Find Plane From 3 Points Calculator

Calculate the Equation of a Plane

Enter the coordinates of three non-collinear points to find the equation of the plane passing through them.

2D Projection of Vectors (v1, v2, normal n) from P1 (origin for vectors)

Coordinates and Vector Components

Point/Vector	X	Y	Z
P1	1	0	0
P2	0	1	0
P3	0	0	1
v1 (P1P2)	-1	1	0
v2 (P1P3)	-1	0	1
n (Normal)	1	1	1

What is a Plane Defined by Three Points?

A plane in three-dimensional space can be uniquely defined by three non-collinear points (points that do not lie on the same straight line). The find plane from 3 points calculator helps determine the equation of this plane. This equation is typically represented in the form Ax + By + Cz + D = 0, where A, B, and C are the components of the normal vector to the plane, and D is a constant.

Anyone working with 3D geometry, such as engineers, architects, physicists, computer graphics programmers, and students of mathematics, might need to find the equation of a plane from three points. For instance, defining a surface in a 3D model or determining the orientation of an object in space often involves this calculation.

A common misconception is that any three points define a unique plane. This is only true if the points are not collinear. If the three points lie on a single line, an infinite number of planes can pass through them, and the find plane from 3 points calculator will indicate this (usually by a normal vector of (0,0,0)).

Plane from 3 Points Formula and Mathematical Explanation

To find the equation of a plane passing through three points P1=(x1, y1, z1), P2=(x2, y2, z2), and P3=(x3, y3, z3), we follow these steps:

1. Form two vectors in the plane:
   • Vector v1 from P1 to P2: v1 = (x2-x1, y2-y1, z2-z1)
   • Vector v2 from P1 to P3: v2 = (x3-x1, y3-y1, z3-z1)
2. Find the normal vector to the plane: The normal vector n = (A, B, C) is perpendicular to the plane and can be found by taking the cross product of v1 and v2: n = v1 x v2 = ((y2-y1)(z3-z1) – (z2-z1)(y3-y1), (z2-z1)(x3-x1) – (x2-x1)(z3-z1), (x2-x1)(y3-y1) – (y2-y1)(x3-x1)) So, A = (y2-y1)(z3-z1) – (z2-z1)(y3-y1), B = (z2-z1)(x3-x1) – (x2-x1)(z3-z1), C = (x2-x1)(y3-y1) – (y2-y1)(x3-x1).
3. Form the plane equation: The equation of the plane is Ax + By + Cz + D = 0. To find D, we substitute the coordinates of one of the points (e.g., P1) into the equation: Ax1 + By1 + Cz1 + D = 0 So, D = -(Ax1 + By1 + Cz1).
4. The final equation is Ax + By + Cz + D = 0, or Ax + By + Cz = -D.

If A, B, and C are all zero, the points are collinear, and no unique plane is defined.

Variables Table:

Variable	Meaning	Unit	Typical Range
(x1, y1, z1)	Coordinates of Point 1	Length units	Any real number
(x2, y2, z2)	Coordinates of Point 2	Length units	Any real number
(x3, y3, z3)	Coordinates of Point 3	Length units	Any real number
v1, v2	Vectors in the plane	Length units	Components are real numbers
n=(A, B, C)	Normal vector to the plane	(Length units)^2	Components are real numbers
D	Constant in the plane equation	(Length units)^3	Any real number

Practical Examples (Real-World Use Cases)

Example 1: 3D Modeling

An artist is creating a 3D model of a ramp. They define three corner points of the ramp's surface: P1(0, 0, 0), P2(5, 0, 0), and P3(5, 3, 1). Using the find plane from 3 points calculator:

• v1 = (5-0, 0-0, 0-0) = (5, 0, 0)
• v2 = (5-0, 3-0, 1-0) = (5, 3, 1)
• n = (0*1 – 0*3, 0*5 – 5*1, 5*3 – 0*5) = (0, -5, 15)
• D = -(0*0 + (-5)*0 + 15*0) = 0
• Equation: 0x – 5y + 15z + 0 = 0, or -5y + 15z = 0, which simplifies to y – 3z = 0 or y = 3z.

The equation of the ramp surface is y – 3z = 0.

Example 2: Surveying

A surveyor measures three points on a relatively flat piece of land to approximate it as a plane: P1(10, 20, 5), P2(50, 25, 5.5), and P3(15, 60, 4.8). Using the find plane from 3 points calculator:

• v1 = (40, 5, 0.5)
• v2 = (5, 40, -0.2)
• n = (5*(-0.2) – 0.5*40, 0.5*5 – 40*(-0.2), 40*40 – 5*5) = (-1 – 20, 2.5 + 8, 1600 – 25) = (-21, 10.5, 1575)
• D = -(-21*10 + 10.5*20 + 1575*5) = -(-210 + 210 + 7875) = -7875
• Equation: -21x + 10.5y + 1575z – 7875 = 0.

This equation approximates the surface of the land.

How to Use This Find Plane from 3 Points Calculator

1. Enter Coordinates: Input the x, y, and z coordinates for each of the three points (P1, P2, P3) into the respective fields. Ensure the points are not collinear for a unique plane.
2. View Results Automatically: The calculator updates in real-time. The primary result is the equation of the plane (Ax + By + Cz + D = 0 or Ax + By + Cz = -D). Intermediate values like vectors v1, v2, the normal vector n, and constant D are also displayed.
3. Check for Collinearity: If the normal vector (A, B, C) is (0, 0, 0), the points are collinear, and no unique plane is defined. The calculator will indicate this.
4. Visualize: The SVG chart provides a 2D projection of the vectors v1, v2 (originating from P1), and the normal vector n.
5. Reset: Click "Reset" to clear the inputs to default values.
6. Copy Results: Click "Copy Results" to copy the plane equation and intermediate values to your clipboard.

The find plane from 3 points calculator is a tool to quickly get the plane's equation. Understanding the normal vector gives you the orientation of the plane.

Key Factors That Affect Plane Equation Results

• Collinearity of Points: If the three points lie on a straight line, they do not define a unique plane. The cross product will be zero, resulting in a (0,0,0) normal vector. Our find plane from 3 points calculator detects this.
• Precision of Coordinates: Small changes in the input coordinates, especially if the points are close together or nearly collinear, can lead to significant changes in the plane's equation and normal vector direction. High precision is needed for accurate results.
• Order of Points (for v1, v2): While the plane itself remains the same, swapping P2 and P3 will reverse the direction of the normal vector (e.g., from (A, B, C) to (-A, -B, -C)) but the plane equation will represent the same plane.
• Coordinate System: The equation depends on the coordinate system used (e.g., right-handed or left-handed, units). Ensure consistency.
• Numerical Stability: When points are nearly collinear, calculations can become numerically unstable. The find plane from 3 points calculator uses standard floating-point arithmetic.
• Scale of Coordinates: If the coordinates are very large or very small, it might affect the numerical precision of the resulting A, B, C, and D values.

Frequently Asked Questions (FAQ)

Q: What if the three points are collinear? A: If the three points lie on the same line, they do not define a unique plane. An infinite number of planes can pass through that line. The find plane from 3 points calculator will show a normal vector of (0, 0, 0) and indicate this.

Q: Does the order of the points matter? A: The order in which you enter P1, P2, and P3 affects the direction of the calculated normal vector (it might point "up" or "down" relative to the plane), but the resulting plane equation (after simplification) represents the same plane.

Q: What does the normal vector represent? A: The normal vector (A, B, C) is a vector perpendicular to the plane. Its direction indicates the orientation of the plane in 3D space.

Q: Can I use this calculator for 2D points? A: This find plane from 3 points calculator is designed for 3D points (x, y, z). For 2D points, you'd be looking for the equation of a line passing through two points.

Q: How can I simplify the plane equation? A: If A, B, C, and D have a common divisor, you can divide the entire equation by it to get a simpler form. For example, 2x + 4y – 2z + 6 = 0 can be simplified to x + 2y – z + 3 = 0.

Q: What are the units of A, B, C, and D? A: If the coordinates are in length units (e.g., meters), A, B, and C have units of (length units)^2, and D has units of (length units)^3, although the equation is often presented unitlessly after simplification.

Q: How do I know if my input values are correct? A: Double-check the coordinates you enter. The calculator performs the math based on your input. Small errors in input can lead to different plane equations.

Q: Can the normal vector be (0,0,0)? A: Yes, if the three points are collinear, the cross product results in a (0,0,0) normal vector, meaning no unique plane is defined by those points using this method. Our find plane from 3 points calculator handles this.