View source for Rose (mathematics)

{{for|the topological usage|Rose (topology)}}
{{refimprove|date=May 2014}}
[[File:7 Petal rose.svg|thumb|200px|right|Rose with 7 petals (''k'' = 7)]]
[[File:8-Petal rose.svg|thumb|200px|right|Rose with 8 petals (''k'' = 4).]]
[[File:Rose-rhodonea-curve-7x9-chart-improved.svg|thumb|200px|right|Rose curves defined by 
<math>r=\cos k \theta</math>, for various values of ''k''=''n''/''d''.]]
<!-- [[File:7-leafed rose.gif|thumb|200px|right|Rose with ''k'' = 7 petals]] -->
<!-- [[File:8-leafed rose.gif|thumb|200px|right|Rose with 8 petals (''k''=4).]] -->

In [[mathematics]], a '''rose''' or '''rhodonea curve''' is a [[sine wave|sinusoid]] plotted in [[polar coordinates]].

== General overview ==
Up to [[Similarity (geometry)|similar]]ity, these curves can all be expressed by a polar equation of the form
:<math>\!\,r=\cos(k\theta)</math><ref>''Mathematical Models'' by [[Martyn Cundy|H. Martyn Cundy]] and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 73.</ref>
or, alternatively, as a pair of Cartesian parametric equations of the form
:<math>\!\,x=\cos(k\theta)\cos(\theta)</math>
:<math>\!\,y=\cos(k\theta)\sin(\theta)</math>
If ''k'' is an integer, the curve will be rose-shaped with
*2''k'' petals if ''k'' is even, and
*''k'' petals if ''k'' is odd.
When ''k'' is even, the entire graph of the rose will be traced out exactly once when the value of theta, ''θ'', changes from 0 to 2<math>\pi</math>.   When ''k'' is odd, this will happen on the interval between 0 and <math>\pi</math>.  (More generally, this will happen on any interval of length 2<math>\pi</math> for ''k'' even, and <math>\pi</math> for ''k'' odd.)

If ''k'' is a half-integer (e.g. 1/2, 3/2, 5/2), the curve will be rose-shaped with 4''k'' petals. Example: ''n''=7, ''d''=2, ''k''= ''n''/''d'' = 3.5, as ''θ'' changes from 0 to 4<math>\pi</math>.

If ''k'' can be expressed as ''n'' ± 1/6, where ''n'' is a nonzero integer, the curve will be rose-shaped with 12''k'' petals.

If ''k'' can be expressed as ''n''/3, where ''n'' is an integer not divisible by 3, the curve will be rose-shaped with ''n'' petals if ''n'' is odd and 2''n'' petals if ''n'' is even.

If ''k'' is [[Rational number|rational]], then the curve is [[Closed (mathematics)|closed]] and has finite length.  If ''k'' is [[Irrational number|irrational]], then it is not closed and has infinite length.  Furthermore, the graph of the rose in this case forms a [[dense set]] (i.e., it comes arbitrarily close to every point in the unit disk).

Since
:<math>\sin(k \theta) = \cos\left( k \theta - \frac{\pi}{2} \right) = \cos\left( k \left( \theta-\frac{\pi}{2k} \right) \right)</math> 
for all <math>\theta</math>, the curves given by the polar equations 
:<math>\,r=\sin(k\theta)</math> and <math>\,r = \cos(k\theta)</math>
are identical except for a rotation of <math>\pi</math>/2''k'' radians.

Rhodonea curves were named by the Italian mathematician [[Guido Grandi]] between the year 1723 and 1728.<ref>{{MacTutor Biography|class=Curves|id=Rhodonea|title=Rhodonea}}</ref>

==Area==
A rose whose polar equation is of the form
:<math>r=a \cos (k\theta)\,</math>
where ''k'' is a positive integer, has [[Polar coordinates#Integral calculus|area]]
:<math>
    \frac{1}{2}\int_{0}^{2\pi}(a\cos (k\theta))^2\,d\theta = \frac {a^2}{2} \left(\pi + \frac{\sin(4k\pi)}{4k}\right) = \frac{\pi a^2}{2}
</math>
if ''k'' is even, and
:<math>
    \frac{1}{2}\int_{0}^{\pi}(a\cos (k\theta))^2\,d\theta = \frac {a^2}{2} \left(\frac{\pi}{2} + \frac{\sin(2k\pi)}{4k}\right) = \frac{\pi a^2}{4}
</math>
if ''k'' is odd.

