Calculator Input
Example Data Table
| Mode | Input Example | Distance Result | Notes |
|---|---|---|---|
| 2D Point to Point | A(1, 2), B(4, 6) | 5 | Uses the 2D Euclidean distance formula. |
| 3D Point to Point | A(1, 2, 3), B(4, 6, 3) | 5 | Includes x, y, and z displacement. |
| Point to Line | P(3, 4), 2x − y − 1 = 0 | 0.447214 | Absolute distance from the point to the line. |
| Point to Plane | P(2, 1, 5), x + 2y + 2z − 9 = 0 | 1.666667 | Projection lands directly on the plane. |
Formula Used
Distance from a point depends on the object you compare it with. Point-to-point problems use Euclidean distance. Point-to-line and point-to-plane problems use projection along the normal direction.
1) 2D point to point
d = √((x₂ − x₁)² + (y₂ − y₁)²)
2) 3D point to point
d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
3) Point to line in 2D
d = |Ax₀ + By₀ + C| / √(A² + B²)
4) Point to plane in 3D
d = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²)
How to Use This Calculator
- Choose the geometry mode that matches your problem.
- Enter the coordinates of your point or both points.
- For line mode, enter coefficients A, B, and C from Ax + By + C = 0.
- For plane mode, enter coefficients A, B, C, and D from Ax + By + Cz + D = 0.
- Click Calculate Distance to see the result above the form.
- Review the steps, closest point, and graph for interpretation.
- Use the CSV or PDF buttons to save the current result.
FAQs
1) What does this calculator measure?
It measures Euclidean distance between points and perpendicular distance from a point to a line or plane. It also shows projections and supporting graph output for interpretation.
2) What is the difference between signed and absolute distance?
Signed distance keeps direction relative to the line or plane normal. Absolute distance removes the sign and gives the shortest non-negative separation value.
3) Why do line and plane formulas use absolute value?
A point can lie on either side of a line or plane. Absolute value ensures the reported distance is always positive when you only need magnitude.
4) Can I use decimals or negative values?
Yes. The calculator accepts integers, decimals, and negative coordinates or coefficients, which is useful for analytic geometry and engineering-style coordinate systems.
5) What happens if the line or plane is invalid?
The page checks the coefficients. A line needs A or B to be non-zero. A plane needs at least one of A, B, or C to be non-zero.
6) Does the graph change with each mode?
Yes. Two-dimensional modes show scatter and line plots. Three-dimensional modes show 3D traces, including a plane surface when plane mode is selected.
7) What is the closest point output?
It is the perpendicular projection of the chosen point onto the line or plane. This point lies on the target object and defines the shortest path.
8) When is point-to-plane distance useful?
It is useful in 3D modeling, robotics, surveying, graphics, structural analysis, and coordinate geometry whenever you need shortest separation from a point to a surface.