Representing lines: The Plücker relations

Recall from the Section 2 that the coordinates of the line passing through two points $\ensuremath{{\bf p}} _1=(X_1,Y_1,W_1)$ and $\ensuremath{{\bf p}} _2=(X_2,Y_2,W_2)$ is given by

$$\ensuremath{{\bf u}} = (Y_1W_2-W_1Y_2, W_1X_2-X_1W_2, X_1Y_2-Y_1X_2).$$

Notice that these three coordinates are just the determinants of the three $2 \times 2$ submatrices of the following matrix:

$$\left[\matrix{\ensuremath{{\bf p}} _1 & \ensuremath{{\bf p}} ... ... \left[\matrix{X_1 & X_2 \cr Y_1 & Y_2 \cr W_1 & W_2}\right],$$

taken in the appropriate order and given the appropriate sign.

The procedure is similar in . The coordinates of the line $\ensuremath{{\bf u}}$ passing through two points $\ensuremath{{\bf p}} _1=(X_1,Y_1,Z_1,W_1)$ and $\ensuremath{{\bf p}} _2=(X_2,Y_2,Z_2,W_2)$ is given by the determinants of the six $3 \times 3$ submatrices of the following matrix:

$$\left[\matrix{\ensuremath{{\bf p}} _1 & \ensuremath{{\bf p}} ... ...x{X_1 & X_2 \cr Y_1 & Y_2 \cr Z_1 & Z_2 \cr W_1 & W_2}\right].$$

In other words, $\ensuremath{{\bf u}} = (l_{41}, l_{42}, l_{43}, l_{23}, l_{31}, l_{12})$, where

$$\begin{eqnarray*}l_{41} & = & W_1X_2 - X_1W_2 \\ l_{42} & = & W_1Y_2 - Y_1W_2 \... ... l_{31} & = & Z_1X_2 - X_1Z_2 \\ l_{12} & = & X_1Y_2 - Y_1X_2. \end{eqnarray*}$$

These coordinates l_ij are called the Plücker coordinates of the line. It is fairly easy to show that, if the points $\ensuremath{{\bf p}} _1$ and $\ensuremath{{\bf p}} _2$ are not ideal (that is, W₁ and W₂ are not zero), then the coordinates have a nice Euclidean interpretation:

$$\begin{eqnarray*}(l_{41}, l_{42}, l_{43}) & = & \ensuremath{{\bf\bar p}} _2 - ... ... \ensuremath{{\bf\bar p}} _1 \times \ensuremath{{\bf\bar p}} _2, \end{eqnarray*}$$

where $\ensuremath{{\bf\bar p}} _i = \frac{1}{W_i}(X_i,Y_i,Z_i)$, i=1,2, are the coordinates of the corresponding Euclidean points. That is, the first three Plücker coordinates describe the direction of the line, and the last three coordinates describe the plane containing the line and the origin and the distance from the origin to the line. Therefore the six Plücker coordinates are sufficient to describe the line. The coordinates are not independent, however, because they always satisfy

l₄₁l₂₃+l₄₂l₃₁+l₄₃l₁₂=0,

which can be derived by noting that the $4 \times 4$ determinant $\vert\ensuremath{{\bf p}} _1, \ensuremath{{\bf p}} _2, \ensuremath{{\bf p}} _1, \ensuremath{{\bf p}} _2\vert$ is identically zero.

Where does the magic number six come from? That is, why do we need six parameters to represent a line in ? Interestingly, it turns out that it takes (ⁿ⁺¹_k) parameters to represent an entity defined by k points in a space requiring n+1 parameters for each point (To see this, count the number of submatrices in the matrix above). For example, in a point requires (³₁) = 3 parameters, and a line (which is defined by two points) also requires (³₂) = 3 parameters. In , a point requires (⁴₁) = 4 parameters, a line (⁴₂) = 6 parameters, and a plane (⁴₃) = 4 parameters.