From 2f4de78325e53e8b4b4f3cb00b3d0d9e7f4870d3 Mon Sep 17 00:00:00 2001
From: John Tsiombikas <nuclear@member.fsf.org>
Date: Fri, 5 Apr 2024 08:27:49 +0300
Subject: [PATCH] started writing the first chapter

---
 gl1/setup.tex  |    2 ++
 gl1/spaces.tex |   39 +++++++++++++++++++++++++++++++++++++++
 2 files changed, 41 insertions(+)

diff --git a/gl1/setup.tex b/gl1/setup.tex
index 707f395..cbbc145 100644
--- a/gl1/setup.tex
+++ b/gl1/setup.tex
@@ -1 +1,3 @@
 \chapter{Setup \& hello world}
+
+% to setup na exei prota clear-color, kai meta ena trigono
diff --git a/gl1/spaces.tex b/gl1/spaces.tex
index 7940ef3..e86b560 100644
--- a/gl1/spaces.tex
+++ b/gl1/spaces.tex
@@ -1 +1,40 @@
 \chapter{Transformations \& coordinate systems}
+
+\section{Overview}
+
+In the previous chapter we saw how to use GLUT to create a window in any
+operating system, and then used OpenGL to fill it with a single color, and draw
+our first 2D triangle onto it. In this chapter we will enter the world of
+3-dimensions, and learn how to go from a single triangle, to rendering a
+polygonal object in 3D space. To do that we'll have to grasp the concept of
+coordinate systems, and how to transform vertices from one to another. OpenGL
+will help us quite a lot in this, with its \em{matrix stacks}, but in
+later chapters we will revisit the underlying mathematics and their
+implementation in more detail.
+
+\section{Mathematical fundamentals}
+
+The fundamental mathematical tools necessary to do anything in 3D graphics, are
+vectors, matrices, and the concept of coordinate systems from linear algebra.
+
+\subsection{Coordinate systems and vectors}
+
+A coordinate system represents a \em{space} which has a certain point as its
+\em{origin}, the point from which every measurement starts, the place where all
+coordinates are defined to be 0. It also has a number of basis vectors, defining
+the axes of the coordinate system. By stepping varying distances along each
+axis, we can reach any point in space. How much we step along each axis to reach
+a certain point is defined by a pair of numbers in 2-space, or a triplet of
+numbers in 3-space, or more generally an \em{n-tuple} for n-dimensional spaces.
+These tuples are what we call \em{vectors}.
+
+Consider the triangle we defined in the previous chapter. We arbitrarily used
+vectors $v_0 = (42, 42)$, $v_1 = (42, 42)$, and $v_2 = (42, 42)$  % TODO
+to
+define the positions in space of each vertex of the triangle. We chose these
+numbers arbitrarily, because they will result in a big clearly visible triangle
+in our case, due to the extends of the screen ranging from $-1$ to $1$ both
+horizontally and vertically. In essence we were dealing with a coordinate
+system, called \em{normalized screen space} or sometimes known by the initials
+\em{NDC} (normalized device coordinates), which has its origin at the center of
+our window, and basis vectors $i = (1, 0)$ and $j = (0, 1)$.
-- 
1.7.10.4