and constants pi and e. A common shorthand is to define the radial distance as r = sqrt(x^2 + y^2) and then write functions of r, such as sin(sqrt(x^2+y^2)) for a ripple pattern.

Classic surfaces to explore

Paraboloid: z = x² + y²

The paraboloid is the 3D version of a parabola. It looks like a bowl. Every horizontal cross-section (a slice at constant z) is a circle, and every vertical cross-section is a parabola. This surface appears naturally in satellite dish shapes, the mirror of a car headlight, and the equations of gravitational potential wells. At NCEA Level 3 and university, you learn that the minimum of this surface is at the origin, where the partial derivatives with respect to x and y are both zero.

Saddle surface: z = x² - y²

The saddle surface curves upward in one direction and downward in another, like the seat of a horse's saddle. The origin is a saddle point: it is a local minimum along one axis and a local maximum along the other. Saddle points are an important concept in multivariable calculus and optimisation because the gradient is zero there but it is neither a maximum nor a minimum.

Ripple: z = sin(r) / r

This function, also called the 3D sinc function, produces concentric rings of oscillating height that diminish in amplitude as you move away from the centre. It is related to the diffraction pattern of light passing through a circular aperture and appears in signal processing, optics, and Fourier analysis. The preset button will load it instantly.

Wave: z = sin(x) cos(y)

This product of two trigonometric functions produces a checkerboard wave pattern. All the saddle points, peaks, and troughs form a regular grid. This surface is used to teach the concept of critical points in two-variable calculus, and classifying them using the second derivative test.

The colour scale

The colour of each point on the surface represents its z height. Low values (valleys) are shown in blue, progressing through green and yellow to red for high values (peaks). The colour bar on the right edge of the graph shows this scale. You can change the colour scheme in the Display settings on the left panel.

Why is 3D graphing useful for NZ maths students?

Three-dimensional graphing becomes relevant in New Zealand's NCEA Level 3 calculus curriculum and in first and second year university mathematics. At Level 3, students work with functions of two variables when studying optimisation problems and rates of change in multiple directions. At university, papers covering multivariable calculus, vector calculus, and differential equations require students to visualise surfaces, gradient vectors, level curves, and tangent planes. Being able to rotate and explore a surface interactively builds intuition that is difficult to get from textbook diagrams. This tool is a free alternative to Wolfram Alpha, MATLAB, and GeoGebra's 3D graphing tool.