Parametric Curve Grapher | x(t) y(t)
Plot parametric curves defined by x(t) and y(t) and download the graph as PNG.
CURVE PRESETS
t as the parameter. Operators: +, -, *, /, ^ (power). Functions: sin, cos, tan, asin, acos, atan, exp, ln, log, sqrt, abs. Constants: pi, e.What Is the Parametric Curve Grapher | x(t) y(t)?
This parametric curve grapher plots any curve defined by x(t) and y(t) as functions of parameter t. Unlike Cartesian y = f(x) plots, parametric curves can double back, self-intersect, and form closed loops, enabling spirals, roses, hearts, and Lissajous figures.
- ›10 built-in presets: Circle, Ellipse, Lissajous, Spiral, Butterfly, Heart, Rose, Astroid, Hypotrochoid, and Cycloid.
- ›Arc length estimation: Numerically approximates total arc length via Euclidean distance sum.
- ›Bounding box: Reports exact x and y extents of the plotted curve.
- ›Sample count slider: Adjust 100–2000 points for speed vs. smoothness tradeoff.
- ›Start/end markers: Green dot = t_min, red dot = t_max, shows curve direction.
- ›Download PNG: Save the plotted curve for reports or presentations.
Formula
| Curve | x(t) | y(t) | t range |
|---|---|---|---|
| Circle | cos(t) | sin(t) | [0, 2π] |
| Heart | 16·sin³(t) | 13cos(t)−5cos(2t)−… | [0, 2π] |
| Astroid | cos³(t) | sin³(t) | [0, 2π] |
| Lissajous 3:4 | sin(3t) | sin(4t) | [0, 2π] |
| Archimedean Spiral | t·cos(t) | t·sin(t) | [0, 6π] |
How to Use
- 1Select a preset or enter your own x(t) and y(t) expressions using t as the parameter.
- 2Set t min and t max for the parameter range.
- 3Drag the sample count slider to adjust smoothness (more = smoother but slower).
- 4Press Enter or click Plot Curve.
- 5Read bounding box and arc length below the plot.
- 6Click Download PNG to save. Click Clear to reset.
Example Calculation
Heart curve
Arc length formula
Understanding Parametric Curve Grapher | x(t) y(t)
What Are Parametric Curves?
In Cartesian graphing, y = f(x) restricts the curve to one y per x. Parametric curves introduce a parameter t, defining x and y independently. As t increases, the point (x(t), y(t)) traces any path, self-intersecting, spiraling, or closed. A circle cannot be a single Cartesian function but is trivially x = cos(t), y = sin(t).
Lissajous Figures
Formed by x = A·sin(at + δ), y = B·sin(bt). The ratio a:b determines shape: 1:1 gives an ellipse, 3:4 gives three vertical loops crossing four horizontal segments. They appear on oscilloscopes when comparing two sinusoidal signals. Integer ratios produce closed shapes; irrational ratios produce never-repeating paths that fill the bounding rectangle over time.
Supported Syntax
| Element | Syntax | Example |
|---|---|---|
| Parameter | t | cos(t) |
| Power | ^ | cos(t)^3 |
| Constants | pi, e | sin(2*pi*t) |
| Trig | sin, cos, tan, asin, acos, atan | sin(t)*cos(2*t) |
| Exponential | exp, ln, log | exp(cos(t)) |
| Other | sqrt, abs, floor, ceil | sqrt(abs(t)) |
Frequently Asked Questions
What is the t parameter in a parametric curve?
The parameter t traces the path. As t increases, the point (x(t), y(t)) moves along the curve, think of it as time. For a circle, t ∈ [0, 2π] traces the full circumference. For a spiral, t keeps increasing and the point spirals outward.
The range [t_min, t_max] determines which portion is plotted. Closed curves need one full period (e.g. [0, 2π]); open spirals can extend indefinitely.
Why does my curve look jagged or have gaps?
Jaggedness means too few samples, increase the slider to 1000+. Gaps occur when the expression is undefined at some t values (division by zero, log of negative, sqrt of negative). The grapher silently skips these. Narrow the t range to avoid singularities or choose a nearby evaluation point.
How is arc length calculated?
By summing straight-line distances between consecutive points:
This approximates the integral L = ∫√((dx/dt)²+(dy/dt)²) dt. With 800 samples, accuracy is typically 4+ significant figures. Increase to 2000 for higher precision on tightly curved paths.
What is the difference between parametric and Cartesian curves?
Cartesian y = f(x) must pass the vertical line test. Parametric curves have no such restriction:
- ›A circle needs two Cartesian equations (top/bottom halves) but one parametric pair.
- ›Lissajous figures self-intersect, impossible as a single Cartesian function.
- ›Spirals extend in both x and y simultaneously.
How do I plot a rose curve?
Rose curves r = cos(nθ) in parametric form: x = cos(n·t)·cos(t), y = cos(n·t)·sin(t).
Odd n gives n petals; even n gives 2n petals. The 3-petal preset is built in.
Can I plot curves in degrees instead of radians?
The trig functions use radians. To work in degrees, multiply t by π/180:
Or keep t in radians with range [0, 6.2832], most parametric formulas are naturally in radians.
What are Lissajous figures and where do they appear?
Formed by x = A·sin(at + δ), y = B·sin(bt). The shape depends on frequency ratio a:b and phase δ:
- ›a:b = 1:1, δ = π/2 → circle or ellipse
- ›a:b = 1:2 → figure-8
- ›a:b = 3:4 → three vertical loops, four horizontal segments
- ›Irrational ratios → space-filling, never-repeating path
They appear on oscilloscopes comparing two AC signals, in music visualisers, and coupled oscillator physics.