Rose Curves
Key Questions
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Answer:
The polar equation of a rose curve is either
r = a cos ntheta or r = a sin ntheta . The number of rose petals will be n or 2n according as n is an odd or an even integer. See explanation.Explanation:
Having seen that there were more than 1 K viewers in a day, I now add more.
The 2-D polar coordinates
P ( r, theta) , r =sqrt (x^2 + y^2 ) >= 0 . It represents length of the position vector< r, theta >. theta determines the direction. It increases for anticlockwise motion of P about the pole O. For clockwise rotation, it decreases. Unlike r,theta admit negative values.r-negative tabular values can be used by artists only.
The polar equation of a rose curve is either
r = a cos ntheta or r = a sin ntheta .n is at your choice. Integer values 2,, 3, 4.. are preferred for easy counting of the number of petals, in a period. n = 1 gives 1-petal circle.
To be called a rose, n has to be sufficiently large and integer + a fraction, for images looking like a rose. For integer values, the petals might be redrawn, when the drawing is repeated over successive periods.
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The period of bothsin ntheta and cos ntheta is2pi/n .The number of petals for the period
[0, 2pi/n] will be n or 2n ( including r-negative n petals ) according as n is odd or even, for0 <= theta <= 2pi . Of course, I maintain that r is length>=0 , and so non-negative. For Quantum Physicists, r > 0.Foe example, consider
r = 2 sin 3theta . The period is2pi/3 and the number of petals will be 3. In continuous drawing. r-positive and r-negative petals are drawn alternately. When n is odd, r-negative petals are same as r-positive ones. So, the total count here is 3.Prepare a table for
(r, theta) , in one period[0, 2pi/3] , fortheta = 0, pi/12, 2pi/12, 3pi/12, ...8pi/12 . Join the points by smooth curves, befittingly. You get one petal. You ought to get the three petals for0 <= theta <= 2pi. .For
r = cos 3theta , the petals rotate through half-petal angle =pi/6 , in the clockwise sense.A sample graph is made for
r = 4 cos 6theta , using the Cartesian
equivalent. It is r-positive 6-petal rose, for0 <=theta <=2pi .graph{(x^2+y^2)^3.5-4(x^6-15x^2y^2(x^2-y^2)-y^6)=0}
A. S. Adikesavan · 2 · Apr 11 2016
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Answer:
r = a sin(n theta) " or " r = a cos (n theta) Explanation:
Given: A rose curve
r = a sin(n theta) " or " r = a cos (n theta) ,where
a = "a constant that determines size" and if
n = "even" you'll get2n petalsand if
n = "odd" you'll getn petalsTo graph a rose curve on a graphing calculator:
select MODE, arrow down to FUNC, arrow over to POL ENTER
select
Y= and enter the following:r_1 = 8 sin 5 theta GRAPH=> graphs a 5 petaled roser_1 = 8 sin 4 theta GRAPH=> graphs a 8 petaled rose
marfre
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1
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Jul 20 2018
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A rose curve has an equation of the form
r=cos(n theta) ,where
n=1,2,3,...
**Examples **
Odd:
n=5 (5 petals)Even:
n=4 (8 petals)Remark: As shown above, if
n is odd, then a rose hasn petals, however, ifn is even, then a rose has2n petals.
I hope that this was helpful.
Wataru
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2
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Dec 7 2014