\( C \) = Point defining the plane position
\( n \) = Vector normal to the plane
\( P \) = Point on the plane

\( L_{0} \) = Point on the line
\( \mathbf{v} \) = Vector that defines the line direction
Parametric equation of a point on the plane A generic point on the plane satisfies the equation: \[ \left( P \cdot \mathbf{\hat{n}} \right) = \left( C \cdot \mathbf{\hat{n}} \right) \] The length of the projection of the vector connecting the point to the space origin along the plane normal \( \mathbf{\hat{n}} \) must be equal to the distance of the plane from the space origin.

Parametric equation of the line A generic line defined by a point \( L_{0} \) and a direction vector \( \mathbf{v} \). It describes all the points along the line as a function of the parameter \( t \). \[ L_{\left( t \right)} = L_{0} + t \mathbf{v} \] Intersection calculation To calculate the intersection between plane and line, substitute the generic point \( P \) on the plane with a generic point on the line \( L_{ \left( t \right) } \). \[ \left[ \left( L_{0} + t \mathbf{v} \right) \cdot \mathbf{\hat{n}} \right] - \left( C \cdot \mathbf{\hat{n}} \right) = 0 \] Apply the distributive property of the dot product. \[ \left( L_{0} \cdot \mathbf{\hat{n}} \right) + \left( \mathbf{v} \cdot \mathbf{\hat{n}} \right) t - \left( C \cdot \mathbf{\hat{n}} \right) = 0 \] Solve for \( t \). \[ t = \frac{\left( C \cdot \mathbf{\hat{n}} \right) - \left( L_{0} \cdot \mathbf{\hat{n}} \right)}{\left( \mathbf{v} \cdot \mathbf{\hat{n}} \right)} \] Intersection points Depending on the values of numerator and denominator of the solution there can be one or no intersections: