3D Midpoint Calculator

What is the 3D Midpoint?

The 3D midpoint is a point in three-dimensional space that is equidistant from two given points. It represents the center of a line segment connecting these two points in 3D space.

How to Calculate the 3D Midpoint

To find the 3D midpoint, we calculate the average of the x, y, and z coordinates of the two given points. This process involves finding the arithmetic mean of each coordinate pair.

Formula

The formula for finding the midpoint (x, y, z) of a line segment with endpoints (x₁, y₁, z₁) and (x₂, y₂, z₂) is:

\[ x = \frac{x_1 + x_2}{2}, \quad y = \frac{y_1 + y_2}{2}, \quad z = \frac{z_1 + z_2}{2} \]

Where:

• (x, y, z) are the coordinates of the midpoint
• (x₁, y₁, z₁) are the coordinates of the first endpoint
• (x₂, y₂, z₂) are the coordinates of the second endpoint

Calculation Steps

1. Identify the coordinates of the two endpoints
2. Add the x-coordinates and divide by 2
3. Add the y-coordinates and divide by 2
4. Add the z-coordinates and divide by 2
5. The resulting (x, y, z) is the 3D midpoint

Example and Visual Representation

Let's find the 3D midpoint of a line segment with endpoints A(1, 2, 3) and B(5, 8, 9):

Step 1: We have (x₁, y₁, z₁) = (1, 2, 3) and (x₂, y₂, z₂) = (5, 8, 9)

Step 2: x = (x₁ + x₂) / 2 = (1 + 5) / 2 = 6 / 2 = 3

Step 3: y = (y₁ + y₂) / 2 = (2 + 8) / 2 = 10 / 2 = 5

Step 4: z = (z₁ + z₂) / 2 = (3 + 9) / 2 = 12 / 2 = 6

Step 5: The 3D midpoint M is (3, 5, 6)

In this 2D representation of our 3D space, you can see the line segment AB in blue, with endpoints A and B in red. The green point M represents the 3D midpoint we calculated. While this 2D diagram can't fully capture the 3D nature, it illustrates how M is positioned centrally between A and B.

Understanding the 3D midpoint formula is crucial in various fields such as computer graphics, 3D modeling, physics simulations, and spatial analysis. It allows us to precisely locate the center point between any two positions in three-dimensional space, forming the basis for more complex 3D calculations and constructions.