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Rose (mathematics)

Roses specified by the sinusoid r = cos() for various rational numbered values of the angular frequency k = n/d.
Roses specified by r = sin() are rotations of these roses by one-quarter period of the sinusoid in a counter-clockwise direction about the pole (origin). For proper mathematical analysis, k must be expressed in irreducible form.

In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728.[1]

General overview

Specification

A rose is the set of points in polar coordinates specified by the polar equation[2]

or in Cartesian coordinates using the parametric equations

Roses can also be specified using the sine function.[3] Since

.

Thus, the rose specified by r = a sin() is identical to that specified by r = a cos() rotated counter-clockwise by π/2k radians, which is one-quarter the period of either sinusoid.

Since they are specified using the cosine or sine function, roses are usually expressed as polar coordinate (rather than Cartesian coordinate) graphs of sinusoids that have angular frequency of k and an amplitude of a that determine the radial coordinate r given the polar angle θ (though when k is a rational number, a rose curve can be expressed in Cartesian coordinates since those can be specified as algebraic curves[4]).

General properties

Artistic depiction of roses with different parameter settings

Roses are directly related to the properties of the sinusoids that specify them.

Petals

Symmetry

All roses display one or more forms of symmetry due to the underlying symmetric and periodic properties of sinusoids.

Roses with non-zero integer values of k

The rose r = cos(4θ). Since k = 4 is an even number, the rose has 2k = 8 petals. Line segments connecting successive peaks lie on the circle r = 1 and will form an octagon. Since one peak is at (1,0) the octagon makes sketching the graph relatively easy after the half-cycle boundaries (corresponding to apothems) are drawn.
The rose specified by r = cos(7θ). Since k = 7 is an odd number, the rose has k = 7 petals. Line segments connecting successive peaks lie on the circle r = 1 and will form a heptagon. The rose is inscribed in the circle r = 1.

When k is a non-zero integer, the curve will be rose-shaped with 2k petals if k is even, and k petals when k is odd.[6] The properties of these roses are a special case of roses with angular frequencies k that are rational numbers discussed in the next section of this article.

The circle

A rose with k = 1 is a circle that lies on the pole with a diameter that lies on the polar axis when r = a cos(θ). The circle is the curve's single petal. (See the circle being formed at the end of the next section.) In Cartesian coordinates, the equivalent cosine and sine specifications are

and

respectively.

The quadrifolium

A rose with k = 2 is called a quadrifolium because it has 2k = 4 petals and will form a square. In Cartesian coordinates the cosine and sine specifications are

and

respectively.

The trifolium

A rose with k = 3 is called a trifolium[9] because it has k = 3 petals and will form an equilateral triangle. The curve is also called the Paquerette de Mélibée. In Cartesian Coordinates the cosine and sine specifications are

and

respectively.[10] (See the trifolium being formed at the end of the next section.)

The octafolium

A rose with k = 4 is called an octafolium because it has 2k = 8 petals and will form an octagon. In Cartesian Coordinates the cosine and sine specifications are

and

respectively.

The pentafolium

A rose with k = 5 is called a pentafolium because it has k = 5 petals and will form a regular pentagon. In Cartesian Coordinates the cosine and sine specifications are

and

respectively.

The dodecafolium

A rose with k = 6 is called a dodecafolium because it has 2k = 12 petals and will form a dodecagon. In Cartesian Coordinates the cosine and sine specifications are

and

respectively.

Total and petal areas

The total area of a rose with polar equation of the form r = a cos() or r = a sin(), where k is a non-zero integer, is[11]

When k is even, there are 2k petals; and when k is odd, there are k petals, so the area of each petal is πa2/4k.

Roses with rational number values for k

In general, when k is a rational number in the irreducible fraction form k = n/d, where n and d are non-zero integers, the number of petals is the denominator of the expression 1/21/2k = nd/2n.[12] This means that the number of petals is n if both n and d are odd, and 2n otherwise.[13]

The Dürer folium

A rose with k = 1/2 is called the Dürer folium, named after the German painter and engraver Albrecht Dürer. The roses specified by r = a cos(θ/2) and r = a sin(θ/2) are coincident even though a cos(θ/2) ≠ a sin(θ/2). In Cartesian coordinates the rose is specified as[17]

The Dürer folium is also a trisectrix, a curve that can be used to trisect angles.

The limaçon trisectrix

A rose with k = 1/3 is a limaçon trisectrix that has the property of trisectrix curves that can be used to trisect angles. The rose has a single petal with two loops. (See the animation below.)

Examples of roses r = cos() created using gears with different ratios.
The rays displayed are the polar axis and θ = π/2.
Graphing starts at θ = 2π when k is an integer, θ = 2 otherwise, and proceeds clockwise to θ = 0.

Roses with irrational number values for k

A rose curve specified with an irrational number for k has an infinite number of petals[18] and will never complete. For example, the sinusoid r = a cos(πθ) has a period T = 2, so, it has a petal in the polar angle interval 1/2θ1/2 with a crest on the polar axis; however there is no other polar angle in the domain of the polar equation that will plot at the coordinates (a,0). Overall, roses specified by sinusoids with angular frequencies that are irrational constants form a dense set (that is, they come arbitrarily close to specifying every point in the disk ra).

See also

Notes

  1. ^ O'Connor, John J.; Robertson, Edmund F., "Rhodonea", MacTutor History of Mathematics Archive, University of St Andrews
  2. ^ Mathematical Models by H. Martyn Cundy and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 73.
  3. ^ "Rose (Mathematics)". Retrieved 2021-02-02.
  4. ^ Robert Ferreol. "Rose". Retrieved 2021-02-03.
  5. ^ Xah Lee. "Rose Curve". Retrieved 2021-02-12.
  6. ^ Eric W. Weisstein. "Rose (Mathematics)". Wolfram MathWorld. Retrieved 2021-02-05.
  7. ^ "Number of Petals of Odd Index Rhodonea Curve". ProofWiki.org. Retrieved 2021-02-03.
  8. ^ Robert Ferreol. "Rose". Retrieved 2021-02-03.
  9. ^ "Trifolium". Retrieved 2021-02-02.
  10. ^ Eric W. Weisstein. "Paquerette de Mélibée". Wolfram MathWorld. Retrieved 2021-02-05.
  11. ^ Robert Ferreol. "Rose". Retrieved 2021-02-03.
  12. ^ Jan Wassenaar. "Rhodonea". Retrieved 2021-02-02.
  13. ^ Robert Ferreol. "Rose". Retrieved 2021-02-05.
  14. ^ Xah Lee. "Rose Curve". Retrieved 2021-02-12.
  15. ^ Xah Lee. "Rose Curve". Retrieved 2021-02-12.
  16. ^ Jan Wassenaar. "Rhodonea". Retrieved 2021-02-02.
  17. ^ Robert Ferreol. "Dürer Folium". Retrieved 2021-02-03.
  18. ^ Eric W. Weisstein. "Rose (Mathematics)". Wolfram MathWorld. Retrieved 2021-02-05.

External links