Parametric Equations Grapher | Animated Parametric Curves

The world's most advanced 2D parametric equations grapher for drawing parametric curves in Cartesian and polar coordinate systems from a starting value t₁ to an ending value t₂ by animating the parametric graphing process.

A parametric equations grapher is a grapher that draws the range of a function p(t) = [f(t), g(t)] on a given domain in a coordinate system. 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).

Utilizing the most sophisticated coordinate systems, this parametric equations grapher uses animation to graph parametric curves.

The animated graph shows how parametric curves are constructed not just in the Cartesian but also in the polar coordinate system — actually, it's the only known grapher that is even capable of graphing parametric curves in the polar coordinate system.

You can start animation by pressing at the bottom of the graphing area (if it's hidden, press the Animate button first).

It starts the animation of the parametric graphing process of the parametric expression in focus. The graph is drawn progressively from the initial value to the final value of t.

You can then press || to pause the animation or press Done to stop it. This also closes the animation interface. To display it again press the Animate button at the top of the parametric grapher.

You can also change the speed of parametric graphing animation by using the slider provided

In addition, it's also the only parametric curve grapher that enables you to rotate any of the coordinate axes and thus graph parametric curves in non-orthogonal Cartesian coordinate systems.

Tips: As you type:
• pi is replaced by π.
• inf (infinity) is replaced by .
MouseMatics! You can use your mouse to Rotate Axis, Translate and Change Scale

In addition to inputting data — by first pressing the gear button — you can use your mouse to perform some functionality unique to this interactive parametric grapher as outlined below.

• Click on (or near) an axis and move your mouse. That will rotate the axis. The graph(s) are re-drawn in the non-orthogonal (non-perpendicular) Cartesian coordinate system. Click again to release the axis.
• Drag the mouse to move the coordinate system together with the graphs.
• Double-click in the canvas to move the origin to where was clicked.
• Hold down Alt key and click on an axis to change the scale (zoom in one direction); the point which was clicked will be labelled "1" (or "-1") and becomes the new unit for that axis.
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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 pop-up menu.

Instruction

It's easy to use the parametric equations grapher; type in a parametric expression in any expression box, for example, p(t) = [3sin(t), 3cos(t)] (the use of the enclosing brackets [ ] is optional). The parametric grapher graphs as you type (default).

• To graph two or more parametric curves press » to display the multi-graph pane. The multi-graph pane consists of expression panels, which can be added or deleted as desired by pressing or × on each panel, respectively. Selecting or deselecting the checkbox for any expression displays or hides the corresponding graph.
• For convenience, the parametric equations grapher appends a suitable interval, dom = (0, 2π), to the parametric expressions and graphs on the specified domain. You can change the end points if desired.
• To draw parametric graphs of the given expressions in the polar coordinate system select the Polar checkbox.
• The parametric graphs are shown immediately as you type. You can Animate the parametric graphing process as described above in both the Cartesian and polar coordinate system.
• You can set the fineness of the curves by selecting the desired option from Graph Fineness drop-down list. In general, the higher the accuracy, the longer it takes the parametric grapher to graph expressions.
• To copy or save graphs first press the Copy/Save graph button. An image of the graphs will appear below the parametric grapher. You can then use your browser's capability to save it or copy it in your documents.
• To evaluate a parametric expression type in a number or a numeric (constant) expression in the box provided; the parametric graphing calculator displays the calculated values with the number of decimal places which can be specified by using the slider provided.

Interesting curves: Graph any of the expression under Interesting Graphs by clicking on it. For best results you may need to select Graph Fineness as "+1" or higher.

You can set the following options by pressing the ⚙ (gear) button at the top right corner of the graph canvas.

• Change the thickness of the graphs by using the slider provided.