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Line-Plane Intersection

A line r = r₀ + t·d intersects plane n·r = k when n·d≠0. Parameter t = (k − n·r₀)/(n·d). If n·d=0: line parallel to plane (no intersection or line in plane).

Concept Fundamentals
r = r₀ + t·d
Line
n·r = k
Plane
(k−n·r₀)/(n·d)
t
n·d=0 ⇒ no or ∞
Parallel

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n·d ≠ 0: unique intersection at t = (k−n·r₀)/(n·d). n·d = 0: line parallel; check if r₀ satisfies plane. Intersection point: r₀ + t·d.

Key quantities
r = r₀ + t·d
Line
Key relation
n·r = k
Plane
Key relation
(k−n·r₀)/(n·d)
t
Key relation
n·d=0 ⇒ no or ∞
Parallel
Key relation

Ready to run the numbers?

Why: Line-plane intersection is fundamental in 3D graphics (ray casting), CAD, and physics. Used for collision detection, shadow computation, and geometric queries.

How: Line: r = r₀ + t·d. Plane: n·r = k. Substitute: n·(r₀+td)=k ⇒ t = (k−n·r₀)/(n·d). If n·d=0, line is parallel to plane.

n·d ≠ 0: unique intersection at t = (k−n·r₀)/(n·d).n·d = 0: line parallel; check if r₀ satisfies plane.
Sources:Wolfram MathWorld — Line-Plane IntersectionKhan Academy — 3D Geometry

Run the calculator when you are ready.

Find IntersectionEnter line (point + direction) and plane (normal + constant)

Sample Examples

Input

Line: r = r₀ + tv

Plane: Ax+By+Cz+D=0

Results

Intersection Point

(1, 1, 1)

t = -0

Step-by-Step

Denominator: Aa+Bb+Cc = 1·1+1·2+1·3 = 6

Numerator: Ax₀+By₀+Cz₀+D = 0

t = -numer/denom = -0

Intersection: P = (1, 1, 1)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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t = (k−n·r₀)/(n·d) when n·d≠0.

— 3D Geometry

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n·d=0 ⇒ line parallel to plane.

— Condition

Key Takeaways

  • Substitute line vecr=vecr0+tvecv\\vec{r} = \\vec{r}_0 + t\\vec{v} into plane Ax+By+Cz+D=0.
  • Solve for t: t=−fracAx0+By0+Cz0+DAa+Bb+Cct = -\\frac{Ax_0+By_0+Cz_0+D}{Aa+Bb+Cc}.
  • Parallel: denominator Aa+Bb+Cc = 0 (no intersection).
  • Line in plane: denominator and numerator both zero.
  • Otherwise: unique intersection point.

Did You Know?

Ray tracing uses this for every pixel-ray intersection.
Robotics uses it for path-obstacle intersection.
n·v = 0 means line parallel to plane.
When line is in plane, every point is an intersection.
Physics uses this for particle-surface collisions.
CAD uses it for clipping and visibility.

Understanding

Substitute x=x₀+at, y=y₀+bt, z=z₀+ct into Ax+By+Cz+D=0. Solve for t.

t=−fracAx0+By0+Cz0+DAa+Bb+Cct = -\\frac{Ax_0+By_0+Cz_0+D}{Aa+Bb+Cc}
P=(x0+at,y0+bt,z0+ct)P = (x_0+at, y_0+bt, z_0+ct)

Expert Tips

Check |n·v| < ε for parallel (numerical).
If parallel, check if point satisfies plane for line-in-plane.
t < 0 means intersection is behind the line point.
For ray casting, use t ≥ 0 for forward direction.

FAQ

Q: When is there no intersection?
A: When line is parallel to plane (n·v=0) and not in it.
Q: Infinite intersections?
A: When line lies in plane.
Q: What if direction is zero?
A: Not a valid line.
Q: Applications?
A: Ray tracing, collision detection, clipping.
Q: How to get t for intersection?
A: t = -(Ax₀+By₀+Cz₀+D)/(Aa+Bb+Cc).
Q: Relation to point-to-plane distance?
A: Distance uses same formula; intersection uses t.
Q: Can t be negative?
A: Yes; intersection is in opposite direction of v.

How to Use

  1. Enter line point (x₀, y₀, z₀) and direction (a, b, c).
  2. Enter plane Ax+By+Cz+D=0.
  3. Get intersection point or special case.

Disclaimer

Direction vector and plane normal must be non-zero.

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