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Linked points lying on spheres

  1. Apr 17, 2009 #1
    Hi to everyone,

    This is my first post here, but I've been a reader for a long time.

    I have this problem which I can't find the solution.

    . There are 3 points on R3, let's call them P1[x1, y1, z1], P2[x2, y2, z2] and P3[x3, y3, z3].

    . These points are equidistant, so they can be considered an equilateral triangle.

    . If it's an eq. triangle the distance from it's geometric center (Pc[xc, yc, zc]) to each vertex is the same.

    . Both X and Y positions of the triangle's center are constrained to 0, and P1 is positioned on the XZ plane. So it's free to move on the Z axis, and free to rotate on X and Y axis.

    . Since it's going to rotate about Pc and P1 is on XZ plane, y2 = -y3

    . Now we have 3 spheres of centers , C2[C2x, C2y, C2z], C1[C2x, C2y, C2z] and radius R1, R2 and R3.

    . Imagine the points above the spheres and start descenting until each one touches one sphere. Each point lie on one spehre.

    . Now this 3 points define a plane, which is what I need. Note that this plane is not tangent to the spheres, the 3 points just lie on the spheres.

    Resuming:

    Sphere 1:
    Center: C1[C1x, C1y, C1z]
    Radius: R1

    Sphere 2:
    Center: C2[C2x, C2y, C2z]
    Radius: R2

    Sphere 3:
    Center: C3[C3x, C3y, C3z]
    Radius: R3

    Point 1:
    Position: [x1, 0, z1]

    Point 2:
    Position: [x2, y2, z2]

    Point 3:
    Position: [x3, -y2, z3]

    Triangle center (Pc):
    Position: [0, 0, zc]

    Sphere centers and radius are constants and I need to determinate P1, P2, P3. Whith them I can determine the Plane I need.

    The closest I got was this:

    -> P1 should lie on sphere 1:
    (x1 - C1x)^2 + (y1 - C1y)^2 + (z1 - C1z)^2 - R1^2 = 0
    (x1 - C1x)^2 + (0 - C1y)^2 + (z1 - C1z)^2 - R1^2 = 0
    (x1 - C1x)^2 + C1y^2 + (z1 - C1z)^2 - R1^2 = 0 (1)

    -> P2 should lie on sphere 2:
    (x2 - C2x)^2 + (y2 - C2y)^2 + (z2 - C2z)^2 - R2^2 = 0 (2)

    -> P3 should lie on sphere 3:
    (x3 - C3x)^2 + (y3 - C3y)^2 + (z3 - C3z)^2 - R3^2 = 0
    (x3 - C3x)^2 + (-y2 - C3y)^2 + (z3 - C3z)^2 - R3^2 = 0 (3)

    -> Distance from center to each vertex is constant (Rcv):

    Rcv^2 = (xc - x1)^2 + (yc - y1)^2 + (zc - z1)^2
    Rcv^2 = (0 - x1)^2 + (0 - 0)^2 + (zc - z1)^2
    Rcv^2 = x1^2 + (zc - z1)^2 (4)


    Rcv^2 = (xc - x2)^2 + (yc - y2)^2 + (zc - z2)^2
    Rcv^2 = (0 - x2)^2 + (0 - y2)^2 + (zc - z2)^2
    Rcv^2 = x2^2 + y2^2 + (zc - z2)^2 (5)

    Rcv^2 = (xc - x3)^2 + (yc - y3)^2 + (zc - z3)^2
    Rcv^2 = (0 - x3)^2 + (0 - (-y2))^2 + (zc - z3)^2
    Rcv^2 = x3^2 + y2^2 + (zc - z3)^2 (6)

    -> Pc is coplanar with P1, P2 and P3, so, it must lie on the plane defined by these points:

    [xc-x1 yc-y1 zc-z1]
    [x2-x1 y2-y1 z2-z1] = 0
    [x3-x1 y3-y1 z3-z1]

    [ -x1 0 zc-z1]
    [x2-x1 y2 z2-z1] = 0
    [x3-x1 -y2 z3-z1]

    Solving for the determinant gives me another equation:

    -x1*y2*z3 - x1*y2*z2 - y2*x2*zc + y2*x2*z1 + 2*y2*x1*zc - y2*x3*zc + y2*x3*z1 = 0 (7)


    So I ended with 7 non linear equations and 8 unknowns (x1, x2, x3, y2, z1, z2, z3 and zc).

    What could I be missing? I think these equations only should be enough to define the points.

    Since all triangle's sides are equal de distance from P1 to P2, P2 to P3 and P3 to P1 is equal. I tried do include one of these equations in the set and them tried to solve on Maple but it took 12 hours calculating and didn't finish it, so I gave up because I think this last equation is redundant since I already included the distance from center to vertex in the set.

    I know this is a big post, but does anyone would have a light for me?

    If anyone is wondering where does this come from... I'm trying to model the swashplate position of my RC helicopter from the servo positions to later, be able to model the it's dynamics.

    Thanks a lot and best regards,
    Rodrigo Basniak
     
  2. jcsd
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