Shortest Distance From Point To Plane Calculator

Distance from Point to Plane Formula:

\[ D = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

Plane Coefficient A:

Plane Coefficient B:

Plane Coefficient C:

Plane Coefficient D:

Point X-coordinate:

Point Y-coordinate:

Point Z-coordinate:

Shortest Distance:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Point-to-Plane Distance?

The shortest distance from a point to a plane is the length of the perpendicular line from the point to the plane. This is a fundamental concept in 3D geometry with applications in computer graphics, physics, and engineering.

2. How Does the Calculator Work?

The calculator uses the point-to-plane distance formula:

\[ D = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

Where:

\( A, B, C, D \) — Coefficients of the plane equation \( Ax + By + Cz + D = 0 \)
\( (x_0, y_0, z_0) \) — Coordinates of the point
\( D \) — Calculated distance from point to plane

Explanation: The formula calculates the perpendicular distance by projecting the vector from any point on the plane to the given point onto the plane's normal vector.

3. Applications of Point-to-Plane Distance

Details: This calculation is used in computer graphics for collision detection, in physics for potential field calculations, in robotics for path planning, and in geography for elevation measurements.

4. Using the Calculator

Tips: Enter the plane equation coefficients (A,B,C,D) and the point coordinates (x,y,z). The plane equation should be in standard form \( Ax + By + Cz + D = 0 \).

5. Frequently Asked Questions (FAQ)

Q1: What if the point is on the plane?
A: The distance will be zero, as the point satisfies the plane equation \( Ax + By + Cz + D = 0 \).

Q2: Does the distance have direction?
A: The distance is always non-negative. To determine which side of the plane the point is on, check the sign of \( Ax_0 + By_0 + Cz_0 + D \).

Q3: How is this different from point-to-line distance?
A: Point-to-line distance is in 2D space, while point-to-plane distance is in 3D space, using an additional dimension.

Q4: Can this be used for higher dimensions?
A: The concept generalizes to n-dimensional spaces with the same formula structure.

Q5: What if the plane coefficients are all zero?
A: This would represent an invalid plane and the calculator will show an error (division by zero).