Results 1 to 6 of 6

Math Help - point on an ellipse

  1. #1
    Senior Member
    Joined
    Jul 2006
    Posts
    364

    Question point on an ellipsoid

    Thanks

    Yes, normal is different to normalized/unit

    I've got a somewhat related question about ellipsoids..

    {x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2} - 1 = 0

    If I'm trying to find a point on its surface that is in a direction v what do I need to do?

    For a sphere centred at p with radius r it is just:

    {p + r{v \over \| v \|}}
    Last edited by scorpion007; September 8th 2006 at 04:35 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Please do not put your all your questions in the same thread.
    -=USER WARNED=-.

    Moderator Note: Thread was split.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by scorpion007
    Thanks

    Yes, normal is different to normalized/unit

    I've got a somewhat related question about ellipses..

    {x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2} - 1 = 0

    If I'm trying to find a point on its surface that is in a direction v what do I need to do?

    For a sphere centred at p with radius r it is just:

    {p + r{v \over \| v \|}}
    As has been said before quite recently for a surface represented by the
    equation:

    F(x,y,z)=0

    any normal to the surface is parallel to: \nabla F

    RonL
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Jul 2006
    Posts
    364
    Perhaps you misunderstood what I was asking or I am misunderstanding your response. I'm not trying to find a normal to the surface but a point on the surface in a specific direction.

    For instance the point on a sphere in a given direction can be found with:

    {\vec{p} + r{\vec{v} \over \| \vec{v} \|}}
    where p is the center and v is the direction vector.

    In this case it is simple since the sphere has uniform dimensions in all directions.
    I need to know how to go about solving this problem for an ellipsoid which can have different dimensions in all axes.
    Last edited by scorpion007; September 8th 2006 at 08:17 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by scorpion007

    {p + r{v \over \| v \|}}
    .
    I presume p is the vector whose components are the center of the circle?

    I presume v is a vector.

    I presume r is a scalar.

    Correct?
    ---
    Next time represent vectors as,
    \bold{v}\mbox{ or }\vec{v}
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Apr 2006
    Posts
    399
    Awards
    1
    Quote Originally Posted by scorpion007
    Thanks

    Yes, normal is different to normalized/unit

    I've got a somewhat related question about ellipsoids..

    {x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2} - 1 = 0

    If I'm trying to find a point on its surface that is in a direction v what do I need to do?

    For a sphere centred at p with radius r it is just:

    {p + r{v \over \| v \|}}
    Write v = (v_x , v_y, v_z). Since the center of the ellipsoid is at the origin, you're looking for the intersection of the ellipsoid and the line defined by (x,y,z) = t(v_x , v_y, v_z). So solve {(tv_x)^2 \over a^2}+{(tv_y)^2 \over b^2}+{(tv_z)^2 \over c^2} - 1 = 0 for t, which yields

     \hat{t} = \pm \sqrt{ \left({v_x^2 \over a^2}+{v_y^2 \over b^2}+{v_z^2 \over c^2}\right)^{-1} }

    and  \hat{t}v is your point on the surface. You can verify this formula gives your same solution for a sphere centered at the origin where  a = b = c = r.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Mid-point of chord on ellipse
    Posted in the Geometry Forum
    Replies: 1
    Last Post: May 13th 2010, 01:40 AM
  2. Replies: 1
    Last Post: April 19th 2010, 12:35 PM
  3. find an edge point of ellipse
    Posted in the Math Software Forum
    Replies: 5
    Last Post: April 11th 2010, 06:25 AM
  4. Replies: 2
    Last Post: August 21st 2009, 05:13 AM
  5. [SOLVED] [SOLVED] Find antoher Point on an Ellipse
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 3rd 2007, 12:11 AM

Search Tags


/mathhelpforum @mathhelpforum