# Parametric Equations Grapher and Animator

Discover our free online parametric equations grapher, world's best 2D parametric graphing calculator, easy-to-use and sophisticated.

## What is a Parametric Equations Grapher?

A parametric equations grapher draws the curve given by the parametric equations `x = f(t)`, `y = g(t)` in the Cartesian or polar coordinate systems by plotting the points `(f(t), g(t))` as `t` varies. In other words, it draws the range of a function `p(t) = [f(t), g(t)]` on a given domain. Such a graph is called the graph of the parametric equations `x = f(t)`, `y = g(t)` or the parametric curve represented by the function p(t).

## Cartesian and Polar Parametric Curves

Our free online parametric curve grapher is unique as it is the only parametric equations graphing calculator that can graph parametric equations in both Cartesian and polar coordinate systems. Our parametric grapher also makes it easy to switch between Cartesian and polar parametric graphs by checking/unchecking the Polar checkbox. This allows you to visualize parametric curves in either coordinate system with ease.

Another unique feature of our parametric equations graphing calculator is that it can graph parametric curves in oblique coordinate systems, where the axes can intersect at any angle.

## Animating Parametric Curves

Our parametric curve grapher is also unique in its ability to animate the construction of parametric graphs in a way that is ideal for teaching and learning how to graph parametric curves in both Cartesian and polar coordinate systems.

The animation shows how to construct a parametric curve in both Cartesian and polar coordinate systems, step-by-step on its domain. You can also control the animation speed, and run, pause, and resume it with the convenient interface. This is useful for students, engineers, and scientists who need to visualize and analyze parametric curves in both graphical representations.

When animating polar parametric curves, our polar parametric equations grapher shows the entire rotating radial axes marked with radial distances. This is a feature that is only available in our polar parametric curve grapher.

You can watch the parametric graphing process in both Cartesian and polar coordinate systems by running the animation feature as described below.

x y
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Label Axes

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Parametric

Lines

[t, 1] dom=(-5, 5) [1,t] dom=(-5, 5) [t, 2t] dom=(-5, 5)

Circles

[4sin(t), 4cos(t)] [3sin(t)+1, 3cos(t)+1]

Ellipses

[4cos(t), 3sin(t)] [3cos(t), 4sin(t)] [4sin(t), 3cos(t)] [3sin(t), 4cos(t)]

Parabolas

[t, t^2] dom=(-4, 4) [t^2, t] dom=(-4, 4)

Hyperbolas

[3sec(t), 4tan(t)] [3tan(t), 4sec(t)]

Other parametric graphs

[5sin(t), 4cos(t)] [5sin(t), 4cos(2t)] [5sin(t), 4cos(3t)] [5sin(2t), 4cos(t)] [5sin(2t), 4cos(3t)] [5sin(2t), 4cos(5t)] [5sin(3t), 4cos(5t)] [5sin(3t), 4cos(7t)] [5sin(5t), 4cos(7t)] [5sin(7t), 4cos(9t)]

Butterfly curve

[sin(t)(e^cos(t)-2cos(4t)-sin(t/12)^5), cos(t)(e^cos(t)-2cos(4t)-sin( t/12 )^5)] dom=(0, 12π)
Parametric – Polar

Lines

[2csc(t), t] [2sec(t), t] [1/(sin(t) - cos(t)), t]

Circles

[1, t] [2, t] [6sin(t), t] [8cos(t), t]

Spirals

[t, t] [t/5, t] dom=(0, 10π) [√(t), t] dom=(0, 10π) [1/t, t] dom=(0, 10π)

Roses

[4sin(3t), t] [4sin(2t), t] [4sin(5t), t] [4sin(4t), t]

Ellipses

[1/(1-.8cos(t)), t] [1/(1-.8sin(t)), t] [1/(1+.8cos(t)), t] [1/(1+.8sin(t)), t]

Parabolas

[1/(1-sin(t)), t] [1/(1+cos(t)), t] [1/(1+sin(t)), t] [1/(1-cos(t)), t]

Hyperbolas

[1/(1+2cos(t)), t] [4/(1+2sin(t)), t] [1/(1-2cos(t)), t] [4/(1-2sin(t)), t]

Cardioids

[3+3cos(t), t] [2+2sin(t), t] [3-3cos(t), t] [2-2sin(t), t]

Limacons

[2+3cos(t), t] [1+2sin(t), t] [2-3cos(t), t] [1-2sin(t), t]

Lemniscates

[√(4sin(2t)), t] [√(4cos(2t)), t]

Other parametric graphs

[5sin(t), 4cos(t)] [5sin(t), 4cos(2t)] [5sin(t), 4cos(3t)] [5sin(2t), 4cos(t)] [5sin(2t), 4cos(3t)] [5sin(2t), 4cos(5t)] [5sin(3t), 4cos(5t)] [5sin(3t), 4cos(7t)] [5sin(5t), 4cos(7t)] [5sin(7t), 4cos(9t)] 🔍+ 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 context menu.

As you type:
• 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 for the Parametric Equations Grapher

This interactive parametric equations grapher has been developed to specifically show how parametric graphs are created in the most intuitive way, ideal for teaching or learning about the process of parametric graphing.

To explore parametric graphs, type a parametric expression in any expression box, for example, [`f(t), g(t)`], or [`rt), θ(t)`] — the use of the enclosing brackets [ ] is optional. The parametric grapher graphs as you type (default) in the selected Cartesian or polar coordinate system.

The parametric equations 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 parametric graphing. The parametric equations grapher automatically changes infinite values to finite values.

### Animating Parametric Curves

This unique parametric equations grapher introduces a proper way to graph parametric equations in both the Cartesianand polar coordinate systems. Specifically, this parametric equations grapher uses a sophisticated animation method to draw parametric graphs on their specified domain. In polar graphing of parametric equations, the polar parametric equations grapher shows the rotating angular axes and radial distances, making it easy to follow the animation.

You can use this interactive animation feature by pressing the Play button at the bottom of the parametric equations grapher (if hidden, press Animate first).

• The parametric equations grapher starts the parametric graphing animation for the focused parametric expression. The graph is drawn progressively from an initial angular value `t1₁` to the final value `t₂`. The parametric graphing animation shows whether any loops or parts of the Cartesian or polar parametric 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 parametric graphing animation speed.

### Graphing Multiple Parametric Expressions

To graph multiple parametric expressions, press » to show the multi-graph pane with expression panels. Add or delete panels by clicking on the + or × buttons. Select or deselect checkboxes to show/hide graphs.

### Other Features

• Select Graph Fineness for the desired curve accuracy. Higher accuracy takes longer to graph.
• To copy or save graphs, first click the Copy/Save graph button. An image of the graphs will appear below the parametric grapher. You can then save or copy the image by right-clicking on it and selecting the appropriate option from the context menu.
• To evaluate a parametric expression, type a number or expression in the provided box. The calculator displays the calculated value with decimal places set by the slider.

Interesting curves: Graph any of the parametric expressions under Interesting Graphs to render some cool parametric graphs by selecting it. To improve the accuracy of the graph, you may need to select a higher Graph Fineness setting.

### Parametric Equations Grapher 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.