Line Coordinates - Formulas

Formulas

The intersection of the lines (l1, m1) and (l2, m2) is the solution to the linear equations

By Cramer's rule, the solution is

The lines (l1, m1), (l2, m2), and (l3, m3) are concurrent when the determinant

\begin{vmatrix} l_1 & m_1 & 1 \\ l_2 & m_2 & 1 \\ l_3 & m_3 & 1 \end{vmatrix}=0.

For homogeneous coordinates, the intersection of the lines (l1, m1, n1) and (l2, m2, n2) is

The lines (l1, m1, n1), (l2, m2, n2) and (l3, m3, n3) are concurrent when the determinant

\begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix}=0.

Dually, the coordinates of the line containing (x1, y1, z1) and (x2, y2, z2) are

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