====== Spirals ======

The equation for a spiral can be given by polar coordinates or by Cartesian coordinates.

In **polar coordinates** $r$ and $\theta$

$$ r = f(\theta)$$

where:\\
  * $\theta$ is an angle in radians, and 
  * $r$ is distance from the origin in units

An angle can be given in either degrees or radians.
1 radian = 57.29 degrees.

$$\pi = 3.141592653589793$$
$$2\pi r = \text{gives the circumference of a circle where r is the radius}$$
$$1 \text{rad} = \frac{360^{\circ}}{2 \pi} = 57.29^{\circ}$$
$$\text{a circle contains} 360^{\circ} \text{and} 2 \pi \text{rad}$$

$$e = 2.718281828459045$$

In python, $e^x$ can be expressed as math.e**x or math.exp(x)

n == math.log(math.exp(n))

logarithm is the inverse of exponent

log() is the inverse of exp()

x^a = log(a,x)

pow(2,5) == 2**5 == 32


**Cartesian coordinates** $x$ and $y$ can be derived from the polar coordinates as

$$x = r  \text{cos}  \theta$$
$$y = r \text{sin} \theta$$

Adding a $z$ dimension turns any spiral into a **helix**.
$$z = \theta$$

clockwise = right-handed
counter-clockwise = left-handed

as you approach a spiral staircase, look up and sight along the z-axis, the steps are going up clockwise, and you would reach out with your right-hand to grab the rail.


====Circle====

In Polar coordinates, $r$ is a constant.

In Cartesian coordinates, $x^{2} + y^{2} = r^{2}$ where r is a constant.

====Arithmetic Spiral====

aka **Archimedes** spiral.

In Polar coordinates, $r = a + b\theta$

In Cartesian coordinates, $f(r,\theta) = \theta$

====Logarithmic Spiral====

aka **Equiangular** spiral and **Bernoulli** spiral.

Often found in nature, as in the shape of the nautilus shell, the arrangement of sunflower seeds in the sunflower...


In Polar coordinates, $r = \theta$ \\  
"...it shall widen and lengthen in the same unvarying proportions."

In Cartesian coordinates, $f(r,\theta) = r^{a * \theta}$, 
where $a$ is between 0 and 1

====Parabolic Spiral====

aka **Fermat's** spiral.

In Polar coordinates, $r = a \sqrt{\theta}$ \\
where $\theta >= 0$

In Cartesian coordinates, $f(r,\theta) = a \sqrt{\theta}$ \\
where $\theta >= 0$

====Hyperbolic Spiral=====
aka **Fermat's** spiral.

In Polar coordinates, $r = \frac{a}{\theta}$ \\
where $\theta \neq 0$

In Cartesian coordinates, $f(r,\theta) = \frac{a}{\theta}$ \\
where $\theta \neq 0$