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Parametric Equations Grapher | Cartesian & Polar Curves

About the Parametric Equations Grapher

Graph parametric curves with our fully interactive parametric equations grapher. Watch the step‑by‑step construction of both Cartesian and polar parametric curves, allowing you to visualize the creation of intricate parametric paths with smooth animation over specified domains. Additionally, our unique Calculators Math™ parametric curve grapher lets you rotate the axes and explore parametric equations in oblique (non‑orthogonal) coordinate systems.

Our powerful and easy-to-use parametric equations grapher draws parametric curves represented by p(t) = [f(t),g(t)] in both Cartesian and polar coordinate systems.

When graphing in the Cartesian coordinate system, this is typically expressed as p(t) = [x(t),y(t)] and in the polar coordinate system as p(t) = [r(t),θ(t)]

Unique Cartesian & Polar Parametric Grapher

Our fully interactive parametric grapher uses a unique, easy-to-follow animation algorithm to illustrate how both Cartesian and polar parametric graphs are drawn. This capability allows you to clearly visualize the construction of parametric curves from start to finish.

Furthermore, our parametric grapher allows you to run, pause, and resume the animation, giving you full control over the speed of the parametric curve graphing process.

Besides being the first ever polar parametric grapher available online, it also has the unique feature of rotating radial axes when constructing polar parametric curves from scratch.

Finally, this parametric grapher allows you to rotate axes and graph parametric equations in an oblique coordinate system, where axes can intersect at any angle and have any orientation.

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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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Your browser does not support the Canvas element, or you need to enable Javascript on your browser to use this parametric equation grapher. Below are images of the grapher.

Parametric Curve Grapher: Graph parametric equations in rectangular coordinate system.
Parametric Equations Grapher: Parametric curves in rectangular Cartesian coordinate system.
Oblique Parametric Curve Grapher: Graph parametric equations in skew (oblique) coordinate system.
Skew Parametric Equations Grapher: Parametric curves in skew (oblique) Cartesian coordinate system.
Polar Parametric Curve Grapher: Graph parametric equations in polar coordinate system.
Polar Parametric Equations Grapher: Parametric curves in polar coordinate system.
Skew Polar Parametric Curve Grapher: Graph parametric equations in skew (oblique) polar coordinate system.
Skew Polar Parametric Equations Grapher: Parametric curves in skew (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 Parametric Expressions into the Cartesian and Polar Parametric Grapher

To explore the graph of parametric equations, type a parametric expression—typically expressed as p(t) = [f(t),g(t)]—into any expression box. For example, you could type [sin(t),cos(t)]; using the enclosing brackets [ ] is optional. The parametric equations plotter graphs as you type (by default) in the selected Cartesian or polar coordinate system.

Our parametric grapher plots on a specified interval (domain). If no interval is specified, the grapher appends dom=(0, 2π). You can change the endpoints, but they must be finite. The parametric grapher automatically replaces infinite values with finite ones.

By default, parametric expressions are graphed in the Cartesian coordinate system. That is, for each t, the components of ordered pairs [f(t),g(t)] will be treated as Cartesian coordinates—typically represented as [x(t),y(t)].

Selecting the Polar checkbox, and our parametric curve grapher will seamlessly switch to its built-in polar coordinate system. This means for each t, the components of ordered pairs [f(t),g(t)] will be treated as polar coordinates—typically represented as [r(t),θ(t)]—and graphed accordingly.

Important Clarification on Graphing p(t) = [f(t),g(t)]

It is crucial to understand that when plotting the parametric equations defining the curve, you are not typically graphing the function p(t) itself. The graph of p(t) would actually be a three-dimensional curve in (t, x, y) space, as it accounts for the independent variable t alongside its two output variables.

Instead, what we plot is the image of p(t): the points with the coordinates (f(t),g(t)). This forms a two-dimensional curve—essentially a projection of the 3D curve onto the xy-plane (or rθ-plane). This is why we refer to it as the graph of the parametric equations/curve represented by p(t) rather than simply "the graph of p(t)".

Parametric Curve Animator

Why use animation? Parametric curves can be intricate, often featuring multiple loops. Unlike functions—which can only loop when graphed in the polar coordinate system—both Cartesian and polar parametric curves can loop extensively. Most graphers display parametric paths instantly, hiding where they start, where they end, or how those loops are traced.

Our parametric graph animator solves this by drawing curves step-by-step, allowing you to clearly visualize their creation across their domains. Equipped with a sophisticated coordinate system, it is the first tool to introduce a proper, controlled animation method for graphing parametric equations.

Watch your parametric graphs take shape in real time!

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