17. 3D Transformations - 1
Dated: 13-05-2025
Distance of 2 3D Points
\[d = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}\]
Parametric Equation of a line
Let \(P_1\) and \(P_2\) be 2 end points on a line1 then
\[P = P_1 + k \times (P_2 - P_1)\]
Where \(P_2 - P_1\) can be thought as a vector2 where \(k \in [0, 1]\) scales it onto a point \(P\), lying between \(P_1\) and \(P_2\).
Homogeneous Coordinates
\[
\begin{bmatrix}
x^\prime \\
y^\prime \\
z^\prime \\
1
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 & t_x \\
0 & 1 & 0 & t_y \\
0 & 0 & 1 & t_z \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x\\
y\\
z\\
1
\end{bmatrix}
\]