1

I have a 2 curves to provide a height for a time:

1: $H=T^2 * R$ - in this curve, R is known and T is not known

2: $h=t^2 * r $ - in this curve, t is known and r is not known

I would like to blend them so that $H+h = Y$ and their slopes are the same. So I know that $S = 2TR$, $s = 2tr$ and $S=s$

And I know that $Y$ is $T^2 * R + t^2 * r$

But given $Y, R,$ and $t,$ how do I get $T$ or $r?$

Ok I got a bit further: Because $s = S$ I can eliminate one of the unknowns $T$:

$T = rt/R$ so

$Y = rt^2 + rt^2 / R$ , so I just need to reshape this into $t = ??$

Ianb
  • 123

1 Answers1

1

From the slopes you obtain: $$r=T\frac{R}{t}$$ You also obtain a quadratic function $r(T)$: $$r(T)=\frac{1}{t^2}\left(R T^2 - Y\right)$$ So now we need to solve it for $T$: $$T\frac{R}{t}=\frac{1}{t^2}\left(R T^2 - Y\right)$$ $$T^2 - tT - \frac{Y}{R}=0$$

We have then $\Delta=t^2+4\frac{Y}{R}$ and $$T = \frac{t \pm \sqrt{t^2+4\frac{Y}{R}}}{2}$$ and $$r = \frac{R}{t} T$$

Jaroslaw Matlak
  • 4,895