Homogeneous Coordinates
Useful in Computer Graphics:
If we want to represent rotations and translations using matrices we can use the homogeneous coordinates. It changes translations in 3D to shear transformation in 4D.
\begin{align*} \begin{bmatrix} 1 & 0 & 0 & T_x \\ 0 & 1 & 0 & T_y \\ 0 & 0 & 1 & T_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{pmatrix} x \\ y \\ z \\ 1 \end{pmatrix} = \begin{pmatrix} x + T_x \\ y + T_y \\ z + T_z \\ 1 \end{pmatrix} \end{align*}Mathematics:
Given a point \((x, y)\) on a plane, and the homogeneous coordinate of the point is \((xZ, yZ, Z)\) for any non-zero real \(Z\). So, there are infinite coordinates for the same point.
Properties:
- Coordinate of points at infinity can be represented using finite coordinates
- Number of coordinates is one more that the projective space being considered
Homogeneous:
Since homogeneous coordinates of a point is not unique, any function \(f(x,y, z)\) doesn't necessarily specify a curve in those coordinates. However, when \(f\) is homogeneous, \(f = 0\) represents a curve in homogeneous coordinate system.
\(f\) is homogeneous when for some \(k\):
\[ f(\lambda x, \lambda y, \lambda z) = \lambda^k f(x, y, z) \]
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