Polar & Parametric Plotter
Roses, spirals, cardioids, Lissajous figures and cycloids — drawn from r(θ) or from x(t) and y(t), with an animated tracer.
theta (or t) for the angle. Radians.
Try one
Try one
About this curve
Arc length and area are numerical estimates from the plotted points — raise the point count for a closer figure. Area uses the shoelace formula and is only meaningful for a closed curve.
How to use this calculator
Step by step
- Choose Polar to describe a curve by its distance from the origin at each angle, or Parametric to give x and y separately as functions of a parameter t.
- Type the expression using
theta(polar) ort(parametric). Angles are in radians throughout. - The range boxes accept expressions, so you can type
2pi,4piorpi/2rather than working out the decimal. - Press Animate to watch the curve being drawn, or drag the slider to step along it by hand. The readout under the graph gives x, y, r and the parameter value at the tracer.
- Tap any card in Try one to load a classic curve. It is the fastest way to see what changing a single coefficient does.
- Fit frames the whole curve. The axes are always kept square here, because a rose or a circle drawn on stretched axes tells you nothing.
Good to know
- For a rose
cos(kθ): an odd k gives k petals, an even k gives 2k petals. Try changing the 2 in3cos(2theta)and watch the count jump. - If a polar curve looks like it stops halfway, extend the θ range. Spirals and many roses need more than 2π to close.
- A negative r is drawn on the opposite side of the origin, which is standard convention and is what produces the extra petals on even-k roses.
- Lissajous figures are the parametric equivalent:
sin(at)againstsin(bt). The ratio a : b decides the shape, and a small phase shift makes it rotate. - The enclosed area figure only means something for a closed curve that does not cross itself — the shoelace formula cancels out overlapping regions.
- Raise the point count for tight spirals; 1200 points spread over 24π is only about 16 points per turn.
The maths behind it
- Polar to Cartesian x = r cos θ, y = r sin θ The conversion done for every plotted point.
- Cartesian to polar r = √(x² + y²), θ = atan2(y, x) The reverse, shown in the readout under the graph.
- Polar area A = ½ ∫ r² dθ The exact form of what the shoelace estimate approximates.
- Parametric arc length L = ∫ √( (dx/dt)² + (dy/dt)² ) dt Estimated here by summing the straight segments between plotted points.
- Cycloid x = a(t − sin t), y = a(1 − cos t) The path traced by a point on a rolling wheel.
Why does my rose have the wrong number of petals?
Count again over a full 2π. For r = cos(kθ), odd k draws k petals because the second half of the sweep retraces the first; even k draws 2k because the negative-r half lands in the gaps. Both are correct — it is the geometry, not the plotter.
Can I plot two curves at once?
Not on this page — one curve at a time keeps the tracer and the measurements unambiguous. For comparing several functions side by side, the graphing calculator handles six at once.
Why is the arc length slightly short?
Because it sums straight lines between plotted points, and a straight line is always shorter than the curve it cuts across. Raising the point count closes the gap quickly — the error falls with the square of the step size.
What is the dashed line to the origin?
The radius vector at the tracer’s position. In polar mode it is literally the r you typed; in parametric mode it is just the distance from the origin to the current point.