Quartic Dilation — Simpler With ‘Designer Ratios’

Author(s): Greg Oliver

Originally published on Towards AI.

A Simpler Way to Dilate Quartics: In One Radial MoveQuartic Dilation-In Red

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Often Polynomial vertical and horizontal dilation are treated separately rather than as a composite radial movement about a common origin.

While vertical dilation is easy to grasp with multiplication by a Scale Factor k of f(x)=kf(x), keeping Roots, Ip(x) and Tp(x) constant, using horizontal f(x/k) is less so. The formula presents as f(x)=k(A(x/k)⁴+C(x/k)²+D(x/k)+E).

Objective of This Post

This post arrives at the same underlying dilation formula with f(x)=(A/k³)x⁴+(C/k)x²+Dx+kE but by using genetic ‘Designer Ratios’ in the Polynomial architecture, it aims to further intuition facilitating function design for ML and robotics etc. The underlying premise being that when the Opposite and Adjacent sides of a Right Angle Triangle are both dilated by the same amount k, the Hypotenuse is similarly dilated by k. Hence the radial perspective.

Do a quick check with the well known 3, 4, 5 triangle using Pythagorus; @ k=1.1 hence: 3.3, 4.4 giving 5.5 result.

This post assumes math at the high School level.

First a Brief on Designer Ratios

These are architectural ratios that I promote as useful for function design, such as the various x² Coefficient C and x⁴ Coefficient A formulations shown in Graph 1 below with a… Read the full blog for free on Medium.

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Published via Towards AI