How do you find the midpoint of several 3D points?

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Dexter Conner · Accepted Answer

Step 1If you are using a computer graphics system that uses homogeneous coordinates (4 values per 3D point) then you can just add all the components together.For example      (                                        x            1                                                            y            1                                                            z            1                                                1                      )    +      (                                        x            2                                                            y            2                                                            z            2                                                1                      )    +      (                                        x            3                                                            y            3                                                            z            3                                                1                      )    =      (                                        x            1                    +                      x            2                    +                      x            3                                                            y            1                    +                      y            2                    +                      y            3                                                            z            1                    +                      z            2                    +                      z            3                                                3                      )  The cartesian coordinates of the result is the &quot;mid-point&quot; (also known as the barycenter).      (                                                                      x                1                            +                              y                1                            +                              z                1                                      3                                                                                          x                2                            +                              y                2                            +                              z                2                                      3                                                                                          x                3                            +                              y                3                            +                              z                3                                      3                                )

Ronnie Porter · Accepted Answer

Step 1This will be the center of gravity, or the average of the points. If the points are   x  =  (      x    1    ,      x    2    ,      x    3    ) ,       y  =  (      y    1    ,      y    2    ,      y    3    ) , and   z  =  (      z    1    ,      z    2    ,      z    3    ) , you want to average each coordinate to obtain the point:  a  =        (                            x          1                +                  y          1                +                  z          1                    3        ,                            x          2                +                  y          2                +                  z          2                    3        ,                            x          3                +                  y          3                +                  z          3                    3        )  This same solution works for any number of points, not just three. For example, for two points, this gives you the midpoint.Note that, in vector arithmetic, we have   x  +  y  +  z  =  3  a . Thus,   (  x  −  a  )  +  (  y  −  a  )  +  (  z  −  a  )  =  0 . We can think of each of those terms as a force from the point a towards one of the three points. Since it adds to 0, the forces cancel each other out, and so a is in equilibrium. This means it works with your grappling hook intuition.

How do you find the midpoint of several 3D points?

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