I have following points (-|b-a|,a), (0,b), (|c-b|,c) with a, b and c as two-dimensional vectors. These should be interpolated component-by-component with a second-degree polynomial p.
My problem now is, that I don't know how the polynomial should look like: $p(-|b-a|)= \alpha(-|b-a|)^2 + \beta(-|b-a|)+\gamma = a$
$p(-|b_x - a_x|) = \alpha(-|b_x-a_x|)^2 + \beta(-|b_x-a_x|)+\gamma = a_x$ $p(-|b_y - a_y|) = \alpha(-|b_y-a_y|)^2 + \beta(-|b_y-a_y|)+\gamma = a_y$
$\cdots$
Is my conversion correct?
No. The polynomial you need is:
$$p(t) = \frac{t(t-|c-b|)}{|b-a|(|b-a|+|c-b|)}a + \frac{(t+|b-a|)(t-|c-b|)}{-|b-a||c-b|} b + \frac{t(t+|b-a|)}{|c-b|(|c-b|+|b-a|)} c.$$