Pure Exchange Economy
As you note in your question the (inner points of the) Pareto set are defined by
$$
x_1^1 = \frac{\alpha_1 r^1 x_1^2}{\alpha_1 x_1^2 - \alpha_2 x_1^2 + \alpha_2 r^2}
$$
In order to better examine this curve, let us treat it as a function. Let
$$
f(x) = \frac{\alpha_1 r^1 x}{\alpha_1 x - \alpha_2 x + \alpha_2 r^2}.
$$
As I stated in my comment
$$
f(0) = 0 \mbox{ and } f(r^2) = r^1,
$$
so the set indeed goes from one corner to the other. In between you have
$$
\frac{d \ f(x)}{d x} = \frac{\alpha_1\alpha_2r^1r^2}{\left(\alpha_1 x - \alpha_2 x + \alpha_2 r^2\right)^2}
$$
and
$$
\frac{d^2 f(x)}{dx^2} = 2 \cdot \frac{(\alpha_1\alpha_2r^1r^2) \cdot (\alpha_2-\alpha_1)}{\left(\alpha_1 x - \alpha_2 x + \alpha_2 r^2\right)^3}.
$$
The first derivative is always positive because $r^2 &gt; x$ and all other parameters are positive. For the same reason, the sign of the second derivative depends on the sign of $\alpha_2-\alpha_1$.  Thus the Pareto set will be a strictly increasing curve, which is convex if agent 2 values good 1 'relatively more', concave if agent 1 values good 1 'relatively more', and a straight line if they value it 'relatively equally'.