Welcome to the PF.Worn_Out_Tools said:
Wait, I don’t understand, how’d you determine that?etotheipi said:
There are two ways you can think about it. Consider the function ##r = 6\cos{(3\theta)}##. If we consider some angle ##\theta_0## where ##r(\theta_0) = r_0##, then we see that$$r \left(\theta_0 + \frac{2\pi}{3} \right) = 6\cos{\left(3(\theta_0 + \frac{2\pi}{3})\right)} = 6\cos{(3\theta_0 + 2\pi)} = 6\cos{(3\theta_0)} = r(\theta_0) = r_0$$which means that both ##\theta_0## and ##\theta_0 + \frac{2\pi}{3}## correspond to equal radial coordinates. The other way to look at it is what I think @berkeman was suggesting, i.e. consider transforming the original function by a constant translation,$$r_2(\theta) := r(\theta + \frac{2\pi}{3})$$so ##r_2## is shifted left of ##r## by the amount ##\Delta \theta = -\frac{2\pi}{3}##. But the periods of both ##r## and ##r_2## are equal to ##\frac{2\pi}{3}##, so we've just shifted the waveform left an entire period. Since ##r## and ##r_2## are then equal everywhere, we have$$r(\theta) = r_2(\theta) = r(\theta + \frac{2\pi}{3})$$And finally, what amounts to a horizontal shift in ##r## vs ##\theta## on Cartesian axes corresponds to a rotation in polar coordinates!Worn_Out_Tools said:
Polar coordinates are a mathematical system used to describe the position of a point in a two-dimensional space. They use a distance from the origin (r) and an angle (θ) to locate a point.
Polar coordinates can be used to describe the shape of a rose petal by plotting points along the polar graph. The distance from the origin (r) represents the length of the petal, while the angle (θ) determines the direction and curvature of the petal.
The equation for a rose petal in polar coordinates is r = a + bcos(nθ), where a and b are constants that determine the size and shape of the petal, and n is the number of petals.
Polar coordinates allow us to easily see the symmetry of a rose petal by showing how the petal is symmetric about the origin. This is because the distance from the origin (r) and the angle (θ) are mirrored on either side of the petal's axis of symmetry.
Yes, polar coordinates can be used to describe a variety of natural shapes, such as flower petals, shells, and spiral galaxies. They are also commonly used in engineering and physics to describe circular and rotational motion.