Plane–plane intersection
In geometry, the intersection of two planes in three-dimensional space is a line or the empty set for parallel planes. == Formulation == The line of intersection between two planes Π 1 : n 1 ⋅ r = h 1 {\displaystyle \Pi _{1}:{\boldsymbol {n}}_{1}\cdot {\boldsymbol {r}}=h_{1}} and Π 2 : n 2 ⋅ r = h 2 {\displaystyle \Pi _{2}:{\boldsymbol {n}}_{2}\cdot {\boldsymbol {r}}=h_{2}} where n i {\displaystyle {\boldsymbol {n}}_{i}} are normalized is given by r = ( c 1 n 1 + c 2 n 2 ) + λ ( n 1 × n 2 ) {\displaystyle {\boldsymbol {r}}=(c_{1}{\boldsymbol {n}}_{1}+c_{2}{\boldsymbol {n}}_{2})+\lambda ({\boldsymbol {n}}_{1}\times {\boldsymbol {n}}_{2})} where c 1 = h 1 − h 2 ( n 1 ⋅ n 2 ) 1 − ( n 1 ⋅ n 2 ) 2 {\displaystyle c_{1}={\frac {h_{1}-h_{2}({\boldsymbol {n}}_{1}\cdot {\boldsymbol {n}}_{2})}{1-({\boldsymbol {n}}_{1}\cdot {\boldsymbol {n}}_{2})^{2}}}} c 2 = h 2 − h 1 ( n 1 ⋅ n 2 ) 1 − ( n 1 ⋅ n 2 ) 2 .