How to use
- Choose point and normal, or three points. Only the fields for that method are used.
- Enter the point coordinates and either a nonzero normal or two further points.
- Calculate to read ax + by + cz + d = 0, the unit normal and the substitution check.
Choose one length unit for all coordinates. Changing the unit label does not convert the numbers. Enter numbers, including scientific notation such as 1e-4; do not enter algebraic expressions.
The drawing updates as soon as the inputs describe valid geometry. If a value is missing or invalid, the drawing stays hidden until the inputs are ready. Editing clears the previous numerical result and disables Copy result. Press Calculate to confirm the current numbers, then copy them as text. Clear empties the numerical inputs while keeping your selections.
Formula and interpretation
A normal is perpendicular to every direction along a plane. With three points, subtract the first point from the other two and take their cross product. The result supplies the normal. Coefficients are scaled so the largest normal component has magnitude 1; the first meaningful component is positive.
n · (X − P) = 0; d = −n · P. For three points, n = (Q − P) × (R − P).
Worked example
Use P = (1, 0, 0), Q = (0, 1, 0) and R = (0, 0, 1). The result is x + y + z − 1 = 0. Substitution of each point gives zero. The unit normal is approximately (0.58, 0.58, 0.58).
For point P = (2, −1, 3) and normal (0, 0, 5), the point-normal expression is 5(z − 3) = 0. The normalized result is z − 3 = 0. Changing the normal to (0, 0, −10) gives the same plane. The point fixes its height, while the normal fixes its orientation.
Uses and limits
This describes an infinite plane for analytic geometry and 3D modeling. The residual is the largest absolute value obtained by substituting the input points into the unrounded output equation; it should be zero or close to zero. A direction angle with sine below 10⁻¹⁰ is treated as numerically collinear.
Numbers are calculated with standard floating-point arithmetic. Ordinary results show up to two decimal places; small nonzero values use scientific notation. Equation coefficients use up to 12 significant digits, so a displayed equation can be approximate. Inputs are limited to zero or magnitudes from 10⁻⁹ to 10⁹, and calculated magnitudes to 10¹⁵.
The drawing is an orthographic view, automatically fitted around the geometry. Perspective on the screen does not preserve every angle or length. The small axis arrows indicate orientation, not the location of the origin; read coordinates and distances from the results.
Frequently asked questions
Why do three points sometimes fail?
Repeated points or points on one straight line do not determine a unique plane. Choose a third point away from that line.
Can a different equation be correct?
Yes. Multiplying every coefficient by a nonzero number gives an equivalent equation. Reversing the order of points can reverse a raw cross product without changing the plane.
Is the arrow a position vector?
No. It shows the unit normal direction starting on the plane. Its drawn length is adjusted for readability; the numerical vector is in the results.
How should I use the copied numbers?
Paste the result into notes or a worksheet as a calculation check. Keep the coordinate unit and original inputs with it. Rounded displayed coordinates are for reading, not exact symbolic substitution. If a calculation is rejected for precision, shift or rescale the coordinates; do not interpret the empty result as zero.