Vector Calculus: Infinity Logic Ray Calculus with Quasi-Quanta Algebra Limits (Rough Draft)

Golden Ratio Angle in Infinity Logic Ray Calculus with Quasi-Quanta Algebra Limits

This document explores the relationship between the golden ratio and ray tracing in the context of Infinity Logic Ray Calculus with Quasi-Quanta Algebra Limits (ILR-QAL). The paper focuses on two key aspects:

1. Densified Sweeping Subnet: This concept involves enriching the sweeping process by incorporating an additional factor into the definition of the sweeping subnet. This results in a denser and more detailed representation of the system dynamics.
2. Golden Ratio Angle: The analysis investigates the conditions under which rays reflected from an obstacle converge to a specific angle related to the golden ratio.

Key Findings:

• The densified sweeping subnet effectively captures the system dynamics and provides a richer representation compared to the traditional sweeping subnet.
• The golden ratio angle emerges under specific conditions related to the radius of the obstacle and the maximum sweep time.
• The analysis provides valuable insights into the behavior of rays in ILR-QAL and sheds light on the potential applications of this framework.

Structure of the Document:

• Introduction: Provides background information on ILR-QAL and outlines the research objectives.
• Densified Sweeping Subnet: Presents the concept of densified sweeping subnets and its construction.
• Golden Ratio Angle: Analyzes the conditions under which reflected rays converge to the golden ratio angle.
• Application: Discusses potential applications of the findings in various fields, such as self-organizing smart on-ramp platooning.
• Conclusion: Summarizes the main results and highlights future research directions.

Further Reading:

• The paper assumes basic knowledge of ILR-QAL and related mathematical concepts.
• References to previous works on densified sweeping subnets and the golden ratio in ray tracing are provided.

Overall, this document offers a valuable contribution to the field of ILR-QAL by exploring the densified sweeping subnet and its connection to the golden ratio. The findings have promising implications for various applications involving ray tracing and network inference.

We choose an arbitrary point~$X_i$ and define~$\vec{x}_i := \pi_{A_r}(X_i)$ and~$\vec{r}_i := \pi_{B_r}(X_i)$. Since $X_i \in A_r \oplus B_r$ we have $\pi_{A_r}(X_i) = X_i - \vec{n}(X_i)$, $\pi_{B_r}(X_j) = X_j + \vec{n}(X_j)$, and we obtain by the triangle inequality

\begin{equation}

\|\vec{r}_i-\vec{x}_i\| = \|X_i + \vec{n}(X_i) - X_i + \vec{n}(X_i)\| \leq 2\|\vec{n}(X_i)\| < 2\xi.

\end{equation}

By Lemma~\ref{lemma:Existence} there exists a lightlike curve from~$\vec{x}_i$ to~$\vec

$\left\{\left\langle\partial \theta \times \vec{r}_{\infty}\right\rangle \cap\left\langle\partial \vec{x} \times \theta_{\infty}\right\rangle\right\} \rightarrow \exists 1$; subnet.

Determining the radius~$r$ of the spheres $\mathcal{S}_r$ is more delicate. For a given radius $r>0$, we define the two submanifolds of $\partial\Omega$,

\begin{equation}

\begin{alignedat}{2}

A_r &:= \{\vec{x} \in \partial\Omega \,\colon\, \exists \theta \,\text{such that}\, \|\partial \theta \times \vec{r}\| \leq 2\xi,\, \|\partial \vec{x} \times \theta\| \leq 2\xi,\, \|\vec{r}-\vec{x}\| < r\},\\

B_r &:= \{ \vec{r} \in \partial\Omega \,\colon\, \exists \vec{x} \,\text{such that}\, \|\partial \theta \times \vec{r}\| \leq 2\xi,\, \|\partial \vec{x} \times \theta\| \leq 2\xi,\, \|\vec{r}-\vec{x}\| < r \},

\end{alignedat}

\end{equation}

for $\xi > 0$, the discretization parameter. We define the sweeping subnet of $\partial\Omega$ in terms of a well-behaved radius~$r$ by

\begin{equation}

\left\{ \left\langle\partial \theta \times \vec{r}_{\infty}\right\rangle \cap\left\langle\partial \vec{x} \times \theta_{\infty}\right\rangle\right\} \rightarrow \left\{ \left(A_r \oplus B_r\right) \cap \mathcal{S}_r^+\right\}.

