Superposition de calques monochromes d'opacit\'es variables

TL;DR

This paper derives and analyzes a formula for the superposition of monochrome layers with variable opacities, providing a simplified theorem, its generalization, and validation against alternative formulas.

Contribution

It introduces a new, simplified theorem for superposing monochrome layers with variable opacities and explores its generalizations and alternative formulations.

Findings

Derived a standard formula for layer superposition with variable opacities.

Proved a simple theorem related to layer superposition.

Validated the theorem with alternative formulas.

Abstract

For a monochrome layer $x$ of opacity $0\le o_x\le1 $ placed on another monochrome layer of opacity 1, the result given by the standard formula is $$\small\Pi\left({\bf C}_\varphi\right)=1+\sum_{n=1}^2\left(2-n-(-1)^no_{\chi(\varphi+1)}\right)\left(\chi(n+\varphi-1)-o_{\chi(n+\varphi-1)}\right),$$ the formula being of course explained in detail in this paper. We will eventually deduce a very simple theorem, generalize it and then see its validity with alternative formulas to this standard containing the same main properties here exposed.