Rose (mathematics)

Summary

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]

Roses specified by the sinusoid for various rational numbered values of the angular frequency k=n/d. Roses specified by 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, must be expressed in irreducible form.

General overviewEdit

SpecificationEdit

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   is identical to that specified by   rotated counter-clockwise by   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   and an amplitude of   that determine the radial coordinate   given the polar angle   (though when   is a rational number, a rose curve can be expressed in Cartesian coordinates since those can be specified as algebraic curves[4]).

General propertiesEdit

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

PetalsEdit

  • Graphs of roses are composed of petals. A petal is the shape formed by the graph of a half-cycle of the sinusoid that specifies the rose. (A cycle is a portion of a sinusoid that is one period   long and consists of a positive half-cycle, the continuous set of points where   and is   long, and a negative half-cycle is the other half where  .)
    • The shape of each petal is same because the graphs of half-cycles have the same shape. The shape is given by the positive half-cycle with crest at   specified by   (that is bounded by the angle interval  ). The petal is symmetric about the polar axis. All other petals are rotations of this petal about the pole, including those for roses specified by the sine function with same values for   and  .[5]
    • Consistent with the rules for plotting points in polar coordinates, a point in a negative half-cycle cannot be plotted at its polar angle because its radial coordinate   is negative. The point is plotted by adding   radians to the polar angle with a radial coordinate  . Thus, positive and negative half-cycles can be coincident in the graph of a rose. In addition, roses are inscribed in the circle  .
    • When the period  of the sinusoid is less than or equal to  , the petal's shape is a single closed loop. A single loop is formed because the angle interval for a polar plot is   and the angular width of the half-cycle is less than or equal to  . When   (or  ) the plot of a half-cycle can be seen as spiraling out from the pole in more than one circuit around the pole until plotting reaches the inscribed circle where it spirals back to the pole, intersecting itself and forming one or more loops along the way. Consequently, each petal forms 2 loops when   (or  ), 3 loops when   (or  ), etc. Roses with only one petal with multiple loops are observed for   (See the figure in the introduction section.)
    • A rose's petals will not intersect each other when the angular frequency   is a non-zero integer; otherwise, petals intersect one another.

SymmetryEdit

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

  • A rose specified as   is symmetric about the polar axis (the line  ) because of the identity   that makes the roses specified by the two polar equations coincident.
  • A rose specified as   is symmetric about the vertical line   because of the identity   that makes the roses specified by the two polar equations coincident.
  • Only certain roses are symmetric about the pole.
  • Individual petals are symmetric about the line through the pole and the petal's peak, which reflects the symmetry of the half-cycle of the underlying sinusoid. Roses composed of a finite number of petals are, by definition, rotationally symmetric since each petal is the same shape with successive petals rotated about the same angle about the pole.

Roses with non-zero integer values of kEdit

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

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

  • The rose is inscribed in the circle  , corresponding to the radial coordinate of all of its peaks.
  • Because a polar coordinate plot is limited to polar angles between   and  , there are   cycles displayed in the graph. No additional points need be plotted because the radial coordinate at   is the same value at   (which are crests for two different positive half-cycles for roses specified by the cosine function).
  • When   is even (and non-zero), the rose is composed of   petals, one for each peak in the   interval of polar angles displayed. Each peak corresponds to a point lying on the circle  . Line segments connecting successive peaks will form a regular polygon with an even number of vertices that has its center at the pole and a radius through each peak, and likewise:
    • The roses are symmetric about the pole.
    • The roses are symmetric about each line through the pole and a peak (through the "middle" a petal) with the polar angle between the peaks of successive petals being   radians. Thus, these roses have rotational symmetry of order  .
    • The roses are symmetric about each line that bisects the angle between successive peaks, which corresponds to half-cycle boundaries and the apothem of the corresponding polygon.
  • When   is odd, the rose is composed of the   petals, one for each crest (or trough) in the   interval of polar angles displayed. Each peak corresponds to a point lying on the circle  . These rose's positive and negative half-cycles are coincident, which means that in graphing them, only the positive half-cycles or only the negative half-cycles need to plotted in order to form the full curve. (Equivalently, a complete curve will be graphed by plotting any continuous interval of polar angles that is   radians long such as   to  .[7]) Line segments connecting successive peaks will form a regular polygon with an odd number of vertices, and likewise:
    • The roses are symmetric about each line through the pole and a peak (through the "middle" a petal) with the polar angle between the peaks of successive petals being   radians. Thus, these roses have rotational symmetry of order  .
  • The rose’s petals do not overlap.
  • The roses can be specified by algebraic curves of order   when k is odd, and   when k is even.[8]

The circleEdit

A rose with   is a circle that lies on the pole with a diameter that lies on the polar axis when  . 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 quadrifoliumEdit

A rose with   is called a quadrifolium because it has 4 petals. In Cartesian Coordinates the cosine and sine specifications are   and  , respectively.

The trifoliumEdit

A rose with   is called a trifolium[9] because it has 3 petals. 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.)

Total and petal areasEdit

The total area of a rose with polar equation of the form

  or  , where   is a non-zero integer, is
 , when   is even; and
 , when   is odd.[11]

When   is even, there are   petals; and when   is odd, there are   petals, so the area of each petal is  .

Roses with rational number values for kEdit

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

  • In the case when both   and   are odd, the positive and negative half-cycles of the sinusoid are coincident. The graph of these roses are completed in any continuous interval of polar angles that is   long.[14]
  • When   is even and   is odd, or visa versa, the rose will be completely graphed in a continuous polar angle interval   long.[15] Furthermore, the roses are symmetric about the pole for both cosine and sine specifications.[16]
    • In addition, when   is odd and   is even, roses specified by the cosine and sine polar equations with the same values of   and   are coincident. For such a pair of roses, the rose with the sine function specification is coincident with the crest of the rose with the cosine specification at on the polar axis either at   or at  . (This means that roses   and   with non-zero integer values of   are never coincident.)
  • The rose is inscribed in the circle  , corresponding to the radial coordinate of all of its peaks.

The Dürer foliumEdit

A rose with   is called the Dürer folium, named after the German painter and engraver Albrecht Dürer. The roses specified by   and   are coincident even though  . 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 trisectrixEdit

A rose with   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   created using gears with different ratios.
The rays displayed are the polar axis and  .
Graphing starts at   when   is an integer,   otherwise, and proceeds clock-wise to  .
 
The circle, k=1 (n=1, d=1). The rose is complete when   is reached (one-half revolution of the lighter gear).
 
The limaçon trisectrix, k=1/3 (n=1, d=3), has one petal with two loops. The rose is complete when   is reached (one and one-half revolution of the lighter gear).
 
The trifolium, k=3 (n=3, d=1). The rose is complete when   is reached (one-half revolution of the lighter gear).
 
The 8 petals of the rose with k=4/5 (n=4, d=5) is each, a single loop that intersect other petals. The rose is symmetric about the pole. The rose is complete at   (five revolutions of the lighter gear).

Roses with irrational number values for kEdit

A rose curve specified with an irrational number for   has an infinite number of petals[18] and will never complete. For example, the sinusoid   has a period  , so, it has a petal in the polar angle interval   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  . Overall, roses specified by sinusoids with angular frequencies that are irrational constants form a dense set (i.e., they come arbitrarily close to specifying every point in the disk  ).

See alsoEdit

NotesEdit

  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 linksEdit

Applet to create rose with k parameter

  • Visual Dictionary of Special Plane Curves Xah Lee
  • Interactive example with JSXGraph
  • Interactive example with p5