Every graph page on this site is an explicit surface: one formula, z equal to an expression in x and y. That form covers a huge amount of calculus, and it has a hard limit. It can only describe shapes where each point above the xy-plane is hit once. A sphere, a torus, a Möbius strip and a helicoid all violate that, and the fix is not a cleverer formula but a different way of describing a surface.
The idea: two parameters, three coordinates
A parametric surface takes a flat rectangle of parameter space, with coordinates u and v, and maps each point in it to a point in 3D. You supply three functions: x(u, v), y(u, v) and z(u, v). Sweeping u and v across their ranges sweeps out the surface, the way a pencil moving along a path draws a curve, except the path is now a two-dimensional patch.
The parameters are not axes. On a torus, u goes around the main ring and v goes around the tube. A point with u = 0, v = 0 is not near the origin in any special sense; it is simply a labelled point on the surface. This is why parametric equations look stranger than z = f(x, y) at first: they describe how to walk on the surface rather than where the surface sits above a grid.
x = (R + r cos v) cos u, y = (R + r cos v) sin u, z = r sin v Reading the torus equations
Take R = 3 and r = 1. The outer radius is R + r = 4 and the inner radius is R - r = 2, so the hole in the middle has radius 2. Fix v: the point (R + r cos v, 0, r sin v) sits on a circle of radius r in a vertical plane, so varying u carries that circle around the z-axis and sweeps the tube. Fix u: varying v traces one cross-section circle of the tube. Neither parameter alone gives a curve you would call x or y.
The surface area follows from the same two radii: 4 * pi^2 * R * r. With R = 3 and r = 1 that is about 118.4 square units. You cannot get that number by integrating a single z = f(x, y) formula over a rectangle, because the torus is not the graph of one function; it would take two, one for the upper half and one for the lower, and the parameterisation gets it in one pass.
Why z = f(x, y) cannot do this
An explicit surface assigns exactly one height to each (x, y). Put a vertical line through the tube of a torus and it crosses the surface twice, once on top of the tube and once underneath; put one through the hole and it crosses nothing. Either way the number of crossings is not one, and the vertical line test, familiar from functions in one variable, is the same obstruction in two: a graph is a surface that no vertical line meets more than once.
You can sometimes work around it by plotting two graphs, upper and lower hemispheres for example, and this calculator supports several surfaces at once for exactly that kind of comparison. What you cannot do is plot the closed shape as one object, which is why parametric equations exist.
Three shapes and what their parameters mean
| Surface | Equations | What each parameter does |
|---|---|---|
| Torus | x = (R + r cos v) cos u, y = (R + r cos v) sin u, z = r sin v | u travels around the ring; v travels around the tube |
| Helicoid | x = u cos v, y = u sin v, z = v | v rotates and lifts; u moves outward from the axis |
| Möbius strip | x = (1 + v cos(u/2)) cos u, y = (1 + v cos(u/2)) sin u, z = v sin(u/2) | u goes around the loop; the cos(u/2) factor twists the strip once |
| Spherical cap | x = R sin u cos v, y = R sin u sin v, z = R cos u | u is polar angle from the top; v is longitude |
The Möbius entry is worth pausing on. The factor cos(u/2) changes sign after u passes 2*pi, so the end of the strip arrives at its own start upside down. That single half-turn is the whole topological oddity: the strip has one side and one edge. No amount of rearranging an explicit formula produces that.
What this calculator actually draws
The live widget here plots explicit surfaces z = f(x, y), with one parameter a available for animation. The torus, Möbius, helicoid and cylinder pages give the full parametric equations and then state, above the widget, which related explicit surface the preview is drawing. The torus page, for example, previews z = sin(sqrt(x^2 + y^2)), a radial wave, and labels it as a preview rather than the torus itself.
That limitation is stated rather than hidden. A widget that silently drew an approximation and called it a torus would teach the wrong thing. The equations on those pages are correct and usable in any parametric plotting tool; the preview exists to show how u and v produce motion, not to claim the shape has been rendered.
Studying parametric surfaces without a 3D parametric plotter
- Write the parameter ranges first. Most of the confusion in parametric problems comes from using 0 to 1 when the equations expect 0 to 2*pi.
- Evaluate at the corners of the parameter rectangle. For the torus at u = v = 0 the point is (R + r, 0, 0); at u = pi it is (-(R + r), 0, 0). Two points already show the ring diameter.
- Fix one parameter and read the resulting curve. Fixing v on the torus gives a circle around the ring; fixing u gives a circle around the tube.
- Check the implicit relationship if one exists. For the torus, (x^2 + y^2 + z^2 + R^2 - r^2)^2 = 4 R^2 (x^2 + y^2) holds for every parameter pair, which is a way to verify a plot without seeing it.
- Use the explicit calculator for cross-sections of shapes that can be graphed explicitly, and use the equations for the rest.
The last step is the honest division of labour. A tool that can plot explicit surfaces is excellent for surfaces that are functions, and parametric equations are a separate skill that should be checked against the algebra rather than only against a picture.