Why Do 3D Coordinates Include Four Parameters?

[marensi]

hello EveryBody,

In the 3D Coordinates I always find 4 parameters instead of 3.

A = (X, Y, Z, 1)

I wonder why?

thank you.

 

[micromass]

Please provide more information. We can't answer your question like this.

 

[HallsofIvy]

You don't. What you have there is NOT "3 D coordinates" but, rather, "3 D projective coordinates" (the name coming from "projective geometry" that I won't go into). In "projective coordinates" the point (x, y, z) is represented by (x, y, z, 1) with the understanding that if any calculation changes that last coordinate to something other than 1, say, a (and a cannot be 0), then we interpret (x, y, z, a) as meaning (x/a, y/a, z/a, 1).

Projective coordinates are often used in computer graphics because they have the property that translations, as well as rotations, can be be written as matrix multiplications.

In ordinary 3D coordinates, a rotation, by angle \theta around the x-axis, is given by
\begin{bmatrix}cos(\theta) & -sin(\theta) & 0 \\ sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}
with obvious changes for rotations around the y and z axes and all rotations given by products of such matrices.

In "projective coordinates" such a rotation would be just
\begin{bmatrix}cos(\theta) & -sin(\theta) & 0 & 0 \\ sin(\theta) & cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z \\ 1\end{bmatrix}

The translation, that moves (x, y, z, 1) to (x+ a,y+ b, z+ c, 1), in projective coordinates, is given by
\begin{bmatrix}1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z \\ 1\end{bmatrix}