The same applies to roses with polar equations of the form
:<math>r=a \sin (k\theta)\,</math>
since the graphs of these are just rigid rotations of the roses defined using the cosine.

==How the parameter ''k'' affects shapes==
In the form ''k''&nbsp;=&nbsp;''n'', for integer ''n'', the shape will appear similar to a flower. If ''n'' is odd, half of these will overlap, forming a flower with ''n'' petals. However, if ''n'' is even, the petals will not overlap, forming a flower with 2''n'' petals.

When ''d'' is a prime number, then ''n''/''d'' is a least common form and the petals will stretch around to overlap other petals. The number of petals each one overlaps is equal to the how far through the sequence of primes this prime is + 1, i.e. 2 is 2, 3 is 3, 5 is 4, 7 is 5, etc.

In the form ''k''&nbsp;=&nbsp;1/''d'' when ''d'' is even, it will appear as a series of ''d''/2 loops that meet at 2 small loops at the center touching (0,&nbsp;0) from the vertical and is symmetrical about the ''x''-axis.
If ''d'' is odd, then it will have ''d''/2 loops that meet at a small loop at the center from either the left (when in the form ''d''&nbsp;=&nbsp;4''n''&nbsp;&minus;&nbsp;1) or the right (''d''&nbsp;=&nbsp;4''n''&nbsp;+&nbsp;1).

If ''d'' is not prime and ''n'' is not 1, then it will appear as a series of interlocking loops.

If ''k'' is an irrational number (e.g. <math>\pi</math>, <math>\sqrt{2}</math>, etc.) then the curve will have infinitely many petals, and it will be [[Dense set|dense]] in the unit disc.

{{multiple image
 | header = Examples of roses created using gears with different ratios
 | align = center

 | image1 = Rose Curve animation with Gears n1 d1.gif
 | width1 = 120
 | caption1 = ''n''=1, ''d''=1, ''k''=1.

 | image2 = Rose Curve animation with Gears n1 d3.gif
 | width2 = 120
 | caption2 = ''n''=1, ''d''=3, ''k''≈0.333

 | image3 = Rose Curve animation with Gears n3 d1.gif
 | width3 = 120
 | caption3 = ''n''=3, ''d''=1, ''k''=3

 | image4 = Rose Curve animation with Gears n4 d5.gif
 | width4 = 120
 | caption4 = ''n''=4, ''d''=5, ''k''=0.8
}}

==Offset parameter==
[[File:Rose offset anim.ogv|thumb|Animating effect of changing offset parameter]]
Adding an offset parameter ''c'', so the polar equation becomes
:<math>\!\,r=\cos(k\theta) + c</math>
alters the shape as illustrated at right. In the case where the parameter ''k'' is an odd integer, the two overlapping halves of the curve separate as the offset changes from zero.

==Programming==
[[R (programming language)|R]] 

<syntaxhighlight lang="OCTAVE">
k <- 4
t <- seq(0, 4*pi, length.out=500)
x <- cos(k*t)*cos(t)
y <- cos(k*t)*sin(t)
plot(x,y, type="l", col="blue")
</syntaxhighlight>

MATLAB and OCTAVE

<syntaxhighlight lang="OCTAVE">
function rose(del_theta, k, amplitude)
% inputs:
%   del_theta = del_theta is the discrete step size for discretizing the continuous range of angles from 0 to 2*pi
%   k = petal coefficient
%      if k is odd then k is the number of petals
%      if k is even then k is half the number of petals
%   amplitude = length of each petal
% outputs:
%   a 2D plot from calling this function illustrates an example of trigonometry and 2D Cartesian plotting
theta = 0:del_theta:2*pi;
x = amplitude*cos(k*theta).*cos(theta);
y = amplitude*cos(k*theta).*sin(theta);
plot(x,y)
</syntaxhighlight>
{{Clear}}

==See also==
*[[Lissajous curve]]
*[[quadrifolium]] – a rose curve where {{nowrap|1=''k'' = 2}}.
*[[Maurer rose]]
*[[Rose (topology)]]
*[[Spirograph]]

==Notes==
{{reflist}}

==External links==
{{commons category|Rose curves}}
* {{MathWorld|Rose|Rose}}
* [https://web.archive.org/web/20070325231158/http://www25.brinkster.com/denshade/PolarFlower.html Applet to create rose with k parameter]
* [http://xahlee.org/SpecialPlaneCurves_dir/Rose_dir/rose.html Visual Dictionary of Special Plane Curves] Xah Lee
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rose Interactive example with JSXGraph]

[[Category:Curves]]
[[Category:Articles containing video clips]]