Where a line meets a plane

Parametric equations

The line through $P_0$ with direction $\mathbf{d}\ne 0$:

The plane with normal vector $\mathbf{n}\ne 0$:

Substituting gives the parameter of the line-plane intersection:

If $\mathbf{n}\cdot \mathbf{d}=0$, the line is parallel to the plane, or the whole line lies in the plane. When a line is fixed by two points $P_0,P_1$, the direction is $\mathbf{d}=\overrightarrow{P_0{P}_1}$.

Interaction

• Intersects at a point / Parallel, no intersection / Line lies in the plane: switch among the three presets and compare $\mathbf{n}\cdot \mathbf{d}$ with $\mathbf{n}\cdot {\mathbf{r}}_0-h$
• Plane tilt: rotate the normal $\mathbf{n}$ and watch when $\mathbf{n}\cdot \mathbf{d}$ passes through $0$, so the intersection disappears
• Plane offset h: translate the plane and watch the intersection slide along the line. The sign of $t$ says which side of the start it falls on
• Line elevation: change the direction of $\mathbf{d}$ and compare an oblique meeting with a parallel line
• Drag the figure to rotate the view and check that the intersection lies on both the line and the plane. Horizontal view and Elevation do the same rotation

What to notice

• When $\mathbf{n}\cdot \mathbf{d}\ne 0$ there is one intersection. The sign of $t$ says which side of the start it is on.
• When $\mathbf{n}\cdot \mathbf{d}=0$ and $\mathbf{n}\cdot {\mathbf{r}}_0\ne h$, the line is parallel to the plane and does not meet it.
• When $\mathbf{n}\cdot \mathbf{d}=0$ and $\mathbf{n}\cdot {\mathbf{r}}_0=h$, the whole line lies in the plane.

Related

• Spatial vector and three plane projections
• Plane normal and distance from a point
• Spatial vectors, planes, and lines

Further reading

• Line–plane intersection (Wikipedia)
• Euclidean plane (Wikipedia)