# Parametric Equations Grapher - Animated Parametric Curves

The world's most advanced parametric equations grapher for graphing parametric equations x = f(t) & y = g(t). This Cartesian and polar parametric equations grapher properly animates the process of parametric curve graphing from a starting value t₁ to an ending value t₂ to help visualize how parametric equations are graphed.

A parametric equations grapher is a grapher that draws the range of a function p(t) = [f1(t), ..., fn(t)] on a given domain in a coordinate system. Such a graph is called the graph of the parametric equations x1 = f1(t), ..., xn = fn(t) or the parametric curve represented by the function p(t).

Utilizing the most sophisticated coordinate systems, this Cartesian and polar parametric equations grapher uses animation to graph parametric curves given by the parametric equations x = f(t) & y = g(t) or represented by the function p(t) = [f(t), g(t)].

The animated parametric curve shows how the Cartesian and polar graphs of parametric equations are constructed progressively on their domain — it's the only known polar parametric equations grapher that is even capable of graphing parametric curves in the polar coordinate system.

The parametric grapher starts animation of the parametric expression in focus when you press at the bottom of the graphing area (if it's hidden, press the Animate button first).

The parametric graph is drawn progressively from the initial value to the final value of t to give you an insight into the process of the parametric graphing.

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

The world's most advanced parametric equations grapher - this is a parametric grapher that helps you visualize, in the most proper way, how the graph of parametric equations are created on a domain by means of animation. You can even pause, resume and stop the animation.

This interactive parametric equations grapher has been developed to graph, and particularly, to show by animation how the graphs of parametric equations are created in the Cartesian and polar coordinate system.

Parametric curves can be very complicated and may have many loops. All other parametric graphers (prior to this parametric graphing calculator – recently some other graphers in partnership with Google and Microsoft have been starting to follow this guideline) display the graph of parametric equations without showing where the graph starts or ends and whether or how the loops, if any, are traced.

This unique parametric curve grapher introduces the most proper way for graphing parametric equations in the Cartesian and polar coordinate systems. Namely, it starts graphing from an initial value t₁ and progressively shows the graphing process up to the final value of t₂, showing whether the loops or any part of the curve re-traced. Moreover, this parametric grapher enables you to change the speed of the parametric graphing process.

It's easy to use the Cartesian and polar 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.