In investigating certain curves it can be more convenient to locate a
point by means of its polar coordinates. The polar coordinates give its
position relative to a fixed reference point O (the pole) and to a given
ray (the polar axis) beginning at O. The diagram to the left shows a standard
polar graph. Points are placed on this graph using a coordinate system where
(r,x) is the ordered pair that represents r = the distance from the origin
and x = the angle measured counterclockwise from the x axis and terminal
arm. An example would be (3,0) this would place a point on the third
ring out at an angle of 0. Another example would be (1,pi/2), this would
place a point directly above the origin on the first ring.
Equations in polar form are relationships between r and x. A few examples would be r = 3, or r = 3 cos(x) or r = 1 + 3 sin(2x).... These types of relationships seem to work out nicely in the polar plane because of the their heavy reliance on angles and radian measure. Below are a few basic examples of equations in polar form and their resulting graph.
r = 2 - all points that are a distance of 2 from the pole. | r = 2 sin(x) - a circle formed with a diameter of 2 | r = 2 cos(x) - a circle formed with a diameter of 2 | r = 2 cos(3x) - a 3 pedal shape is form that extends out to max. |
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for all values of x, r = 2 thus the circle. | r = 0 when x = 0, thus the orientation | r = 1 when x = 0, thus the orientation | value of r = 2. |
To investigate the "n- leaf roses" I will be looking at the equation r = a + b cos(kx) where r = the distance from the pole and x = the angle as it varies from 0 to 2 pi. a, b, and k are variables that I will at some times hold constant while at other times, manipulate depending on the circumstance which is to be developed. Before we can looking at how a graph changes with respect to our manipulations we must first understand the standard form of the graph.
r = 0 + 1cos(x) - a circle formed with a diameter of 1 | r = 0 + 3cos(x) - a circle formed with a diameter of 2 | r = 1 + 1cos(x) - a circle formed with a diameter of 1 | r = 2 + 1cos(x) - a circle formed with a diameter of 1 |
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As discussed earlier a right orientated circle appears with this equation. As b variable is manipulated we see the size of the the circle increasing by that factor. This is because the max. value of cos(x) = 1, thus 3*1 = 3 and the cos(pi/2) value = 0, thus 0*1 = 0. | (up left) A cardioid is formed from this equation. Notice the max. value is 2 = 1 + 1 (cos(0)) and the min. value is 0 = 1 - 1 (cos(pi)). (up right) Hopefully you can now extrapolate why the cardioid has taken on this shape. |
Some general finding that I will not investigate here concerning r = a + bcos(kx) are:
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I'm not going to go over why the max and min values occurred, hopefully that has been understand previously. The number of "leaves" is what becomes interesting, as k increases we see a direct relationship with the number of leaves formed. It is also interesting to notice the orientation difference between an even k and the odd k's. An even k value can have many axises of symmetry but x = pi/2 (y axis) will always be one of them whereas for an odd k, x = pi/2 (y axis) will never be an axis of symmetry as long as k > 1. |
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This relationship between a and b create a double leafing effect. The symmetry of these 3 leaf flowers is quite beautiful. I will stop my investigation here, although there is much more to discover and explore. |