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src/content/blog/computer-graphics.md

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+---
+title: Computer Graphics - An Overview
+pubDate: 2025-02-06T19:00:00Z
+description: A brief overview of Computer Graphics
+---
+
+# Basics
+Before we dive into the details of computer graphics, let us talk about some basic concepts all around light.
+
+## Plenoptic Function
+The plenoptic function describes all light rays at all points in space at any point in time.
+
+![Plenoptic Function](/images/posts/computer-graphics/plenoptic_function.svg)
+
+It operates in a 6-dimensional space consisting of
+- Position: $V_x, V_y, V_z$
+- Direction: $\Phi, \Theta$
+- Time: $t$
+
+In photography and film the time is fixed at each frame and the position is either fixed or follows from the time. Computer graphics often simulates that.
+
+So the problem is reduced to a 2-dimensional space consisting simply of the directions $\Theta$ and $\Phi$.
+
+## Global Illumination
+$$
+L(x, \omega) = L_e(x, \omega) + \int_\Omega f_r(x, \omega', \omega) L(x', \omega') cos(\omega', n) d\omega'
+$$
+
+- $L(x, \omega)$
+: describes the light from an object $x$ in direction $\omega$
+- $L_e(x, \omega)$
+: light emission if x is a light source
+- $f_r(x, \omega', \omega)$
+: reflection function given an object $x$ and an incoming angle $\omega'$ and outgoing angle $\omega$.
+- $cos(\omega', n)$ weakening factor; the greater the distance between the incident angle $\omega'$ away from the surface normal $n$, the greater the area that is lit and the weaker the light at a particular point
+
+# Graphics Pipeline
+
+# Geometry
+We build geometry out of two distinct constructs: **points** and **vectors**.
+
+![Points and Vectors](/images/posts/computer-graphics/points_vectors.svg)
+
+**Points** are coordinates with respect to a coordinate origin  
+**Vectors** are movement statements from a point P to a point Q
+
+## Vector Space $\mathbb{R}^n$
+Contains **scalars** and **vectors** with...
+
+### Vector Addition
+$$
+\begin{bmatrix}
+  v_1 \\
+  v_2 \\
+  \vdots \\
+  v_n
+\end{bmatrix}
++
+\begin{bmatrix}
+  w_1 \\
+  w_2 \\
+  \vdots \\
+  w_n
+\end{bmatrix}
+=
+\begin{bmatrix}
+  v_1 + w_1 \\
+  v_2 + w_2 \\
+  \vdots \\
+  v_n + w_n 
+\end{bmatrix}
+$$
+
+### Scalar Multiplication
+$$
+a \cdot \begin{bmatrix}
+  v_1 \\
+  v_2 \\
+  \vdots \\
+  v_n
+\end{bmatrix}
+=
+\begin{bmatrix}
+  a \cdot v_1 \\
+  a \cdot v_2 \\
+  \vdots \\
+  a \cdot v_n
+\end{bmatrix}
+$$
+<br/>
+<br/>
+We encounter the following problems:
+- there are **no coordinates**
+- there are **no distances** or the like
+
+## Affine Space $A^n$
+Contains **scalars**, **vectors** and **points**.
+
+For each pair of points $(P, Q)$ with $P, Q \in A^n$ there is a vector $v \in V^n$ such that $v$ points from $P$ to $Q$.
+
+Elements of $A^n$ are **points**.
+
+$0$ is the coordinate origin and $v = (OP)$ is the location vector for $P \in A^n$.
+
+**Problem:** We still have no concept of lengths.