Point to Plane Distance
This calculator finds the shortest distance from a point P(xâ‚€, yâ‚€, zâ‚€) to a plane defined by the general equation Ax + By + Cz + D = 0.
The Formula:
Distance = |A·x₀ + B·y₀ + C·z₀ + D| / √(A² + B² + C²)
Where:
- (xâ‚€, yâ‚€, zâ‚€) are the coordinates of the point.
- A, B, C are the coefficients of x, y, and z in the plane equation (representing the normal vector to the plane).
- D is the constant term in the plane equation.
How to Use:
- Enter the coordinates (xâ‚€, yâ‚€, zâ‚€) of the point.
- Enter the coefficients (A, B, C, D) of the plane equation Ax + By + Cz + D = 0.
- Ensure that A, B, and C are not all zero, as this would not define a valid plane for this formula.
- Click “Calculate Distance”.
The result is the perpendicular distance from the point to the plane.
Calculate the shortest distance from a point to a 3D plane.
Distance from Point to Plane
This is the shortest (perpendicular) distance from the point to the plane.
Distance from Point to Plane: Complete 3D Geometry Guide
What is Point-to-Plane Distance?
The distance from a point to a plane is the shortest perpendicular distance between a point in 3D space and a geometric plane. This fundamental concept in analytic geometry has applications in computer graphics, engineering, and physics.
The Mathematical Formula
The distance formula for a point (xâ‚€,yâ‚€,zâ‚€) to plane Ax + By + Cz + D = 0 is:
d = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²)
Key Components Explained
Plane Equation
Ax + By + Cz + D = 0 defines the plane in 3D space
Normal Vector
Vector [A,B,C] perpendicular to the plane
Absolute Value
Ensures distance is always positive
Practical Applications
- Computer graphics rendering
- Robotics path planning
- Architectural design
- Physics collision detection
- Geographic information systems (GIS)
Frequently Asked Questions
What if the point lies on the plane?
The distance will be zero, indicating the point satisfies the plane equation.
How is this different from 2D distance?
This extends the 2D line distance concept to 3D space using vector mathematics.
Can distance be negative?
No, the absolute value in the formula ensures non-negative results.
Calculation Steps
- Substitute point coordinates into plane equation
- Calculate absolute value of numerator
- Compute normal vector magnitude
- Divide numerator by denominator
Special Cases
| Scenario | Result |
|---|---|
| Parallel coordinate plane | Distance equals coordinate value |
| Zero normal vector | Invalid plane equation |