$$C$$ = 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:

• if $$\left( \mathbf{v} \cdot \mathbf{\hat{n}} \right) \neq 0$$
There is one intersection at: $L_{int} = L_{0} + t \mathbf{v}$
• if $$\left( \mathbf{v} \cdot \mathbf{\hat{n}} \right) = 0$$
The line is parallel to the plane.

• if $$\left( P \cdot \mathbf{\hat{n}} \right) - \left( L_{0} \cdot \mathbf{\hat{n}} \right) \neq 0$$
The line does not lie in the plane, there are no intersections.

• if $$\left( P \cdot \mathbf{\hat{n}} \right) - \left( L_{0} \cdot \mathbf{\hat{n}} \right) = 0$$
The line lies in the plane, there are infinite intersections.