Polar Function Grapher | Animated Polar Graphs

The world's most advanced polar function grapher, capable of animating the polar graphing process, uniquly displaying the rotating radial axes. It shows how the polar graph of a function is constructed progressively on its domain.

A polar function grapher is a function graphing calculator that draws the graph of a function on a given domain in the polar coordinate system. Such a graph is called the polar graph or the polar curve of a given function.

The process of graphing in the polar coordinate system and rendering it by using a function graphing calculator is fundamentally different from graphing in the Cartesian coordinate system. This is due to the fact that a polar graph needs to be drawn progressively so that one can visualize how a polar graph is constructed on its domain.

This function grapher, deploying the most sophisticated Cartesian & polar coordinate systems, is the only known graphing tool that can graph in non-perpendicular Cartesian coordinate system and also properly helps visualize how a polar curve is progressively constructed by means of polar graphing animation. This way, you can watch how all your cool polar graphs are drawn step by step.

You can use this tremendously useful feature by pressing at the bottom of the function graphing calculator (if it's hidden, press the Animate button first).

It starts the animation of the polar graphing process of the function in focus. The graph is drawn progressively from the initial value θ₁ to the final value θ₂ of the radial axis. The animated polar graph shows the rotating angular axes (radial axes) and the radial distances.

You can then press to pause the animation or press Done to stop it. This also closes the animation interface. To display it again press the Animate button at the top of the polar grapher.

You can also change the speed of polar graphing animation by using the slider provided

This free online polar function graphing calculator also draws polar graphs with the polar axis rotated.

Tips: 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 .

To graph piecewise defined functions type in each piece with the corresponding subinterval as a single function.

The quickest way to type dom=(0, 2π) or dom=(-∞, ∞) is by deleting the domain entirely, including dom=.

MouseMatics! You can use your mouse to Rotate Axes, Translate and Change Scale

In addition to inputting data — by first pressing the gear button — you can use your mouse to perform some functionality unique to this interactive function grapher as outlined below.

• Click on (or near) an axis and move your mouse. That will rotate the axis. The graph(s) are re-drawn in the non-perpendicular Cartesian coordinate system or generalized polar coordinate system. Click again to release the axis.
• Drag the mouse to move the coordinate system together with the graphs.
• Double-click in the canvas to move the origin to where was clicked.
• Hold down Alt key and click on an axis to change the scale (zoom in one direction); the point which was clicked will be labelled "1" (or "-1") and becomes the new unit for that axis.
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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)
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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 pop-up menu.

Instruction

This interactive polar function Grapher has been developed to graph, and particularly, to show by animation how the graph of a function is created in the polar coordinate system. Polar curves can be very complicated and may have many loops. All other polar graphers display the polar graph of a function without showing you where the curve starts or ends and whether or how the loops, if any, are traced. This unique polar function graphing calculator introduces the proper way for graphing functions in the polar coordinate systems. Namely, it starts graphing from the initial value of an angular coordinate, θ₁, and progressively shows the graphing process up to the final value of θ₂, showing whether the loops or any part of the curve re-traced. Moreover, this polar function grapher enables you to change the speed of the polar graphing process.

It's easy to use the Cartesian or polar function grapher; type in a function in any expression box, for example,f(x) or r(θ). The function grapher graphs as you type (default) in the selected coordinate system. (Don't worry about which variable you use, the function grapher automatically changes the variables according to the selected coordinate system.)

• To graph two or more functions on the same Cartesian or polar coordinate system press » to display the multi-graph pane. The multi-graph pane consists of expression panels, which can be added or deleted as desired by pressing or × on each panel, respectively. Selecting or deselecting the checkbox for any expression displays or hides the corresponding graph.
• For convenience, the polar function grapher or the Cartesian function grapher appends a suitable interval, dom = (0, 2π) or dom = (-∞, ∞), respectively, to the function expressions and graphs on the specified domain. You can change the end points if desired. The end points must be finite for polar graphing. The polar function grapher automatically changes infinities, if any, to finite values.
• The polar graphs or Cartesian graphs are shown immediately as you type. You can Animate the polar graphing process as described above.
• You can set the fineness of the Cartesian or polar curves by selecting the desired option from the Graph Fineness drop-down list. In general, the higher the accuracy, the longer it takes the function grapher to graph functions.
• To copy or save graphs first press the Copy/Save graph button. An image of the graphs will appear below the function grapher. You can then use your browser's capability to save it or copy it in your documents.
• To evaluate a function type in a number or a numeric (constant) expression in the box provided; the function grapher displays the calculated function values with the number of decimal places which can be specified by using the slider provided.

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

You can set the following options by pressing the ⚙ (gear) button at the top right corner of the graph canvas.

• Change the thickness of the graphs by using the slider provided.
• Select the angle mode (radiansdefault, degrees or grades).
• If you deselect the Graph as you type option, you will have to press Graph selected expressions, which then appears at the bottom of the calculator, to update the graphs whenever you make any changes to the expressions or coordinate plane (i.e., move the origin, rotate axes, etc.).
• Optionally display the controls on the top of the function grapher that enable you to start/pause and stop rotation of any or all axes.
• The polar function grapher remembers the function expressions you enter and retrieves them between visits. You can clear them by pressing the Reset Calculator button.