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2

$begingroup$

I want to find out $(u,v)$ on the ellipse $$fracu^2a^2+fracv^2b^2=1$$ for a point $(x,y)$ inside ellipse, which will denote shortest distance from $(x,y)$ to the ellipse boundary. I expressed this problem as a constrained optimization problem by this Lagrangian:

beginalign
L(u,v,lambda) &= (u-x)^2+(v-y)^2-lambdaleft(fracu^2a^2 + fracv^2b^2-1right)\[1ex]
fracpartial L(u,v,lambda)partial u&=2u-2x-frac2lambda ua^2\[1ex]
fracpartial L(u,v,lambda)partial v&=2v-2y-frac2lambda vb^2\[1ex]
fracpartial L(u,v,lambda)partial lambda&=fracu^2a^2+fracv^2b^2-1\[1em]
u&=fraca^2xa^2-lambdatag1\[1ex]
v&=fracb^2yb^2-lambdatag2\[1ex]
fracu^2a^2&+fracv^2b^2-1=0tag3\
endalign

After substituting $u$ and $v$ in equation $(3)$, I get
beginaligned
fraca^2x^2(a^2-lambda)^2+fracb^2y^2(b^2-lambda)^2-1=0\
endaligned
I am getting a 4th degree equation of $lambda$. I am facing difficulties in finding algebraic solution of the equation.

optimization lagrange-multiplier

share|cite|improve this question

edited Mar 30 at 3:56

David M.

2,198421

asked Mar 29 at 11:26

cseju19cseju19

215

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am afraid that any one would be facing the same difficulties.
  $endgroup$
  – Claude Leibovici
  Mar 30 at 4:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your problem is equivalent to find a normal passing through a given point. There're two to four normals which concurrent to the given point. See another answer of mine here.
  $endgroup$
  – Ng Chung Tak
  Apr 1 at 10:38
  
  

  
  
  
  

add a comment | 

2

$begingroup$

I want to find out $(u,v)$ on the ellipse $$fracu^2a^2+fracv^2b^2=1$$ for a point $(x,y)$ inside ellipse, which will denote shortest distance from $(x,y)$ to the ellipse boundary. I expressed this problem as a constrained optimization problem by this Lagrangian:

beginalign
L(u,v,lambda) &= (u-x)^2+(v-y)^2-lambdaleft(fracu^2a^2 + fracv^2b^2-1right)\[1ex]
fracpartial L(u,v,lambda)partial u&=2u-2x-frac2lambda ua^2\[1ex]
fracpartial L(u,v,lambda)partial v&=2v-2y-frac2lambda vb^2\[1ex]
fracpartial L(u,v,lambda)partial lambda&=fracu^2a^2+fracv^2b^2-1\[1em]
u&=fraca^2xa^2-lambdatag1\[1ex]
v&=fracb^2yb^2-lambdatag2\[1ex]
fracu^2a^2&+fracv^2b^2-1=0tag3\
endalign

After substituting $u$ and $v$ in equation $(3)$, I get
beginaligned
fraca^2x^2(a^2-lambda)^2+fracb^2y^2(b^2-lambda)^2-1=0\
endaligned
I am getting a 4th degree equation of $lambda$. I am facing difficulties in finding algebraic solution of the equation.

optimization lagrange-multiplier

share|cite|improve this question

edited Mar 30 at 3:56

David M.

2,198421

asked Mar 29 at 11:26

cseju19cseju19

215

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am afraid that any one would be facing the same difficulties.
  $endgroup$
  – Claude Leibovici
  Mar 30 at 4:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your problem is equivalent to find a normal passing through a given point. There're two to four normals which concurrent to the given point. See another answer of mine here.
  $endgroup$
  – Ng Chung Tak
  Apr 1 at 10:38
  
  

  
  
  
  

add a comment | 

$begingroup$

I want to find out $(u,v)$ on the ellipse $$fracu^2a^2+fracv^2b^2=1$$ for a point $(x,y)$ inside ellipse, which will denote shortest distance from $(x,y)$ to the ellipse boundary. I expressed this problem as a constrained optimization problem by this Lagrangian:

beginalign
L(u,v,lambda) &= (u-x)^2+(v-y)^2-lambdaleft(fracu^2a^2 + fracv^2b^2-1right)\[1ex]
fracpartial L(u,v,lambda)partial u&=2u-2x-frac2lambda ua^2\[1ex]
fracpartial L(u,v,lambda)partial v&=2v-2y-frac2lambda vb^2\[1ex]
fracpartial L(u,v,lambda)partial lambda&=fracu^2a^2+fracv^2b^2-1\[1em]
u&=fraca^2xa^2-lambdatag1\[1ex]
v&=fracb^2yb^2-lambdatag2\[1ex]
fracu^2a^2&+fracv^2b^2-1=0tag3\
endalign

After substituting $u$ and $v$ in equation $(3)$, I get
beginaligned
fraca^2x^2(a^2-lambda)^2+fracb^2y^2(b^2-lambda)^2-1=0\
endaligned
I am getting a 4th degree equation of $lambda$. I am facing difficulties in finding algebraic solution of the equation.

optimization lagrange-multiplier

share|cite|improve this question

edited Mar 30 at 3:56

David M.

2,198421

asked Mar 29 at 11:26

cseju19cseju19

215

$endgroup$

share|cite|improve this question

edited Mar 30 at 3:56

David M.

2,198421

asked Mar 29 at 11:26

cseju19cseju19

215

share|cite|improve this question

edited Mar 30 at 3:56

David M.

2,198421

asked Mar 29 at 11:26

cseju19cseju19

215

edited Mar 30 at 3:56

David M.

2,198421

edited Mar 30 at 3:56

David M.

2,198421

asked Mar 29 at 11:26

cseju19cseju19

215

asked Mar 29 at 11:26

cseju19cseju19

215