Polar Function Grapher | Animated Polar Graphs

Visualize functions in a new way with our free online polar function grapher! Compare function graphs in Cartesian and polar coordinate systems seamlessly, or explore the unique world of graphs of function in oblique coordinate systems - all in one powerful tool.

This dedicated polar function grapher uses a unique animation algorithm to visualize the step-by-step construction of the graph of a function in the polar coordinate system like no other grapher. With its ability to rotate radial axes, it helps you understand the polar graphing process for functions in stunning animation.

Our polar grapher makes it easy to switch between polar and Cartesian coordinate systems by checking or unchecking the Polar checkbox. This allows you to visualize the polar graph and Cartesian graph of a given function in either coordinate system with remarkable ease.

How Our Polar Function Grapher Works

Our polar function grapher plots graphs of functions in the polar coordinate system, similarly to how you would graph them on paper.

For each value of θ, a temporary radial axis is drawn, making an angle of θ with the polar axis. The polar function graphing calculator computes the signed distance r(θ) and locates that point along the radial axis.

The polar function grapher then connects this point to the next point located using the same method with a slightly larger value of θ. The online polar function graphing calculator thus completes the polar graph of the given function.

Our polar function grapher allows you to watch the entire animated graphing process, as detailed below.

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Label Axes

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Functions

Lines

1 x+1 2x

Semi-circles

√(9-x^2) -√(9-x^2)

Semi-ellipses

√(9-x^2/3) √(9-x^2/3)

Parabolas

x^2 0.5x^2-4x+1 -(0.5x^2-4x+1)

Semi-hyperbolas

√(x^2-4) -√(x^2-4)

Other graphs

√(4sin(2x)) √(4cos(2x))
Functions – Polar

Lines

2csc(θ) 2sec(θ) 1/(sin(θ) - cos(θ))

Circles

1 2 6sin(θ) 8cos(θ)

Spirals

θ θ/5 dom=(0, 10π) √(θ) dom=(0, 10π) 1/θ dom=(0, 10π)

Roses

4sin(3θ) 4sin(2θ) 4sin(5θ) 4sin(4θ)

Ellipses

1/(1-.8cos(θ)) 1/(1-.8sin(θ)) 1/(1+.8cos(θ)) 1/(1+.8sin(θ))

Parabolas

1/(1-sin(θ)) 1/(1+cos(θ)) 1/(1+sin(θ)) 1/(1-cos(θ))

Hyperbolas

1/(1+2cos(θ)) 4/(1+2sin(θ)) 1/(1-2cos(θ)) 4/(1-2sin(θ))

Cardioids

3+3cos(θ) 2+2sin(θ) 3-3cos(θ) 2-2sin(θ)

Limacons

2+3cos(θ) 1+2sin(θ) 2-3cos(θ) 1-2sin(θ)

Lemniscates

√(4sin(2θ)) √(4cos(2θ))

Butterfly curve

e^sin(θ)-2cos(4θ)+sin((2θ-π)/24)^5 dom=(0, 12π)
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To copy or save graphs right click on the image of a saved graph below and select "Copy image" or "Save image" from the context (pop-up) menu.


As you type:
  • ..t is replaced by θ. (You can also use x or t; they are internally replaced by θ).
  • pi is replaced by π.
  • inf (infinity) is replaced by .
More tips

MouseMatics: Learn how you can use your mouse to rotate axes (graph in oblique coordinate system), change scales, and move coordinate system.

Instructions

This interactive online function grapher can be used to graph functions in the Cartesian (both rectangular and oblique), and polar coordinate systems. In particular, this unmatched function grapher is designed to show, in a clear and intuitive way, how polar graphs are created, making it ideal for teaching or learning about polar graphing.

To explore function graphs, type a function into any expression box, for example, f(x), or r(θ). The grapher will update the graph as you type (default) in the selected coordinate system

Cartesian Function Grapher
Cartesian function grapher: Function graphs in Cartesian coordinate system.
Polar Function Grapher
Polar function grapher: Function graphs in polar coordinate system.
Function Grapher - Oblique Coordinate System
Oblique function grapher: Function graphs in oblique coordinate system.

The grapher appends a suitable interval to function expressions and graphs them on the specified domain. You can change the endpoints, but they must be finite for graphing functions in the polar coordinate system. The polar function grapher automatically changes infinite values to finite ones. It uses a sophisticated interactive animation method to draw the polar graph on the specified domain, dom=(θ1, θ2), allowing you to see the rotating radial axes and radial distances.

Using Polar Graphing Animation Feature

To activate the animation feature, press at the bottom of the polar function grapher (if hidden, press Animate first).

Graphing Multiple Functions

To graph multiple functions, press the » button to show the multi-graph pane with expression panels.

Graph Accuracy Setting

Select Graph Fineness for the desired curve accuracy. Higher accuracy takes longer to graph.

Copying or Saving Graphs

  1. Click the Copy/Save graph button.
  2. An image of the graphs will appear below the grapher.
  3. Right-click on the image and select the appropriate option from the context menu.

Evaluating Functions

  1. To evaluate a function, type a number or expression in the provided box.
  2. The function graphing calculator will display the calculated value, rounded to the number of decimal places set by the slider.

Interesting Curves

Interesting curves: Graph any of the expressions under Interesting Graphs to render some cool polar graphs by selecting it. For best results, you may need to select Graph Fineness as "+1" or higher.

Settings

Press the ⚙ (gear) button to set options (if the button is hidden, first click on the icon at the top right of the canvas):