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ToPolarCoordinates[{x,y}]

gives the {r,θ} polar coordinates corresponding to the Cartesian coordinates {x,y}.

ToPolarCoordinates[{x1,x2,…,xn}]

gives the hyperspherical coordinates corresponding to the Cartesian coordinates {x1,x2,…,xn}.

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ToPolarCoordinates

ToPolarCoordinates[{x,y}]

gives the {r,θ} polar coordinates corresponding to the Cartesian coordinates {x,y}.

ToPolarCoordinates[{x1,x2,…,xn}]

gives the hyperspherical coordinates corresponding to the Cartesian coordinates {x1,x2,…,xn}.

Details

  • ToPolarCoordinates produces points in the standard range , in two dimensions, and , , in higher dimensions.
  • ToPolarCoordinates automatically threads over arrays of coordinate lists.

Examples

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Basic Examples  (3)

Convert a general point:

Wolfram Language code: ToPolarCoordinates[{x, y}]

A specific point:

Wolfram Language code: ToPolarCoordinates[{1, 1}]

A conversion in dimension 3:

Wolfram Language code: ToPolarCoordinates[{x, y, z}]

Scope  (3)

Convert several points:

Wolfram Language code: ToPolarCoordinates[{{x, y}, {1, 0}, {-2, 0}, {0, 1}}]

A matrix of points:

Wolfram Language code: ToPolarCoordinates[{{{x, y}, {1, 0}}, {{-2, 0}, {0, 1}}}]

Five-dimensional hyperspherical coordinates:

Wolfram Language code: ToPolarCoordinates[{0, (1/Sqrt[2]), (1/2), (1/2), 0}]

Properties & Relations  (6)

ToPolarCoordinates preserves the shape of arrays:

Wolfram Language code: ToPolarCoordinates[{{{1, -1}}}]

This includes empty arrays:

Wolfram Language code: ToPolarCoordinates[{{} , {}}]

ToPolarCoordinates[pt] is a special case of CoordinateTransform:

Wolfram Language code: CoordinateTransform["Cartesian" -> "Polar", {x, y}] == ToPolarCoordinates[{x, y}]
Wolfram Language code: CoordinateTransform["Cartesian" -> "Hyperspherical", {x, y, z}] == ToPolarCoordinates[{x, y, z}]

ToPolarCoordinates inverts FromPolarCoordinates:

Wolfram Language code: Simplify[ToPolarCoordinates[FromPolarCoordinates[{r, θ}]], r > 0 && -Pi < θ < Pi]
Wolfram Language code: Simplify[ToPolarCoordinates[FromPolarCoordinates[{r, θ, φ}]], r > 0 && 0 < θ < Pi && -Pi < φ < Pi]

ToPolarCoordinates converts pairs of real numbers to their polar representation:

Wolfram Language code: ToPolarCoordinates[{1, 2}]

AbsArg converts complex numbers to their polar representation:

Wolfram Language code: AbsArg[1 + 2I]

ToPolarCoordinates[{x,y,z}] uses spherical coordinates about the axis:

Wolfram Language code: ToPolarCoordinates[{x, y, z}]

ToSphericalCoordinates[{x,y,z}] uses spherical coordinates about the axis:

Wolfram Language code: ToSphericalCoordinates[{x, y, z}]

ToPolarCoordinates changes the coordinate values of points:

Wolfram Language code: ToPolarCoordinates[{x, y}]

TransformedField changes the coordinate expressions for fields:

Wolfram Language code: TransformedField["Cartesian" -> "Polar", {x, y}, {x, y} -> {r, θ}]//Simplify
Wolfram Research (2015), ToPolarCoordinates, Wolfram Language function, https://reference.wolfram.com/language/ref/ToPolarCoordinates.html.

Text

Wolfram Research (2015), ToPolarCoordinates, Wolfram Language function, https://reference.wolfram.com/language/ref/ToPolarCoordinates.html.

CMS

Wolfram Language. 2015. "ToPolarCoordinates." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ToPolarCoordinates.html.

APA

Wolfram Language. (2015). ToPolarCoordinates. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ToPolarCoordinates.html

BibTeX

@misc{reference.wolfram_2026_topolarcoordinates, author="Wolfram Research", title="{ToPolarCoordinates}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/ToPolarCoordinates.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_topolarcoordinates, organization={Wolfram Research}, title={ToPolarCoordinates}, year={2015}, url={https://reference.wolfram.com/language/ref/ToPolarCoordinates.html}, note=[Accessed: 13-June-2026]}