Bezier-Approximation-Plus

Applied ideas on using Bezier curves & hybrid tetrations to approximate & fit curves in data. Authored by Onri Jay Benally.

Bezier curves, being the useful geometric tools that they are, can be described by a Bernstein basis polynomial. They can be modified to follow objects that bend using hidden control handles and anchor points placed on an existing curve of interest. With that in mind, I thought of the introducing the polynomial for the Bezier curve with a tetration to form a hybrid approach which may compensate for very sharp and large changes in curve data as well. Some interesting results are provided in this repository. Check files for the code provided in a Google Colab notebook.

Click here to render the notebooks in the browser:

Defining What a Bezier Curve is Doing Mathematically

Concept	Equation / Explanation
General Bézier Curve	$B(t) = \sum_{i=0}^{n} B_i^n(t) P_i$
Bernstein Basis Polynomial	$B_i^n(t) = \binom{n}{i} (1-t)^{n-i} t^i$
Binomial Coefficient	$\binom{n}{i} = \frac{n!}{i!(n-i)!}$
Curve Properties	- $B(t)$ represents the position on the curve for $t \in [0,1]$. - $P_i$ are the control points that influence the shape. - The curve starts at $P_0$ and ends at $P_n$. - The shape is controlled by the intermediate points $P_1, P_2, \dots, P_{n-1}$.
Linear Bézier Curve ($n = 1$)	Straight line between two points.
Quadratic Bézier Curve ($n = 2$)	$B(t) = (1-t)^2 P_0 + 2(1-t)t P_1 + t^2 P_2$
Cubic Bézier Curve ($n = 3$)	$B(t) = (1-t)^3 P_0 + 3(1-t)^2 t P_1 + 3(1-t)t^2 P_2 + t^3 P_3$
Applications	Cubic Bézier curves are commonly used in computer graphics and font design.

Roles of Anchor Points and Control Points

Term	Definition
Anchor Point	A point that lies directly on the Bézier curve and determines its start and end positions. For example, in a cubic Bézier curve, the first and last points are anchor points.
Control Point	A point that influences the curve's shape but does not necessarily lie on the curve itself. These act as "handles" that pull the curve towards them, affecting its direction and curvature.

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Key Differences Between Anchor and Control Points

Aspect	Anchor Point	Control Point
Definition	A point that lies on the curve and marks its start or end.	A point that influences the curve's shape but does not necessarily lie on it.
Function	Defines the endpoints of the curve (or intermediate points in composite curves).	Determines the direction and curvature of the curve.
Presence in Quadratic Bézier Curve	2 anchor points	1 control point
Presence in Cubic Bézier Curve	2 anchor points	2 control points
Higher-Degree Bézier Curves	Typically 2 anchor points (unless part of a composite curve).	The number of control points is one more than the curve’s degree.

Example of a Bezier Curve with 6 Anchor Points

Component	Details
Curve Definition	A Bézier curve with 6 anchor points is a fifth-degree (quintic) Bézier curve because the number of control points ($n+1$) determines the degree ($n$).
Control Points	$P_0, P_1, P_2, P_3, P_4, P_5$
Parametric Equation	$$B(t) = \sum_{i=0}^{5} B_i^5(t) P_i$$
Bernstein Polynomial	$$B_i^5(t) = \binom{5}{i} (1-t)^{5-i} t^i$$
Binomial Coefficient	$$\binom{5}{i} = \frac{5!}{i!(5-i)!}$$
Expanded Equation	$$B(t) = (1-t)^5 P_0 + 5(1-t)^4 t P_1 + 10(1-t)^3 t^2 P_2 + 10(1-t)^2 t^3 P_3 + 5(1-t)t^4 P_4 + t^5 P_5$$
Parameter Range	$t \in [0, 1]$

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Towards Hybrid Bezier Curves for Approximation

Concept	Equation / Explanation
Polynomial Definition	A polynomial consists of a series of terms involving powers of a variable, typically expressed as: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ where the exponents are added sequentially.
Tetration Definition	Tetration is a form of repeated exponentiation, written as: $^n a = a^{a^{a^{\dots}}}$ where the exponentiation stacks instead of adding.
Comparison to a Series	- A series consists of a sum of terms. - A polynomial is a finite sum of powers of $x$. - Exponentiation is an iterative multiplication operation. - Tetration is an iterative exponentiation operation.
Growth Difference	Unlike a polynomial, a tetration does not consist of a sum of terms; instead, it is an iterated power tower, which grows much faster.
Can Tetration Be Expressed as a Series?	Tetration does not naturally expand into a power series like a polynomial. However, in some cases, it can be approximated using: - Logarithmic expansions (breaking it down via $a^{a^{a^x}}$). - Power series representations (like Taylor series) for small values. But in general, tetration does not behave like a polynomial series because it is based on hierarchical exponentiation rather than summation.

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Concept	Equation / Explanation
Hybrid Polynomial-Tetration Possibility	A polynomial power series can be appended or modified as a hybrid with tetration, depending on how the two mathematical structures are combined.
1. Direct Summation (Appending a Tetration Term)	A tetration term is added to a polynomial power series: $H(x) = \sum_{n=0}^{\infty} a_n x^n + c \cdot {}^m x$ where: - $\sum_{n=0}^{\infty} a_n x^n$ is a traditional polynomial or power series, - ${}^m x$ is the tetration term, - $c$ is a scaling coefficient. Blends polynomial growth with tetration's extreme growth.
2. Recursive Hybridization (Tetration Within a Polynomial)	Instead of adding tetration separately, we embed it into the polynomial: $H(x) = a_n ({}^m x)^n + a_{n-1} ({}^m x)^{n-1} + \dots + a_1 ({}^m x) + a_0$ Amplifies the polynomial’s growth through tetration.
3. Series Expansion Involving Tetration (Power Series Approximation)	For small $x$, tetration can be approximated using a Taylor or power series expansion: ${}^m x = e^{x + x^2 + \frac{x^3}{3} + \dots}$ This allows for: $H(x) = \sum_{n=0}^{\infty} b_n ({}^m x)^n$ where $b_n$ are coefficients to moderate tetration’s extreme growth.
4. Logarithmic Transformation (Taming Tetration Growth)	To prevent tetration from dominating a polynomial, we introduce logarithmic damping: $H(x) = \sum_{n=0}^{\infty} a_n x^n + d \log({}^m x)$ Controls tetration's rapid growth by applying a logarithm.
Challenges of Hybridizing a Polynomial with Tetration	1. Growth Rate Disparity: Tetration grows much faster than polynomial terms. Scaling is necessary. 2. Analytic Continuation Issues: Tetration is not always well-defined for non-integer heights, requiring super-exponential extensions. 3. Computational Stability: Tetration grows hyper-exponentially, which can cause numerical instability.
Conclusion	A hybrid polynomial-tetration function is possible with different formulations depending on the desired properties: - Controlled growth: Use logarithmic damping or power series approximations. - Ultra-fast growth: Use direct summation or embed tetration inside a polynomial.

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Examples of the Hybrid Bezier Approximation Applied to RF/MW Resonance Peaks