A Power Function $f(x)=x^n$, where $x\in[0,1]$ and $n\ge0$.
A Cubic Bezier with points $P_0, P_1, P_2, P_3$ such that $P_0=(0,0)$ and $P_3=(1,1)$. The Cubic Bezier function is $B(t)=(1-t)^3P_0+3(1-t)^2tP_1+3(1-t)t^2P_2+t^3P_3$.

The $n$ in the power function.

How can I get $P_1$ and $P_2$? An approximate solution within about 1% would be acceptable.

  • $\begingroup$ You can't get an exact form for all $n$ - this is somewhat complicated to show, but should be intuitively fairly obvious. If $x$ and $y$ are both cubic polynomials in $t$, then their resultant is a polynomial $P(x,y)$ of degree at most 9; this isn't enough to represent any power functions of higher order than that. If you want an approximation, it would be good to specify your criteria for goodness of approximation. $\endgroup$ Dec 22 '15 at 5:32
  • $\begingroup$ I thought that might be the case. An approximation would be acceptable if there is an approximate solution. Within about 1% would be more than good enough. $\endgroup$
    – Yay295
    Dec 22 '15 at 5:41

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