Parametric Grapher
A parametric curve gives both coordinates as functions of a parameter t. Enter x(t) and y(t) and a t range; the calculator samples 600 points along the interval and plots the curve with an equal aspect ratio so circles look round.
Curves a function cannot describe
Writing a curve as y = f(x) forces one value of y for each x, ruling out anything that doubles back — a circle, a loop, a spiral. Parametric equations remove that restriction by giving both coordinates as functions of a third quantity and letting it run.
The parameter is usually called t, and reading it as time is natural: the curve is the path traced by a point whose position is given at every instant. This grapher plots such a path over an interval of t.
The gain is not only that closed and self-crossing curves become expressible. The description also carries more than the shape: two parametrisations can trace the same path at different speeds or in opposite directions, and the equations distinguish them where the drawing cannot.
The endpoints are reported for that reason. They say where the path starts and where it finishes, which is information the picture alone loses entirely on a closed curve.
How to use this calculator
- Enter x(t) and y(t) Both in terms of t, using the usual operators and named functions. Other letters are read as unknown function names, not as the parameter.
- Set the range of t Two comma-separated numbers. This is an interval of the parameter, not of either coordinate — how far the curve reaches is a consequence, not a setting.
- Read the endpoints The positions at the first and last values of t are reported. On a closed curve they coincide, which is the confirmation the loop actually closed.
- Check the drawn shape against the aspect The axes are equally scaled, so a circle looks circular. A shape that appears stretched is genuinely stretched rather than an artefact of the plot.
How the curve is drawn
Both expressions are compiled against t as their variable, and the interval is divided into 600 equal steps. At each step both coordinates are evaluated and the resulting point kept, provided both values came out finite — a step where either fails is skipped rather than aborting the plot.
The points are then joined in the order the parameter produced them. That ordering is what allows the curve to cross itself or return to its starting point without ambiguity: the path is a sequence, not a set.
The plot bounds come from the actual minimum and maximum of the sampled coordinates, padded by a tenth of each span. There is no percentile trimming, because a parametric curve is normally bounded and every sampled point genuinely belongs to it.
The aspect ratio is fixed at equal, so one unit horizontally occupies the same space as one unit vertically. Without that, a circle would be drawn as an ellipse whenever the plot area was not square, and the most common parametric curve of all would be misrepresented.
What each input means
- x(t) Horizontal coordinate — form field “x(t)”
- A function of the parameter giving the position along the x-axis at each value of t.
- y(t) Vertical coordinate — form field “y(t)”
- The corresponding vertical position. The two are evaluated at the same t to give one point.
- [tMin, tMax] Parameter interval — form field “t range”
- How far the parameter runs. It controls how much of the curve is traced, not the size of the region drawn.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
The unit circle
Cosine for the horizontal coordinate and sine for the vertical, with t running once around from zero. The archetypal parametric curve.
Inputs x(t) = cos(t), y(t) = sin(t), t range = 0, 6.28
- x(t) cos(t)
- y(t) sin(t)
- Range t ∈ [0, 6.28]
- Endpoints t = 0 → (1, 0), t = 6.28 → (0.999995, -0.0031853)
Result Parametric curve sampled at 601 points
The path returns almost exactly to where it began, since the interval stops just short of a full turn — 6.28 being slightly less than 2π. The reported endpoints make that small gap visible where the drawing cannot.
This curve cannot be written as y = f(x) at all, because every x between −1 and 1 corresponds to two points. That is the limitation parametric form exists to remove.
A Lissajous figure
Sine of twice the parameter against sine of three times it. Different frequencies in the two coordinates produce a woven closed curve.
Inputs x(t) = sin(2*t), y(t) = sin(3*t), t range = 0, 6.28
- x(t) sin(2*t)
- y(t) sin(3*t)
- Range t ∈ [0, 6.28]
- Endpoints t = 0 → (0, 0), t = 6.28 → (-0.00637057, -0.00955578)
Result Parametric curve sampled at 601 points
The curve crosses itself several times. Each crossing is a pair of different parameter values that happen to give the same point — the path visits that spot twice, which a set of points could not express.
