Volume of a truncated ellipsoidMy objective is to find the volume of the intersection of the ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 ≤ 1and A x + B y + C z ≤ Dwhere a , b , c , A , B , C , D ∈ R . I have done several attempts but it seems like I have to consider a lot of different cases. My approach is to give a parametrization of the intersection and then finding the volume through solving an integral. However, I can't seem to find a good enough parametrization.

Question

Volume of a truncated ellipsoidMy objective is to find the volume of the intersection of the ellipsoid            x      2              a      2        +            y      2              b      2        +            z      2              c      2        ≤  1and   A  x  +  B  y  +  C  z  ≤  Dwhere   a  ,  b  ,  c  ,  A  ,  B  ,  C  ,  D  ∈      R  . I have done several attempts but it seems like I have to consider a lot of different cases. My approach is to give a parametrization of the intersection and then finding the volume through solving an integral. However, I can&#039;t seem to find a good enough parametrization.

Emmalee Reilly · Accepted Answer

Step 1Define the &quot;step function&quot;   θ  (  x  )  =  {  1  (  x  &amp;gt;  0  )  ;  0  (  x  ≤  0  )  }.So the volume you want is   V  =  ∫  ∫  ∫  θ  (  1  −  (            x      2              a      2        +            y      2              b      2        +            z      2              c      2        )  )  θ  (  D  −  (  A  x  +  B  y  +  C  z  )  )  d  x  d  y  d  zChange variables to   X  =  x      /    a,   Y  =  y      /    b,   Z  =  z      /    c (w.l.o.g.   a  ,  b  ,  c  &amp;gt;  0). Then  V  =  a  b  c  ∫  ∫  ∫  θ  (  1  −  (      X    2    +      Y    2    +      Z    2    )  )  θ  (  D  −  (  A  a  X  +  B  b  Y  +  C  c  Z  )  )  d  X  d  Y  d  ZStep 2Now this is the abc times the volume of a unit sphere cut by a plane. The rest which is cut off the sphere is called a spherical cap. So you need the distance d from the plane to the origin. This can be computed at the locus where the plane   D  −  (  A  a  X  +  B  b  Y  +  C  c  Z  ) meets the plane&#039;s normal vector   (  X  ,  Y  ,  Z  )  =  d  (  A  a  ,  B  b  ,  C  c  )      /        (    A    a          )      2        +    (    B    b          )      2        +    (    C    c          )      2      .So you get   d  =  D      /        (    A    a          )      2        +    (    B    b          )      2        +    (    C    c          )      2      .

Find the volume of the intersection of the ellipsoid x^2/a^2+y^2/b^2+z^2/c^2 leq 1 and Ax+By+Cz leq D.

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