Cylindrical and Spherical Coordinates

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Non-Cartesian Systems

You can use Cartesian coordinates in both 2D and 3D. In many cases, however, it can be more helpful to describe the location of a point using distance and direction.

For polar coordinates in 2D, \(r\) is the distance from the origin to the point, and θ is the direction or the angle of rotation from the positive \(x\)-axis. You can write the polar coordinates of a point as \((r,θ)\), with θ usually given in radians. Although the Graphing Calculator doesn’t plot polar points, it considers any equation or inequality written in terms of \(r\) and \(θ\) to be in polar form, and it will plot these equations/inequalities as a polar curve or region. Learn more in our Polar Graphing article.

 

Cylindrical Coordinates

If you extend the idea of polar coordinates to 3D, you get cylindrical coordinates. If we add a \(z\)-coordinate, a cylindrical becomes \((r, θ, z)\) and combines polar coordinates in the \(xy\)-plane with the Cartesian coordinate \(z\) in 3D. The 3D Calculator cannot plot these points directly, but it will recognize equations and inequalities written in terms of \(r\), \(θ\), and \(z\) to be cylindrical. For example, \(r = 2\) will graph a circle with radius of 2. To view the cylinder, check the Extend to 3D. Explore more in the Cylinder in Cylindrical Coordinates example.

Screenshot in Desmos 3D of a point with coordinates (r, theta, phi) showing the length r, angle from the z axis phi, and the angle from the x axis theta.

 

Spherical Coordinates

You can also use spherical coordinates to describe the location of a point using distance and direction in 3D. You can write a spherical point in the form \((ρ, θ, \phi)\). Rho (\(ρ\)) is the distance between the origin and the point, rather than the distance from the origin to the projection of the point onto the \(xy\)-plane like with cylindrical coordinates. Phi (\(φ\)) is a second angle (rather than the Cartesian \(z\)), and \(\phi\) is the angle between the ray from the origin and the positive \(z\)-axis as shown.

Screenshot in Desmos 3D of a point with coordinates (rho, theta, phi) showing the length rho, angle from the z axis phi, and the angle from the x axis theta.

Similar to cylindrical coordinates, the 3D calculator cannot directly plot points in this form. However, it will recognize equations and inequalities written in terms of \(ρ\), \(θ\), and \(φ\) to be spherical. Try typing \(ρ=2\) to graph a sphere with radius a radius 2. Explore more in the Sphere in Spherical Coordinates example.

 

Not all disciplines use the same convention for spherical coordinates. For example, physics texts commonly use spherical coordinates in the form \((ρ, θ, φ)\) where \(θ\) is the polar angle to the \(z\)-axis and \(φ\) is the azimuthal angle. This is switched from what Desmos 3D uses.

 

Graphing with Spherical Coordinates in Desmos

When you write an equation in the form ρ equals a function of \(θ\) and \(φ\), the 3D Calculator will use spherical coordinates. For example, try plotting the magnitude of the real spherical harmonic function \(Y_{2}^{2}\).

To use the \(ρ\) symbol, type the word rho. Similarly, for the \(θ\) symbol, type the word theta, and for the \(φ\) symbol, type phi.

The default domain is that \(θ\) plots between \(0\) and \(2\pi\), and \(φ\) plots between \(0\) and \pi. You can change these bounds, but the defaults are set to best display unique spherical coordinates. Explore more in the Spherical Harmonic Function example.

Screenshot in Desmos 3D of the spherical harmonic function.

 

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