# Polar Function Grapher and Animator

Explore our free online polar function grapher, world's best polar function graphing calculator, easy-to-use and sophisticated.

## What is a Polar Function Grapher?

A polar function grapher is a function grapher that draws the graph of a function in the polar coordinate system on its domain of definition. A function graph in the polar coordinate system is called the polar graph or polar curve of the function.

## Features of Our Polar Function Grapher

• Animated Polar Graphs: Experience the most captivating polar graph animation. Our online polar function grapher uses a unique animation algorithm to visualize the step-by-step construction of polar graphs of functions. This makes it easy to see how even the most complicated polar graphs are created.
• Rotating Angular (Radial) Axes: In animating the construction of polar graphs, our online polar function graphing calculator is unmatched in its ability to rotate radial axes. This allows you to create stunning visualizations that help you understand the intricacies of polar graphing of functions.
• Seamless Switching Between Polar and Cartesian Graphs: Our polar function grapher makes it easy to switch between polar and Cartesian coordinate systems by checking/unchecking the Polar checkbox. This allows you to visualize the polar graph and Cartesian graph of a given functions in either coordinate system with remarkable ease.

## How Our Polar Function Grapher Works

Our polar function grapher plots function graphs in the polar coordinate system in a similar way to how you would graph them on paper.

The polar function graphing calculator does this by drawing angular (radial) axes for the values of `θ` that increase incrementally.

For each value of `θ` a radial axis is drawn making an angle of `θ` with the polar axis. The polar function graphing calculator calculates 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.

You can watch the entire polar graphing process by running the animation feature as described below.

x y
message
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4
Label Axes

Rotate Axes

ResultsHide
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) 🔍+ 1 🔍

Make this Transparent
Graph Thickness
Angle Mode
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Done

To copy or save graphs right click on the image of a saved graph below and select "Copy image" or "Save image" from the 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, change scales, and move coordinate system.

## Instructions

This interactive online polar function grapher is designed to show how polar graphs are created in a clear and intuitive way, 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

The grapher appends a suitable interval to expressions and graphs them on the specified domain. You can change the end points, but they must be finite for polar graphing. The polar function grapher automatically changes infinite values to finite values.

This unique polar function grapher introduces a proper way to graph functions in the polar coordinate system. Specifically, this polar function plotter uses a sophisticated animation method to draw the polar graph on the specified domain, showing the rotating angular axes and radial distances.

### Using Animation Feature

You can use the animation feature by pressing at the bottom of the polar function grapher (if hidden, press Animate first).

• The polar grapher starts the polar graphing animation for the focused function. The graph is drawn progressively from an initial angular value `θ₁` to the final value `θ₂`. The polar graphing animation shows whether any loops or parts of the polar curve are traced multiple times.
• You can press to pause the animation or Done to stop it. This also closes the animation interface. To display the animation interface again, press Animate.
• Use the slider to change the polar graphing animation speed.

### Graphing Multiple Functions

• To graph multiple functions, press the » button to show the multi-graph pane with expression panels.
• Add or delete panels with the + or × buttons if necessary.
• Select or deselect checkboxes to show or hide graphs.

### Graph Accuracy Setting

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

### Copying or Saveing 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 with 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):

• Change graph thickness using the slider.