The pattern closes because the two frequencies are whole-number multiples of a common base. An irrational ratio would produce a curve that never quite repeats, gradually filling a region instead.
A cycloid
The path traced by a point on the rim of a rolling wheel, over two full revolutions.
Inputs x(t) = t - sin(t), y(t) = 1 - cos(t), t range = 0, 12.56
- x(t) t - sin(t)
- y(t) 1 - cos(t)
- Range t ∈ [0, 12.56]
- Endpoints t = 0 → (0, 0), t = 12.56 → (12.5664, 0.0000202923)
Result Parametric curve sampled at 601 points
The horizontal coordinate advances steadily while the vertical oscillates, so the curve moves along rather than closing. Two arches appear, one per revolution of the wheel.
At the bottom of each arch the curve comes to a sharp point — a cusp, where the rim point is momentarily stationary. The parametrisation handles it without difficulty, though the curve has no tangent there.
Reading the result
Direction and speed are in the equations, not the picture
Two parametrisations tracing the same shape can differ in where they start, which way they go and how the point is distributed along the path. The plot shows only the shape; the endpoints and the equations carry the rest.
A self-crossing is not a contradiction
The same point being visited at two different parameter values is entirely ordinary, and it is precisely what a function of x cannot represent. Loops and crossings are the reason to use this form.
Equal aspect matters here more than elsewhere
Because so many parametric curves are circles or near-circles, distorting the axes would misrepresent them badly. The equal scaling means a shape that looks round is round, and one that looks stretched genuinely is.
When you would use this
Describing motion along a path
Where a point moves in the plane, giving each coordinate as a function of time is the natural description. Projectile paths and orbital tracks are written this way as a matter of course.
Drawing closed and looping curves
Circles, ellipses, spirals and figure-eights all fail the test for being a function of x. Parametric form expresses them directly, with no need to split the curve into upper and lower halves.
Assumptions and limitations
What this calculator assumes
- Both coordinate functions use t as their variable.
- The parameter interval is finite, and the curve is sampled uniformly across it.
- Points where either coordinate is not finite are omitted rather than ending the plot.
- The axes are equally scaled, so distances are comparable in both directions.
Where it stops being the right tool
- One curve at a time; two parametric paths cannot be overlaid.
- No direction arrows are drawn, so the sense of travel has to be inferred from the endpoints.
- No arc length, speed or tangent is computed from the parametrisation.
Common mistakes
Writing the functions in terms of x
Why it happens. Every other grapher on the site takes a function of x, so the habit is strong. Here x is an output, not the variable, and a stray x is read as an unknown name.
How to avoid it. Use t throughout both fields. The error message names the offending field, which localises it immediately.
Expecting the t range to control the visible window
Why it happens. On a function plot the range does exactly that, so the same field is assumed to set the axes here.
How to avoid it. Think of the interval as how long the point moves for. The window is then whatever region the resulting path happens to occupy.
Assuming a closed-looking curve has closed
Why it happens. At plot resolution a path that stops just short of its starting point looks identical to one that reached it — the gap is far smaller than a pixel.
How to avoid it. Compare the two reported endpoints. Using 6.28 for a full turn leaves a small shortfall that the numbers show and the drawing does not.
Frequently asked questions
What variable should I use?
Use t in both fields. It is the parameter both coordinates depend on, and any other letter is treated as an unknown function name rather than as a variable.
How are the axes scaled?
With an equal aspect ratio, so one unit horizontally covers the same distance as one unit vertically. That keeps circles circular, which matters because so many parametric curves are built from sine and cosine.
Can the t range exceed 2π?
Yes. For a closed curve the extra range simply retraces the same path, while for an open one such as a cycloid or a spiral it continues the curve further. The range is a duration, not a window.
Why use parametric form at all?
Because it describes curves that are not functions of x — anything that loops, crosses itself or doubles back. It also records the direction and pacing of the traversal, which the shape alone does not carry.