3D Line-Plane Intersection and Angle Calculator

Plane Line Intersection in 3D — Figure 1. The intersection of a plane and a line

Mathematical Formulas

Line in 3D (parametric form):

\[ \mathbf{r}(t) = \mathbf{P_0} + t\mathbf{v} = (x_0, y_0, z_0) + t(l, m, n) \]

where \((l, m, n)\) are the direction numbers of the line.

Plane \( P \) equation:

\[ ax + by + cz = d \]

Intersection point: Substitute line into plane equation and solve for \(t\):

\[ t = \frac{d - (ax_0 + by_0 + cz_0)}{a\cdot l + b\cdot m + c\cdot n} \] \[ \text{Intersection point: } (x_0 + t\cdot l,\; y_0 + t\cdot m,\; z_0 + t\cdot n) \]

Angle between line and plane:

\[ \phi = \arcsin\left(\frac{|a\cdot l + b\cdot m + c\cdot n|}{\sqrt{a^2+b^2+c^2}\sqrt{l^2+m^2+n^2}}\right) \]

where \(\phi\) is the angle between the line and its projection onto the plane.

Angle between line and normal to plane P: \(\theta = 90° - \phi\)

plane-line-intersection-angle.gif

Line-Plane Intersection

Find the intersection point and angle between a line and a plane in 3D space

Line input format:

Point A (x₁, y₁, z₁)

x₁

y₁

z₁

Point B (x₂, y₂, z₂)

x₂

y₂

z₂

Point P₀ (x₀, y₀, z₀)

x₀

y₀

z₀

Direction Vector v = (l, m, n)

Point (x₀, y₀, z₀)

x₀

y₀

z₀

Direction numbers (l, m, n) from (x-x₀)/l = (y-y₀)/m = (z-z₀)/n

Note: l, m, n are the direction numbers (denominators) in the symmetric form

Plane: a·x + b·y + c·z = d

Decimal Places

Intersection Point --

Angle (Line & Plane) φ -- °

Angle (Line & Normal) θ -- °

About the Calculations

Line Forms

Two Points: Line through points A and B: \( \vec{v} = B - A \)
Parametric: \(P_0 + t\cdot\vec{v}\) where \(\vec{v} = (l, m, n)\) is direction vector
Symmetric: \(\dfrac{x - x_0}{l} = \dfrac{y - y_0}{m} = \dfrac{z - z_0}{n}\) where \((l, m, n)\) are direction numbers

Intersection Cases

Line intersects plane: Unique intersection point when \( \vec{n} \cdot \vec{v} \neq 0 \)
Line parallel to plane: No intersection when \( \vec{n} \cdot \vec{v} = 0 \) and point not in plane
Line lies in plane: Infinite intersection points when \( \vec{n} \cdot \vec{v} = 0 \) and point satisfies plane equation

Angles

φ (phi): Angle between the line and its projection onto the plane (\(0° \leq \phi \leq 90°\))
θ (theta): Angle between the line and the plane's normal vector (complementary to φ)

Note: \(\phi + \theta = 90°\)