Polar Rose Curves: Equation, Petals and Math Flowers

A one-line equation can draw a flower. Plot the cosine of a multiple of an angle, but measure the result as a distance from the center instead of a height above a line, and the wave curls into petals. The Italian mathematician Guido Grandi studied these curves between 1723 and 1728 and named them rhodonea, after the rose¹. Below: the equation, the rule that counts the petals, a worked area, and the loop that draws one on a screen.

What is the polar equation for a rose curve?

A rose, or rhodonea curve, is a sinusoid with no phase angle, specified by either the cosine or the sine function and plotted in polar coordinates¹. The usual form is r = a cos(kθ), or r = a sin(kθ) for the sine version¹. Two numbers shape the flower. The constant a sets the size, because the rose is inscribed in the circle r = a and every petal tip lies on that circle¹. The constant k, the angular frequency, sets the number of petals¹.

The curve is a plain wave drawn in a different space. Rose curves are drawn using polar coordinates, not Cartesian ones⁴. The wave swings between positive and negative values, and the radius follows it out and back in.

Polar coordinates in one paragraph

The polar coordinate system is an alternative way of mapping points to ordered pairs⁵. The first coordinate is the radial coordinate and the second is the angular coordinate⁵. The center point is the pole, or origin, and the ray running to the right is the polar axis⁵. Positive angles are measured counterclockwise and negative angles clockwise⁵. A curve is graphed in the same way as in rectangular coordinates, from a function r = f(θ)⁵. Start from a list of values of θ, compute r for each, and plot the resulting ordered pairs⁵.

How many petals does a rose curve have?

Count by the type of k. Graphs of roses are composed of petals, and a petal is the shape formed by one half-cycle of the sinusoid¹.

The named whole values follow one pattern. A rose with k = 1 is a circle on the pole with its diameter on the polar axis, a single petal¹. The next ones are named for their petals¹:

Connecting the tips of the petals gives a regular polygon¹. Whole k gives petals that do not cross each other, and any other k gives petals that intersect¹.

Why do even and odd k give different counts?

Every petal has a tip where r reaches a or −a⁴. The peaks of the petals are the points furthest from the pole⁴. A point with negative radius is plotted on the opposite side of the pole, so a negative half-cycle can land on top of a positive one⁴¹.

Worked example (our arithmetic): peaks of the cosine rose over one full turn

For even k, the 8 peaks of k = 4 sit in 8 different directions, one petal each¹. For odd k, the negative peaks flip onto the directions of the positive ones, so the 6 peaks of k = 3 collapse into 3 petals¹. The negative-radius convention does the work: a point with negative radius is the same point as one with the same positive radius in the opposite direction from the pole⁴.

For odd k, the positive and negative half-cycles coincide, so only one kind needs plotting to form the full curve¹.

How to tell a sine rose from a cosine rose

The sine rose r = a sin(kθ) is the same rose as r = a cos(kθ), rotated counterclockwise¹. Both are the same curve turned, so the petal count is identical. The difference is the direction.

• The cosine rose is symmetric about the polar axis¹.
• The sine rose is symmetric about the vertical line through the pole¹.
• The cosine orientation gives a petal oriented to the right².

For a whole number k, the cosine and sine roses are never coincident¹. All petals of a rose are rotations of one petal about the pole¹, and the petals share one shape because each is a half-cycle of the same wave¹.

Fractions and irrational values of k

When k is a rational number, the curve is closed, meaning it eventually ends where it started⁵. When k is irrational, the curve never closes⁵. Roses with irrational k form a dense set, coming arbitrarily close to every point of the disk r ≤ a¹.

For a fraction k = n/d in lowest terms, the number of petals is n when n and d are both odd, and twice n otherwise¹.

Worked example (our arithmetic): petals for rational k

Each count was checked against a numeric plot of the curve (peaks on the circle r = a, grouped by direction), and all 4 agree with the rule (our arithmetic)¹.

Two fractions have names. A rose with k = 1/2 is the Dürer folium, a trisectrix, a curve that can be used to trisect angles¹. The curve with k = 3 is also called the Paquerette de Mélibée¹.

Worked example: how much area does a rose cover?

Counting petals is one way to size a rose, and area is another. The popular game of “he loves me, he loves me not” could be played on a rose by calculating its area rather than counting petals¹. OpenStax finds the area of one petal of the rose r = 3 sin 2θ⁶. Here is the same calculation set out in steps.

Worked example (our arithmetic): area of one rose, from the OpenStax setup

This rose covers exactly half of the circle it is inscribed in (our arithmetic)¹.

Make your own flower in code

A short program can draw these curves. The Ohio State project starts from one idea: make the radius depend on the angle⁷. Each pass through the loop moves θ forward by a small step and plots a point⁷. The shapes that come out look like roses, and some look like daisies⁷.

Worked example (our arithmetic): how long a full turn takes in that program

A larger step draws faster and a smaller one draws smoother, and the petal counts can be compared with the rule above.

Where did the name come from?

Grandi named the curves rhodonea because they looked like roses³. He was a member of the Camaldolite order, professor of philosophy at the University of Pisa from 1700 and professor of mathematics there from 1714³. His geometry work looked at analogies of the circle and the equilateral hyperbola³. The curves have fascinated mathematicians since⁴. Another name for them is Grandi’s rose, or the multifolium².

Where to go next

The polar plotting rules are in the open textbook sections on polar coordinates and area in polar coordinates. The history is at MacTutor, and the code project is at Ohio State.

The same sine and cosine waves power the other curves in the Math in the world section. Pi appears in every area above, as covered on what pi is. For another case where one scale factor changes shape, see how much Godzilla would eat, and for waves in a different setting, why power lines use high voltage.