Distance Between Plane And Point Calculator

Distance 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 (x₀):

Point Y-coordinate (y₀):

Point Z-coordinate (z₀):

Distance Between Point and Plane:

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1. What is the Distance Between Plane And Point?

The distance between a point and a plane is the shortest distance from the point to the plane. It's calculated using the perpendicular distance from the point to the plane.

2. How Does the Calculator Work?

The calculator uses the distance formula:

\[ d = \frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}} \]

Where:

\( ax + by + cz + d = 0 \) — Equation of the plane
\( (x_0, y_0, z_0) \) — Coordinates of the point
\( d \) — Calculated distance

Explanation: The numerator is the absolute value of the plane equation evaluated at the point coordinates. The denominator normalizes this by the magnitude of the plane's normal vector.

3. Importance of Distance Calculation

Details: Calculating the distance between a point and a plane is fundamental in computer graphics, physics simulations, robotics, and geometric modeling.

4. Using the Calculator

Tips: Enter the plane coefficients (a, b, c, d) and the point coordinates (x₀, y₀, z₀). 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.

Q2: Does the order of coefficients matter?
A: The relative signs matter, but the overall scale doesn't (you can multiply all coefficients by a non-zero constant).

Q3: What does a negative distance mean?
A: Distance is always non-negative. The formula uses absolute value to ensure this.

Q4: Can this work in 2D?
A: Yes, for a line (ax + by + c = 0) and point (x₀, y₀), use similar formula without z terms.

Q5: What if all plane coefficients are zero?
A: This isn't a valid plane, and the distance is undefined (division by zero).