Equation of the ellipse between 2 points (its vertices)Start point and end point: A ( x , y ) B ( x , y )Ellipse width: w = 2.5Midpoint or center of the ellipse: M = ( A x + B x 2 , A y + B y 2 ) The angle (that would be the angle between points A and B): θ = ?Parametric equation of the ellipse with rotation: x = a ∗ c o s ( t ) ∗ c o s ( θ ) + b ∗ s i n ( t ) ∗ s i n ( θ ) y = b ∗ s i n ( t ) ∗ c o s ( θ ) − a ∗ c o s ( t ) ∗ s i n ( θ )

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Equation of the ellipse between 2 points (its vertices)Start point and end point:  A  (  x  ,  y  )    B  (  x  ,  y  )Ellipse width:  w  =  2.5Midpoint or center of the ellipse:  M  =      (                            A          x                +                  B          x                    2        ,                            A          y                +                  B          y                    2        )  The angle (that would be the angle between points A and B):  θ  =  ?Parametric equation of the ellipse with rotation:  x  =  a  ∗  c  o  s  (  t  )  ∗  c  o  s  (  θ  )  +  b  ∗  s  i  n  (  t  )  ∗  s  i  n  (  θ  )  y  =  b  ∗  s  i  n  (  t  )  ∗  c  o  s  (  θ  )  −  a  ∗  c  o  s  (  t  )  ∗  s  i  n  (  θ  )

Josie Stephenson · Accepted Answer

Step 1If the end points of one axis are A,B, and the other semi-axis has length w, then the center of the ellipse is  C  =            1      2        (      A    x    +      B    x    ,      A    y    +      B    y    )The first semi-axis vector is      V    1    =            1      2        (      A    x    −      B    x    ,      A    y    −      B    y    )And the second semi-axis vector is      V    2    =            w              ‖                  V          1                ‖                        1      2        (      B    y    −      A    y    ,      A    x    −      B    x    )Now the parametric equation of the ellipse is  p  (  t  )  =  C  +      V    1    cos  ⁡  t  +      V    2    sin  ⁡  t

Equation of the ellipse between 2 points (its vertices) Start point and end point: A ( x

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