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Point-to-Plane Distance

The shortest distance from a point to a plane is along the perpendicular. Formula: d = |n·(P − P₀)| / |n| where n is the normal and P₀ is a point on the plane. Used in collision detection and machine learning.

Concept Fundamentals
d = |n·(P−P₀)|/|n|
Distance
P − d_signed·n̂
Foot
Side of plane
Signed
Generalizes 2D
3D

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When point is on plane, distance is zero. Signed distance: positive on one side, negative on the other. SVMs use point-to-hyperplane distance for classification margin.

Key quantities
d = |n·(P−P₀)|/|n|
Distance
Key relation
P − d_signed·n̂
Foot
Key relation
Side of plane
Signed
Key relation
Generalizes 2D
3D
Key relation

Ready to run the numbers?

Why: Point-to-plane distance is essential in 3D collision detection, SVM margin calculation, robotics obstacle clearance, and graphics for shadow volumes. Signed distance indicates which side of the plane.

How: Project the vector (P − P₀) onto the unit normal. Distance d = |n·(P−P₀)|/|n|. Foot point: P_foot = P − d_signed·(n/|n|).

When point is on plane, distance is zero.Signed distance: positive on one side, negative on the other.
Sources:Khan Academy — 3D GeometryWolfram MathWorld — Point-Plane Distance

Run the calculator when you are ready.

Find Distance to PlaneEnter point, plane point, and normal

Sample Examples

Input

Point P

Point on Plane P₀

Normal Vector n

Results

Distance

6.93

Foot of Perpendicular

(-1, -0, 1)

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

🧮 Fascinating Math Facts

⟂

Perpendicular is shortest path to plane.

— 3D Geometry

F

Foot is closest point on plane.

— Property

Key Takeaways

  • d = |n·(P - P₀)| / |n| where P is point, P₀ on plane, n is normal
  • Alternative: d = |Ax+By+Cz+D| / √(A²+B²+C²) for plane Ax+By+Cz+D=0
  • Foot of perpendicular: P_foot = P - (signed_dist)(n/|n|)
  • Signed distance indicates which side of plane
  • Normal must be non-zero

Did You Know?

  • SVMs use point-to-hyperplane distance for margin
  • Collision detection uses this for penetration depth
  • Signed distance: positive = same side as n, negative = opposite
  • Robotics uses it for obstacle clearance
  • Graphics uses it for shadow volumes
  • Physics uses it for contact forces

Understanding

The shortest distance from a point to a plane is along the perpendicular. Project (P-P₀) onto the normal.

d=∣n⃗⋅(P⃗−P⃗0)∣∣n⃗∣d = \frac{|\vec{n} \cdot (\vec{P} - \vec{P}_0)|}{|\vec{n}|}
P⃗foot=P⃗−dsignedn⃗∣n⃗∣\vec{P}_{foot} = \vec{P} - d_{signed} \frac{\vec{n}}{|\vec{n}|}

Expert Tips

  • Use normalized n for simpler formulas
  • Signed distance: useful for half-space tests
  • When point on plane, d=0, foot=point
  • For Ax+By+Cz+D=0: d = |Ax+By+Cz+D|/√(A²+B²+C²)

FAQ

Q: Why absolute value?
A: Distance is always positive; sign indicates side.
Q: What is foot point?
A: Closest point on plane to P.
Q: When is distance zero?
A: When point lies on the plane.
Q: Applications?
A: Collision, SVM, robotics, graphics.
Q: Relation to line-plane intersection?
A: Line through P with dir n hits plane at foot.
Q: Can normal be zero?
A: No; invalid plane.
Q: Signed vs unsigned?
A: Unsigned = length; signed = side info.

How to Use

  1. Enter point P (x,y,z)
  2. Enter point on plane P₀ and normal n
  3. Get distance and foot of perpendicular

Disclaimer

Normal vector must be non-zero.

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