Parametric Curve Grapher | Cartesian & Polar Graphs

Explore our easy-to-use and powerful Parametric Curve Grapher. Graph and visualize Cartesian and polar parametric graphs with stunning animation.

Our unique parametric grapher can draw parametric curves represented by p(t)=[f(t),g(t)] in both Cartesian and polar coordinate systems, allowing for easy switching between the two..

It also has the unique ability to graph these curves in oblique coordinate systems, where the axes can be freely rotated and angled.

Animating Parametric Curves

This superior Parametric Curve Grapher introduces the most proper way to graph parametric equations in both Cartesian and polar coordinate systems. Specifically, it uses a sophisticated interactive animation method to draw parametric graphs on their specified domain.

Notably, when animating polar parametric curves, it has the distinct ability to rotate radial axes. In doing so, it shows the entire rotating radial axes marked with radial distances, making it easy to follow the animation. This is a feature that is only available in our polar parametric equations grapher.

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

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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)]
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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 context menu.

As you type:
  • pi is replaced by π.
  • inf (infinity) is replaced by .
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MouseMatics: Learn how you can use your mouse to rotate axes (graph in oblique coordinate system), change scales, and move coordinate system.

Instructions for Our Parametric Curve (Parametric Equations) Grapher

This interactive Parametric Curve Grapher has been developed to specifically show how a parametric curve represented by the function p(t)=[f(t),g(t)] (or equivalently defined by parametric equations x=f(t) and y=g(t)) are graphed in both Cartesian and polar coordinate systems using animation, 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, [x(t),y(t)], or [r(t),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.

Cartesian Parametric Equations Grapher - Cartesian Curve
Parametric equations grapher: Parametric curve in Cartesian coordinate system.
Polar Parametric Equations Grapher - Polar Curve
Polar parametric equations grapher: Parametric curve in polar coordinate system.
Parametric Equations Grapher - Oblique Coordinate System
Parametric equations grapher: Parametric curve in oblique coordinate system.

Our Parametric Curve 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 grapher automatically changes infinite values to finite values.

This parametric equations grapher uses a unique animation algorithm to visualize the step-by-step construction of Cartesian and polar parametric graphs like no other grapher.

Animating Parametric Curves

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).

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

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):