\documentclass[a4paper,twocolumn]{article}

\title{Toribash Head Texture Vertical Pixel Stretch Factor}
\author{Anthony Cox}

\usepackage{amsmath,amsthm}

\providecommand{\abs}[1]{\lvert#1\rvert}

\usepackage[british]{babel}

\usepackage[pdfusetitle]{hyperref}

\begin{document}

\twocolumn[
	\begin{@twocolumnfalse}
		\maketitle
		\begin{abstract}
			When you map a planar texture onto a sphere, the
			pixels are stretched vertically with more
			distortion towards the poles of the spheres.
			This document describes a formula to calculate
			the vertical stretching of the pixels.

			This document was written as a memorandum for a
			friend and not as a definitive piece of
			mathematical work. Use of the content of this
			document is entirely at your own risk.
		\end{abstract}
	\end{@twocolumnfalse}
]

\section{Introduction}
\label{sec:introduction}

The formulas and methods described in this document relate to head
textures in the game Toribash. As such, they refer to planar textures of
128px x 128px mapped onto a sphere. These formulas can be used for other
texture sizes, but changes will have to be made for the constant values
in them.

\subsection{Motivation}
\label{subsec:motivation}

Through the use of these algorithms it is possible to more accurately
map head textures for use in Toribash.

\section{Formulas}
\label{sec:formulas}

As the pixel stretch factor is the same along an entire row of pixels on
the planar texture, it is possible to model with a circle rather than a
sphere for the sake of simplicity.

The x coordinate of a point on the circumference of a circle with the y
coordinate corresponding with the row of pixels we are interested in on
the planar texture first needs to be calculated. A simple application of
pythagoras' theorem will yield the result since the radius of the circle
is known. The $y$ value used in \autoref{eqn:pythagoras} is measured in
texture coordinates (0 at the top) rather than from the origin of the
circle.

\begin{equation}\label{eqn:pythagoras}
	\begin{split}
		h^2 &= x^2 + y_1^2\\
		x &= \sqrt{h^2 - y_1^2}\\
		&= \sqrt{64^2 - (64 - y)^2}
	\end{split}
\end{equation}

The angle of the gradient at this point on the sphere can be used for
the pixel stretch factor and consists of calculating the derivative of
the circle.

\begin{equation}\label{eqn:derivative}
	\begin{split}
		\frac{dy}{dx} &= \frac{a-x}{y-b}\\
		&= \frac{-x}{y - 64}
	\end{split}
\end{equation}

The derivative from \autoref{eqn:derivative} is now made to correspond
to the amount of vertical stretching applied to the pixels when mapping
to the sphere simply by calculating the inverse.

\begin{equation}\label{eqn:stretch}
	\delta = \frac{1}{\abs{\frac{dy}{dx}}} + 1
\end{equation}

$\delta$ is the pixel stretch factor at the given y coordinate. This
value ranges from $1$ at the equator to $\infty$ at the pole of the
sphere. At $45^\circ$ the stretch factor will be 2 and corresponds to a
doubling in the height of the pixels when viewed from a point collinear
with the poles of the sphere rather than from a point which lies on the
equatorial plane.

The given formulas have a range of $0 \leq y < 64$. For $64 < y \leq 128$
apply the function $y = 128 - y_1$. When $y = 64$, $\delta \equiv 1$


\end{document}