Polar Grapher
A polar curve gives the radius r as a function of the angle θ. Enter r(θ) and a θ range; the calculator samples the radius, converts each (r, θ) to Cartesian coordinates via (r·cos θ, r·sin θ), and plots the result.
Curves organised by rotation
Some curves are awkward in Cartesian coordinates and effortless in polar ones. A circle centred on the origin needs a square root and two branches as y = f(x); in polar form it is r = constant. A four-petalled flower has no Cartesian expression anyone would want to write, and in polar form it is one cosine.
The trade is that the describing variable becomes an angle rather than a horizontal position. This grapher plots r as a function of that angle over a range you choose.
Polar coordinates suit anything whose natural symmetry is rotational. Petals, spirals and shapes with a distinguished centre all become short expressions, because sweeping the angle is exactly the motion the curve is built around.
The plot is drawn with equal scaling for the same reason. A polar curve's symmetry is its defining feature, and stretching one axis relative to the other would destroy precisely the property that made polar form worth using.
How to use this calculator
- Enter the radius as a function of the angle Either θ or the word theta is accepted for the variable, and both are rewritten internally to a single ASCII letter before parsing.
- Set the angle range Two comma-separated numbers, in radians. A full turn is roughly 6.28; some curves need more than one turn to close and others close in less.
- Read the conversion note It states the transformation applied to every sampled point. The plot is Cartesian underneath — the polar description is converted, not drawn directly.
- Adjust the range if the curve looks incomplete Petals appear one at a time as the angle sweeps. A shape that seems to be missing a section usually needs a wider range.
How the curve is sampled and drawn
The expression is first normalised: any Greek theta or the spelled-out word is replaced by a plain letter the tokeniser can handle, and the result compiled against that variable. If compilation fails it is retried as a function of x, so an input written the other way still works.
The angle range is then divided into 720 equal steps — twice the density used for ordinary function plots, because polar curves frequently double back on themselves and a coarse sweep would visibly flatten the petals of a rose.
At each step the radius is evaluated and the point converted to Cartesian coordinates by multiplying the radius by the cosine and the sine of the angle. Every plotted point goes through that conversion; nothing is drawn in polar coordinates directly. A step where the radius comes out non-finite is skipped rather than ending the plot.
The window is then centred on the midpoint of the sampled extent, with the same half-width used in both directions and padded by fifteen per cent. Forcing a square window rather than fitting each axis independently is what keeps a circle circular and a rose symmetric.
What each input means
- r(θ) Radius function — form field “r(θ)”
- The distance from the origin as a function of the angle. It may be negative, which places the point on the opposite ray.
- [θMin, θMax] Angle range — form field “θ range”
- How far the angle sweeps, in radians. It controls how much of the curve is traced rather than the size of the window.
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.
A cardioid
One plus the cosine of the angle, swept through a full turn. The radius varies from two down to zero and back.
Inputs r(θ) = 1 + cos(θ), θ range = 0, 6.28
- Polar function r(θ) = 1 + cos(θ)
- Range θ ∈ [0, 6.28]
- Conversion Each polar point (r, θ) is plotted in Cartesian coordinates as (r·cos θ, r·sin θ).
Result Polar curve sampled at 721 points
The radius reaches its maximum at an angle of zero and falls to nothing at half a turn, which produces the characteristic dimple pointing back towards the origin. The heart shape is entirely the consequence of that single cosine.
The whole curve is traced in one turn and closes on itself. Extending the range further would simply retrace the same outline.
A rose with an odd petal count
The cosine of three times the angle. Multiplying the angle inside the cosine is what creates petals.
Inputs r(θ) = cos(3*θ), θ range = 0, 6.28
- Polar function r(θ) = cos(3*θ)
- Range θ ∈ [0, 6.28]
- Conversion Each polar point (r, θ) is plotted in Cartesian coordinates as (r·cos θ, r·sin θ).
Result Polar curve sampled at 721 points
Three petals appear rather than six, which is the surprise of odd multipliers. The radius goes negative for part of each cycle, and a negative radius plots on the opposite ray — placing those points on top of petals already drawn rather than beside them.
