Parametric Equations Grapher | Animated Parametric Curves

About the Parametric Equations Grapher

Our parametric equations grapher (aka parametric curve 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 Cartesian and polar parametric equations grapher uses a unique and easy-to-follow animation algorithm to illustrate how Cartesian and polar parametric graphs are drawn. This capability lets you to clearly visualize the construction of parametric curves from beginning to end.

Additionally, our parametric grapher allows you to run, pause, and resume the animation and easily control the speed of the parametric curve graphing process.

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)].

However, simply select the Polar checkbox, and our parametric 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.

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.

Additionally, this parametric graphing calculator allows you to rotate axes and graph parametric equations in oblique coordinate systems, where axes can be rotated to any angle and have any orientation.

Entering Parametric Expressions into the Cartesian and Polar Parametric Graphing Calculator

To explore the graph of parametric equations, type a parametric expression—typically expressed as p(t) = [f(t),g(t)]—in 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 systems.

Remark: The parametric expression you enter represents either the parametric equations x = f(t) y = g(t) or r = f(t) θ = g(t)

These notations for parametric equations are used based on whether you are graphing them in Cartesian or polar coordinate systems, respectively. It is also customary to represent a parametric expression correspondingly as either p(t) = [x(t),y(t)] or p(t) = [r(t),θ(t)]

Important Clarification on Graphing p(t)

It's crucial to understand that when graphing parametric equations represented by p(t), you are not graphing the function p(t) itself. The function p(t) defines a three-dimensional graph (curve), considering the independent variable t alongside its two output variables. Instead, you are graphing the range of p(t), which forms a two-dimensional curve—the projection of the three-dimensional curve onto the xy-plane or rθ-plane.

This is precisely why we say "the graph of the parametric equations represented by p(t)" or "the graph of the parametric expression defined by p(t)", and not simply "the graph of p(t)". The distinction highlights that we're visualizing the path traced by the outputs, rather than the input-output relationship including the parameter t.

Our parametric graphing calculator graphs 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.