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Polar and Cartesian Function Grapher

About the Polar Function Grapher

Graph functions in the polar coordinate system with our fully interactive polar function grapher. Watch how polar curves are constructed step-by-step through smooth, controlled animation. This powerful Calculators Math™ polar function plotter allows you to run, pause, or resume the animation and control the plotting speed. Our polar grapher is the perfect tool to visualize the detailed graphing process of polar curves such as roses, cardioids, spirals, limacons, lemniscates, and other complex curves—automatically and progressively from start to finish.

Additionally, with its unique ability to display the rotating radial axes (the axes used in polar graphing, generated by rotating the polar axis), the grapher helps you intuitively understand how polar curves are formed through engaging animation.

Comprehensive Function Visualization: Polar, Cartesian, and Oblique Systems

Our grapher plots functions in both Cartesian and polar coordinate systems. To demonstrate how the same function is graphed in the Cartesian coordinate system, simply switch coordinate systems—by deselecting the Polar checkbox. This mathematically sound approach—exclusive to our grapher—allows users to visualize and compare the polar and rectangular graphs of the same function with remarkable ease.

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

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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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Your browser does not support the Canvas element, or you need to enable JavaScript in your browser to use this polar function grapher. Below are images of the grapher.

Polar Function Grapher - Graph functions in polar coordinate system.
Polar Function Grapher: Function graphs in the polar coordinate system.
Oblique Polar Function Grapher - Graph functions in oblique polar coordinate system.
Oblique Polar Function Grapher: Function graphs in the oblique polar coordinate system.
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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.

Entering Function Expressions into the Function Grapher

To explore graphs 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; the grapher will internally replace these with θ as you type. (If you hover your mouse over the expression label, you will see the characters change to θ in your input box).

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.

How Our Polar Function Grapher Works

This unique interactive polar function grapher plots functions r(θ) directly in the polar coordinate system, mimicking how you would graph them on paper—without the need for Cartesian conversion.

  1. For each value of angular coordinate θ, the grapher draws a temporary radial axis at that specific angle relative to the polar axis. The calculator then computes the signed distance r(θ) and plots the point along that radial axis.
  2. The polar plotter then connects this point to the next point plotted using the same method with a slightly larger value of θ. This process continues until the full polar graph of the function is complete.

Our polar grapher also offers an animated graphing process, allowing you to visualize this sequence in real time, as detailed below.

Polar Function Graph Animator

Why animation? Polar curves can be intricate, often having multiple loops or overlapping paths. Most other graphers display the polar graph of a function instantly, without showing where it starts, where it ends, or the direction in which loops are traced.

To address this, our tool draws the graph step-by-step, allowing for a clear visualization of its creation across its entire domain. Our polar graph animator, equipped with a sophisticated polar coordinate system, is specifically designed for this.

It is the first tool to introduce the proper method for graphing functions in the polar coordinate system through a controlled animation. This way, you can watch your polar graphs take shape in real time!

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