Line of Intersection of Two Planes Calculator

Line of Intersection of Two Planes

In three-dimensional Euclidean space $\mathbb{R}^3$, two non-parallel planes intersect along a single straight line. Finding the equation of this line is a core problem in spatial geometry, computer graphics, mechanical CAD modeling, and vector calculus.

Mathematical Derivation

Let the two planes $\pi_1$ and $\pi_2$ be defined in standard Cartesian form:

$$\pi_1: A_1 x + B_1 y + C_1 z = D_1$$

$$\pi_2: A_2 x + B_2 y + C_2 z = D_2$$

Their normal vectors are $\mathbf{n}_1 = \langle A_1, B_1, C_1 \rangle$ and $\mathbf{n}_2 = \langle A_2, B_2, C_2 \rangle$.

1. Determining the Direction Vector

Because the line of intersection lies simultaneously in both planes, it is perpendicular to both normal vectors $\mathbf{n}_1$ and $\mathbf{n}_2$. Therefore, the line's directional vector $\mathbf{v}$ is given by the vector cross product:

$$\mathbf{v} = \mathbf{n}_1 \times \mathbf{n}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \end{vmatrix} = \langle B_1 C_2 - C_1 B_2,\; C_1 A_2 - A_1 C_2,\; A_1 B_2 - B_1 A_2 \rangle$$

If $\mathbf{v} = \mathbf{0}$, the normal vectors are collinear, meaning the planes are parallel (either disjoint with no intersection or coincident).

2. Finding a Specific Point on the Line

To complete the line's equation, choose any particular point $P_0(x_0, y_0, z_0)$ satisfying both plane equations. A standard approach is setting one convenient coordinate to zero (e.g., $z_0 = 0$) and solving the remaining 2x2 system of linear equations for $x_0$ and $y_0$.

3. Parametric and Symmetric Forms

With point $P_0$ and direction vector $\mathbf{v} = \langle v_x, v_y, v_z \rangle$, the vector equation of the intersection line is:

$$\mathbf{r}(t) = P_0 + t \mathbf{v}$$

In parametric scalar form:

$$x(t) = x_0 + v_x t, \quad y(t) = y_0 + v_y t, \quad z(t) = z_0 + v_z t$$

In symmetric form (for non-zero direction components):

$$\frac{x - x_0}{v_x} = \frac{y - y_0}{v_y} = \frac{z - z_0}{v_z}$$

Dihedral Angle Between the Planes

The angle $\theta$ between the two intersecting planes is defined as the acute angle between their normal vectors:

$$\cos(\theta) = \frac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{|\mathbf{n}_1| |\mathbf{n}_2|} = \frac{|A_1 A_2 + B_1 B_2 + C_1 C_2|}{\sqrt{A_1^2 + B_1^2 + C_1^2}\sqrt{A_2^2 + B_2^2 + C_2^2}}$$

Frequently Asked Questions

Can two distinct planes intersect at a single point?

No. In three-dimensional space, two distinct non-parallel planes always intersect along an infinite straight line. You need at least three planes to intersect at a single isolated point.

What happens when two planes are parallel?

When planes are parallel, their normal vectors point in the same (or opposite) direction, so their cross product $\mathbf{n}_1 \times \mathbf{n}_2 = \mathbf{0}$. If they have different constant terms, they never intersect. If they have proportional constants, they are coincident (the same plane).

How do I find a common point P₀ if z = 0 fails?

If setting $z = 0$ produces a singular matrix ($v_z = 0$), the intersection line is parallel to the xy-plane. In that case, set $y = 0$ (if $v_y \neq 0$) or $x = 0$ (if $v_x \neq 0$) instead.

Are parametric line equations unique?

No. You can pick any point lying anywhere on the line as $P_0$, and any non-zero scalar multiple of the direction vector $\mathbf{v}$. All such parametric equations describe the exact same geometric line in space.

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Line of Intersection of Two Planes Calculator