# Foundation

## Input: unit circle

The exercise deals with transformations of the unit circle, $$(\cos \theta, \sin \theta)$$. We start with the unit circle with Hue coloring over the angular range of $$(-\pi, \pi)$$ as seen below. The green shading denotes a region of right-handed chirality. (As you travel from $$-\pi$$ to $$\pi$$, the interior is on the left.)

## Transform in 2D

The transformation in two dimensions is $$\left( \begin{array}{c} \sin \theta\\ \cos \theta \end{array} \right) \Longrightarrow \frac{\sin \theta} {1 + \cos^2 \theta} \left( \begin{array}{c} 1\\ \cos \theta \\ \end{array} \right)$$

The transformation twists the circle into this racetrack figure:

The two lobes separate regions of right-handed chirality (green, left) and left-handed chirality (red, right). The point of the exercise is to show how the unit circle was twisted and mark the chirality of the connected domains.

## Transform in 3D

The transformation in three dimensions is $$\left( \begin{array}{c} \sin \theta\\ \cos \theta \end{array} \right) \Longrightarrow \frac{\sin \theta} {1 + \cos^2 \theta + \cos^4 \theta} \left( \begin{array}{c} 1\\ \cos \theta \\ \cos ^2 \theta \end{array} \right)$$

In Mathematica the transformation is cast as:

fields = Sin[\[Theta]]/(1 + Cos[\[Theta]]^2 + Cos[\[Theta]]^4) {1, Cos[\[Theta]],Cos[\[Theta]]^2};


This code snippet produces the plot, a gently rolling racetrack:

ParametricPlot3D[fields, {\[Theta], -\[Pi], \[Pi]},
ColorFunction -> (Hue[#4 0.8] &),
PlotRange -> {1.05 {-1, 1}, 1.05 {-1, 1}},
SphericalRegion -> True,
PlotStyle -> Thickness[0.01]]


# Question

How do I shade the two interior zones (left in red, right in green) bound by the curve?

#### Reference

This post looked promising, but ultimately, I could not use this to solve the problem.

Shading the surface of the 3D plot

f[θ_] :=
Sin[θ]/(1 + Cos[θ]^2 + Cos[θ]^4) {1,
Cos[θ], Cos[θ]^2};
ParametricPlot3D[
f[θ]*(1 - s) +
f[-θ + π] s, {s, θ} ∈
ImplicitRegion[{0 <= s <= 1,
0 <= θ <= π/2 ||
3 π/2 <=  θ <= 2 π}, {s, θ}],
ColorFunction -> (Hue[#4 0.8] &), SphericalRegion -> True]


The method can be illustrate by the follow figure,

Show[ParametricPlot3D[
f[θ]*(1 - s) + f[-θ + π] s, {s, 0, 1}, {θ,
0, π/2}, MeshFunctions -> {#1 &}, PlotStyle -> None,
MeshStyle -> Red],
ParametricPlot3D[f[θ], {θ, 0, π/2},
PlotStyle -> Green],
ParametricPlot3D[f[-θ + π], {θ, 0, π/2},
PlotStyle -> Cyan]]


we draw the f[θ], {θ, 0, π/2}(green),and reverse part of f[θ] by f[-θ + π], {θ, 0, π/2}( cyan),and then draw the red segment from green to cyan by f[θ]*(1 - s) + f[-θ + π]