Two 3 dimensional points. A [ x 1 , y 1 , z 1 ] and B [ x 2 , y 2 , z 2 ]. Need to find a transformation matrix which when multiplied to A will give me B and when multiplied by B give me A. The transformation needs to be a reflection against the plane that's perpendicular to the middle of the AB segment and passing through the midpoint of the AB.

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Two 3 dimensional points.   A  [      x    1    ,      y    1    ,      z    1    ] and   B  [      x    2    ,      y    2    ,      z    2    ]. Need to find a transformation matrix which when multiplied to A will give me B and when multiplied by B give me A. The transformation needs to be a reflection against the plane that&#039;s perpendicular to the middle of the AB segment and passing through the midpoint of the AB.

taweirrvb · Accepted Answer

Midpoint   C  [      x    c    ,      y    c    ,      z    c    ]  =  [  (      x    1    +      x    2    )  /  2  ,  (      y    1    +      y    2    )  /  2  ,  (      z    1    +      z    2    )  /  2  ]Vector   A  B  [      x          a      b        ,      y          a      b        ,      z          a      b        ]  =  [      x    2    −      x    1    ,      y    2    −      y    1    ,      z    2    −      z    1    ]Now, to find plane perpendicular to the vector intersecting the midpoint. Plane has canonical form       A    p    ∗  x  +      B    p    ∗  y  +      C    p    ∗  z  +      D    p    =  0 where       A    p    ,      B    p    ,      C    p   describe the vector so they are equal to        x          a      b        ,      y          a      b        ,      z          a      b       and       D    p   is   −      x          a      b        ∗      x    c    −      y          a      b        ∗      y    c    −      z          a      b        ∗      z    c  The reflection matrix is      (                            −          2          ∗                      A            p                    ∗                      A            p                    +          1                          −          2          ∗                      B            p                    ∗                      A            p                                    −          2          ∗                      C            p                    ∗                      A            p                                    0                                      −          2          ∗                      A            p                    ∗                      B            p                                    −          2          ∗                      B            p                    ∗                      B            p                    +          1                          −          2          ∗                      C            p                    ∗                      B            p                                    0                                      −          2          ∗                      A            p                    ∗                      C            p                                    −          2          ∗                      B            p                    ∗                      C            p                                    −          2          ∗                      C            p                    ∗                      C            p                    +          1                          0                                      −          2          ∗                      A            p                    ∗                      D            p                                    −          2          ∗                      B            p                    ∗                      D            p                                    −          2          ∗                      C            p                    ∗                      D            p                                    1                      )

Two 3 dimensional points. A [ x 1 </msub> , y 1 </msub> ,

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