\label{eq:DensifiedSweepingSubnetToS}

\end{equation}

We now determine the thickness of the intersection in~\eqref{eq:DensifiedSweepingSubnetToS}. Let~$X_i$ be an arbitrary point in~$A_r \oplus B_r \cap \mathcal{S}_r^+$ satisfying~$\|X_i-\vec{x}_i\| = r$. We define~$\vec{x}_i := \pi_{A_r}(X_i)$ and~$\vec{r}_i := \pi_{B_r}(X_i)$. Since $X_i \in A_r \oplus B_r$ we have $\pi_{A_r}(X_i) = X_i - \vec{n}(X_i)$, $\pi_{B_r}(X_j) = X_j + \vec{n}(X_j)$, and we obtain by the triangle inequality

\begin{equation}

\|\vec{r}_i-\vec{x}_i\| = \|X_i + \vec{n}(X_i) - X_i + \vec{n}(X_i)\| \leq 2\|\vec{n}(X_i)\| < 2\xi.

\end{equation}

Therefore, the intersection~$A_r \oplus B_r \cap \mathcal{S}_r^+$ has a maximal thickness~$\xi$, which is independent of~$r$.

We can now prove that a sequence of points~$\{X_i\} \in \left(A_r \oplus B_r\right) \cap \mathcal{S}_r^+$ always traces a ray, or a line segment if at least one point of~$\{X_i\}$ becomes light--like.

\begin{lemma}\label{lemma:RayIntegral}

If a sequence of points~$\{X_i\} \in \left(A_r \oplus B_r\right) \cap \mathcal{S}_r^+$ fulfills~$\forall i: X_{i+1} \neq X_i$ and~$\liminf \|X_{i-1}-X_i\| = 0$, then it is contained in a ray, or a line segment (case~$\limsup \|X_{i+1}-X_i\| = 0$). The line segment connects two points~$\vec{p}, \vec{q} \in \partial \Omega$.

\end{lemma}

\begin{proof}

We choose an arbitrary point~$X_i$ and define~$\vec{x}_i := \pi_{A_r}(X_i)$ and~$\vec{r}_i := \pi_{B_r}(X_i)$. Since $X_i \in A_r \oplus B_r$ we have $\pi_{A_r}(X_i) = X_i - \vec{n}(X_i)$, $\pi_{B_r}(X_j) = X_j + \vec{n}(X_j)$, and we obtain by the triangle inequality

\begin{equation}

\|\vec{r}_i-\vec{x}_i\| = \|X_i + \vec{n}(X_i) - X_i + \vec{n}(X_i)\| \leq 2\|\vec{n}(X_i)\| < 2\xi.

\end{equation}

By Lemma~\ref{lemma:Existence} there exists a lightlike curve from~$\vec{x}_i$ to~$\vec{r}_i$ contained in a sphere of radius~$r$ around~$\vec{r}_i$. Assuming~$r < \|\vec{r}_i - \vec{x}_i\|$ we obtain a contradiction, since there must be a point on this curve that surrounds $\vec{r}_i$ more closely than~$\vec{x}_i$.

\end{proof}

The lighter shade of Figure~\ref{fig:SweepingPaths} visualizes the union of the sweeping subnets defined in Equation~\ref{eq:DensifiedSweepingSubnetToS}. In particular, the line segments are rays that start from~$\vec{x}$, and the darker crosshairs on~$\mathcal{S}_r$ demonstrate the limitations of these rays in terms of maximum sweep time.