# Power Function as a Cubic Bezier Curve

With:
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 $B(t)=(1-t)^3P_0+3(1-t)^2tP_1+3(1-t)t^2P_2+t^3P_3$.

Given:
The $n$ in the power function.

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

• 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. Dec 22 '15 at 5:32
• 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. Dec 22 '15 at 5:41