Polar Function Grapher | Polar Graphing Calculator

Explore our polar function grapher—the ultimate polar function graphing calculator—to visualize how polar graphs of functions are drawn. It shows the step-by-step construction via animation, allowing users to run, pause, or resume the animation and control the polar function plotting speed.

About the Polar Function Grapher

Our polar function grapher enables users to visualize the construction of polar graphs of functions by animation. This unique animation method clearly shows how polar graphs of functions are drawn automatically and progressively—from start to finish—in the polar coordinate system.

Additionally, with its unique ability to show the rotating radial axes (the axes used in polar graphing, obtained by rotating the polar axis), this polar function grapher helps you understand how polar graphs of functions are drawn through engaging animation.

Comprehensive Function Visualization

This grapher plots function in both Cartesian and polar coordinate systems. To demonstrate how the same function is graphed in the Cartesian coordinate system, this function graphing calculator accomplishes this in a unique way and with remarkable ease: simply by switching coordinate systems—by deselecting the Polar checkbox. This mathematically sound approach allows users to visualize and compare the polar and Cartesian graphs of a given function.

Moreover, this versatile and oblique function grapher enables users to rotate axes and graph functions in oblique coordinate systems, providing a powerful all-in-one visualization tool.

Try our Graphing Calculator, which in addition to graphing functions can graph parametric equations, equations containing variables on both sides, including implicit functions, and points.

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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)
RAD
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Polar Function Grapher
Polar function grapher
Oblique Polar Function Grapher
Oblique polar function grapher
Function Grapher
Function grapher
Oblique Function Grapher
Oblique function grapher
Graph Thickness
Angle Mode
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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.

Instructions for Using the function grapher

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 ∞.
More tips

MouseMatics: Find out how to use your mouse to rotate axes, change scales, and translate coordinate systems.

Entering Function Expressions into the Function Graphing Calculator

To explore graph of functions, note that, by default, the Polar checkbox is selected, meaning the function grapher will draw graphs in the polar coordinate system. In this case, r(θ) is used to denote the function. To type θ in a function expression, type "..t". You can also simply use t or even x. They will be replaced by θ while typing.

The function graphing calculator graphs on a specified interval (domain). If no interval is specified, the grapher appends a suitable interval to function expressions. For polar graphing, it uses dom=(0, 2π). You can change the endpoints, but they must be finite for graphing functions in the polar coordinate system. The polar function grapher automatically adjusts infinite values to some finite ones.

On the other hand, the grapher uses dom=(-∞, ∞) for graphing functions in the Cartesian coordinate system by default; you can change these endpoints if needed.

You can switch between polar and Cartesian coordinate systems by checking or unchecking the Polar checkbox. This action will redraw the graph of the function as either a polar or Cartesian graph accordingly.

Note: As the variable t is automatically replaced by θ in polar graphing, when typing trigonometric functions such as tan, users will initially see θa. However, upon completing the input, the letter θ will revert to t, as expected, and tan will be displayed correctly—replacing θan.

How Our Polar Function Grapher Works

This unique interactive polar function grapher plots functions r(θ) directly in the polar coordinate system, similarly to how you would graph them on paper—without converting to Cartesian coordinates.

  1. For each value of angular coordinate 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.
  2. It then connects this point to the next point located using the same method with a slightly larger value of θ. This continues until the complete polar graph of the function is drawn.

Our polar grapher also offers an animated graphing process, as detailed below.

Polar Function Graph Animator

Why animation? Polar curves can be intricate, often featuring multiple loops. Most other graphers display the polar graph of a function instantly, without showing where it starts or ends, or how any loops—if present—are traced.

To address this, it's crucial to draw a polar graph step-by-step, allowing for a clear visualization of its creation on its domain. Our polar graph animator, equipped with a sophisticated polar coordinate system, is specifically designed for this.

It is the first to introduce the proper method for graphing functions in the polar coordinate systems through a controlled animation.

This way, you can watch your polar graphs take shape in real time!

Insert on the bottom of multi-input panel: