Finding a simple spline-like interpolating function

Question

I am looking for a continuous function $y=f(x,\alpha)$ for the interval $0\le x \le 1$ such that $0\le y \le 1$ and $y(0,\alpha)=0$ and $y(1,\alpha) = 1$ and $y(\alpha,\alpha) = 1-\alpha$ and $dy/dx|_{x=\alpha}=1$. It is effectively symmetrical about the line $y=1-x$. Typically $0.2\le \alpha \le 0.8$.

It looks like a parabola but rotated so that the new y axis is along $y=1-x$, but since I need to be able to compute $y$ given $x$, I have not yet been able to work it out.

The exact form is not too important, but since this is to go in a OpenGL vertex shader I am trying to come up with something that is as simple as possible in terms of math operations.

Answer 1

The solution by @bubba is quite nice, but if by "It looks like a parabola but rotated", you don't mean that it necessarily is exactly a parabola, then you can get a slightly simpler expression by using a rotated hyperbola instead. The general form for a hyperbola with axis-aligned asymptotes is $$y=p\frac{x+q}{x+r}.$$ Plugging in your conditions gives $$p=\frac{(1-\alpha)^2}{1-2\alpha}, \quad q=0, \quad r=\frac{\alpha^2}{1-2\alpha},\\ y=\frac{(1-\alpha)^2x}{(1-2\alpha)x+\alpha^2}.$$ Here are the graphs of the function for $\alpha=0.1,0.2,\ldots,0.9$:

Answer 2

I constructed a quadratic Bezier curve (somewhat like your b-spline idea), and then converted it to $y = f(\alpha, x)$ form. The result is:

$$y = \frac{1 + 4 \alpha (x-1)-2 x+\sqrt{(1-4 \alpha)^2+8 (1-2 \alpha) x}}{2-4 \alpha}$$

My guess is that this isn't very good for your purposes because of the square root.

Also, this formula only works for values of $\alpha$ with $0.25 \le \alpha \le 0.75$.

Here are the graphs of these functions for $\alpha = 0.25, 0.35, 0.45, 0.55, 0.65, 0.75$

They are all (rotated) parabolas.

The $0.25 \le \alpha \le 0.75$ restriction could be lifted by using a rational quadratic curve, instead. But the square root won't go away, so I want to know if you can live with that before I spend time working out the details of the rational case.