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Math Help - Two equations with two unknowns: how to rearrange or solve the equations

  1. #1
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    Two equations with two unknowns: how to rearrange or solve the equations

    I have to equations:
    y = T*Cx*U and x = T*Cy*U where

    x and y are scalar values that are known,
    Cx and Cy are both 4x4 matrices that are known.
    T is a 1x4 vector : T=[1 t t^2 t^3] where t is unknown and
    U is a 4x1 vector : U=[1 u u^2 u^3]' where ' means transpose and u is unknown.

    So basically I have two equations and two unknowns, t and u. I would like to figure out a way to represent the equation so that
    the unknowns are a function of the knowns
    so that t = f(x,y,Cx,Cy) and u = f(x,y,Cx,Cy) or T = f(x,y,Cx,Cy) and U = f(x,y,Cx,Cy).

    I imagine that getting the final numerical solution might involve getting the roots of a cubic but I don't know how to get that far.

    I've tried to figure out a way to substitute one equation into the other to eliminate one of the unknowns (i.e., U or T) but I get stuck because I'm not sure what to do about dividing by a vector (i.e., finding the inverse of a vector?).
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Two equations with two unknowns: how to rearrange or solve the equations

    can you post the matrix C ? and the values x,y?
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  3. #3
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    Re: Two equations with two unknowns: how to rearrange or solve the equations

    Quote Originally Posted by jakncoke View Post
    can you post the matrix C ? and the values x,y?
    I can give you example numbers but they are floating point values that I'm pulling from a program I'm working on so I don't know if they would be helpful. In my program I can go from u,t to x,y but I would like to go the other way so I am trying to invert the equation. For this example, t = 0.3624 and u = 0.4950. t and u are always between 0 and 1. (I am trying to invert a bicubic interpolation fn. )

    x = 1022.9;
    y = 495.9297;

    Cx =

    930.8436 12.3171 36.8271 -18.4136
    220.5397 -2.7589 -8.5661 4.2831
    2.2148 0.3214 0.6427 -0.3214
    -0.4175 -0.0662 -0.1324 0.0662

    Cy =

    400.3912 194.3285 -5.4094 2.7047
    0.3092 1.2141 0.4406 -0.2203
    -0.0395 -0.1551 -0.0583 0.0291
    0.0066 0.0139 0.0013 -0.0006
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