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Section 2.7 Implicit Surfaces

Implicit Surfaces allow you to graph surfaces defined by any implicit equation in one of the three supported coordinate systems.

Examples:

See the list of example Implicit Surfaces on the Examples submenu of the CalcPlot3D main menu. See more about the Examples menu in Section 4.6.

To add an implicit surface to the plot, select the option Implicit Surface on the Add to graph drop-down menu. You'll see an object dialog appear like the following:

Implicit surface object dialog
Figure 2.7.1. Implicit surface object dialog
Implicit surface plot
Figure 2.7.2. Implicit surface from equation: \(x^2 + y^2 - z^2 = 1\)

Now let's plot the default implicit surface.

Clear Plot button

Clear the plot by clicking the clear plot button.

Then check the checkbox in the upper-left corner of this object to plot the default implicit surface for the implicit equation, \(x^2 + y^2 - z^2 = 1\text{.}\) It should look like Figure 2.7.2.

Implicit surfaces can be defined by implicit equations in any of the following coordinate systems. Just use the corresponding variables.

Cartesian Coordinates

Enter equations using the variables \(x\text{,}\) \(y\text{,}\) and \(z\text{.}\)

Cylindrical Coordinates

Enter equations using the variables \(r\text{,}\) \(\theta\text{,}\) and \(z\text{.}\) Note that you can click on the button labeled \(\theta\) to enter this variable in your implicit equations.

You can also enter theta as a word.

Spherical Coordinates

Enter equations using the variables \(\rho\text{,}\) \(\theta\text{,}\) and \(\phi\text{.}\) Note that you can click on the buttons for \(\rho\text{,}\) \(\theta\text{,}\) and \(\phi\) to enter these variables in your implicit equations.

You can also enter rho, theta or phi as words.

Depending on which variables are in your implicit equation, ranges will be given for you to specify for each variable.

The number of cubes per axis tells CalcPlot3D how many cubes to use in the marching cubes algorithm that is used to approximate the implicit surface. As this value gets larger, the surface will look more and more smooth, but it will also take longer and longer to generate and even to rotate.

Use the settings button ⚙ to adjust surface settings. For a discussion of the options, see Section 2.11.

Parameters are a great option to use with implicit surfaces. See the Solution, Method 2 in the section on Level Surfaces in Section 5.6.