An even multiplier behaves differently: the negative excursions land in fresh directions and the petal count doubles. The parity of the multiplier, not its size alone, decides the shape.
A spiral
The radius equal to the angle itself, swept through two full turns. The curve never closes.
Inputs r(θ) = θ, θ range = 0, 12.56
- Polar function r(θ) = θ
- Range θ ∈ [0, 12.56]
- Conversion Each polar point (r, θ) is plotted in Cartesian coordinates as (r·cos θ, r·sin θ).
Result Polar curve sampled at 721 points
Each turn moves the curve steadily outward, since the radius grows in step with the angle. Consecutive turns are equally spaced, which is what distinguishes this spiral from ones that grow multiplicatively.
Doubling the range would add two more turns rather than retracing. This is the case where the angle range genuinely determines how much curve exists, not merely how much is drawn.
Reading the result
A negative radius is not an error
It places the point half a turn from the direction the angle names. That convention is what allows a single smooth expression to trace petals on both sides of the origin, and it is applied automatically.
How much of a turn a curve needs
Some close in one turn, some in two, and spirals never close at all. If the drawn shape looks truncated or lopsided, widening the range is the first thing to try.
Equal scaling is not cosmetic here
Rotational symmetry is the whole reason to use polar form, and unequal axes would break it — turning circles into ellipses and making petals of the same size look different. The square window preserves what the coordinate system was chosen for.
When you would use this
Drawing curves with rotational symmetry
Roses, cardioids, limaçons and lemniscates are all short polar expressions and long Cartesian ones. The coordinate system does most of the work.
Antenna and acoustic patterns
Directional response is naturally described as a strength at each angle, which is exactly a polar function. The plot then shows the pattern in the orientation it physically has.
Assumptions and limitations
What this calculator assumes
- The variable is the angle, written as θ or theta, measured in radians.
- The angle range is finite and sampled uniformly.
- Points where the radius is not finite are omitted rather than ending the plot.
- The window is square and centred on the curve, so both axes share a scale.
Where it stops being the right tool
- One curve at a time; two polar equations cannot be overlaid to find intersections.
- No area, arc length or tangent is computed from the polar form.
- Degrees are not accepted for the range — it is interpreted in radians throughout.
Common mistakes
Entering the range in degrees
Why it happens. Angles are commonly quoted in degrees, and a range of 0 to 360 looks like the natural way to ask for a full turn. It is read as 360 radians — nearly sixty turns.
How to avoid it. Use radians. A full turn is about 6.28, and the resulting plot is unmistakably different from an over-swept one.
Expecting the petal count to match the multiplier
Why it happens. A multiplier of three suggests three cycles of the cosine and therefore six petals, which is what happens for even multipliers but not odd ones.
How to avoid it. Check the parity. An odd multiplier gives that many petals; an even one gives twice as many, because the negative-radius excursions fall in different directions.
Treating a negative radius as invalid
Why it happens. A distance from the origin ought to be non-negative, so a formula producing negative values looks like it has gone wrong.
How to avoid it. Accept it as the standard convention. The point is plotted on the opposite ray, which is what produces several of the classic polar shapes.
Frequently asked questions
What can I use for the variable?
Either θ or the word theta — both are recognised and rewritten internally to a plain letter before the expression is compiled. If neither is found, the expression is retried against x, so an input written that way still works.
What does a negative radius mean?
The point is placed on the ray opposite the one the angle names, so a radius of −2 at some angle sits where a radius of 2 would be half a turn away. The conversion handles it automatically, and it is what produces the petals of an odd-order rose.
Why is the aspect ratio equal?
Because polar curves are built around rotational symmetry, and unequal axes destroy it — circles would be drawn as ellipses and petals of equal size would appear different. The window is made square and centred on the curve to preserve it.
How wide should the angle range be?
A full turn, roughly 6.28, closes most classic curves. Some need two turns, and spirals never close, so extending the range genuinely extends the curve rather than retracing it.