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Math Help - Parametric Cubic Curve Middle Coordinates

  1. #1
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    Parametric Cubic Curve Middle Coordinates

    If you have two end points of x0,y0,z0 and x1, y1, z1 respectively of a parametric cubic curve and their known derivatives how do you find its middle coordinates x0.5, y0.5, z0.5

    Any help that would be great thanks.
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  2. #2
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    First, write the general formula...

    x(t)=at^3+bt^2+ct+d
    x'(t)=3at^2+2bt+c

    For vector variables x,a,b,c

    s=x(0)=d
    t=x'(0)=c
    u=x(1)=a+b+c+d
    v=x'(1)=3a+2b+c

    Given vectors s,t,u,v, you should be able to easily solve this system for a,b,c,d...

    a=2s+t-2u+v
    b=-3s-2t+3u-v
    c=t
    d=s

    All of these are vectors, so you now have your general parametric equation x(t)=at^3+bt^2+ct+d

    Plug and chug.
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