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construction of a point

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  • Bernard Gibert
    Dear friends, given two points p:q:r and u:v:w in barycentrics, I wish to construct the point c^2*q*u - b^2*r*u - a^2*r*v - b^2*r*v + c^2*r*v + a^2*q*w -
    Message 1 of 3 , Mar 31, 2010
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      Dear friends,

      given two points p:q:r and u:v:w in barycentrics, I wish to construct the point

      c^2*q*u - b^2*r*u - a^2*r*v - b^2*r*v + c^2*r*v + a^2*q*w - b^2*q*w + c^2*q*w :

      a^2*r*u + b^2*r*u - c^2*r*u - c^2*p*v + a^2*r*v + a^2*p*w - b^2*p*w - c^2*p*w :

      -a^2*q*u + b^2*q*u - c^2*q*u - a^2*p*v + b^2*p*v + c^2*p*v + b^2*p*w - a^2*q*w

      I already know one line passing through this point but cannot find another one.

      Any idea ?

      Thank you very much

      Bernard
    • chris.vantienhoven
      [BG] ... Dear Bernard, Did you try splitting the barycentrics in terms? Generally spoken a point P with barycentrics P (P1x+P2x+P3x+P4x : P1y+P2y+P3y+P4y :
      Message 2 of 3 , Apr 2, 2010
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        [BG]
        > given two points p:q:r and u:v:w in barycentrics, I wish to construct the point
        >
        > c^2*q*u - b^2*r*u - a^2*r*v - b^2*r*v + c^2*r*v + a^2*q*w - b^2*q*w + c^2*q*w :
        >
        > a^2*r*u + b^2*r*u - c^2*r*u - c^2*p*v + a^2*r*v + a^2*p*w - b^2*p*w - c^2*p*w :
        >
        > -a^2*q*u + b^2*q*u - c^2*q*u - a^2*p*v + b^2*p*v + c^2*p*v + b^2*p*w - a^2*q*w
        >
        > I already know one line passing through this point but cannot find another one.
        >
        > Any idea ?
        Dear Bernard,

        Did you try splitting the barycentrics in terms?
        Generally spoken a point P with barycentrics
        P (P1x+P2x+P3x+P4x : P1y+P2y+P3y+P4y : P1z+P2z+P3z+P4z)
        can be split into:
        P12 (P1x+P2x : P1y+P2y : P1z+P2z) and P34(P3x+P4x : P3y+P4y : P3z+P4z)
        and also
        P13 (P1x+P3x : P1y+P3y : P1z+P3z) and P24(P2x+P4x : P2y+P4y : P2z+P4z)
        and also
        P14 (P1x+P4x : P1y+P4y : P1z+P4z) and P23(P2x+P3x : P2y+P3y : P2z+P3z).
        Now P = P12.P34 ^ P13.P24 ^ P14.P23.

        When P12, P13, P14, P23, P24, P34 are difficult to construct a further split can be done.
        P1(P1x: P1y : P1z), P2(P2x : P2y : P2z), P3(P3x : P3y : P3z), P4(P4x : P4y : P4z).
        Now P12 is on line P1.P2,etc.

        The point you mentioned has barycentrics:
        (2 SB.q.w - 2 SC.r.v + c^2.q.u - b^2.r.u :
        2 SC.r.u - 2 SA.p.w + a^2.r.v - c^2.p.v :
        2 SA.p.v - 2 SB.q.u + b^2.p.w - a^2.q.w)
        Because the barycentrics consist of 4 terms it can be split as described.

        Best regards,

        Chris van Tienhoven
      • Bernard Gibert
        Dear Chris, ... That s what I did for the line I ve found which corresponds to your P1.P2 but I m still stuck ! Thanks Bernard [Non-text portions of this
        Message 3 of 3 , Apr 2, 2010
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          Dear Chris,

          > Did you try splitting the barycentrics in terms?
          > Generally spoken a point P with barycentrics
          > P (P1x+P2x+P3x+P4x : P1y+P2y+P3y+P4y : P1z+P2z+P3z+P4z)
          > can be split into:
          > P12 (P1x+P2x : P1y+P2y : P1z+P2z) and P34(P3x+P4x : P3y+P4y : P3z+P4z)
          > and also
          > P13 (P1x+P3x : P1y+P3y : P1z+P3z) and P24(P2x+P4x : P2y+P4y : P2z+P4z)
          > and also
          > P14 (P1x+P4x : P1y+P4y : P1z+P4z) and P23(P2x+P3x : P2y+P3y : P2z+P3z).
          > Now P = P12.P34 ^ P13.P24 ^ P14.P23.
          >
          > When P12, P13, P14, P23, P24, P34 are difficult to construct a further split can be done.
          > P1(P1x: P1y : P1z), P2(P2x : P2y : P2z), P3(P3x : P3y : P3z), P4(P4x : P4y : P4z).
          > Now P12 is on line P1.P2,etc.
          >
          > The point you mentioned has barycentrics:
          > (2 SB.q.w - 2 SC.r.v + c^2.q.u - b^2.r.u :
          > 2 SC.r.u - 2 SA.p.w + a^2.r.v - c^2.p.v :
          > 2 SA.p.v - 2 SB.q.u + b^2.p.w - a^2.q.w)
          > Because the barycentrics consist of 4 terms it can be split as described.

          That's what I did for the line I've found which corresponds to your P1.P2 but I'm still stuck !

          Thanks

          Bernard

          [Non-text portions of this message have been removed]
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