I would like to find the value of parameter $t$ of a cubic Bezier curve for a given point $x, y$ lying on the curve. In other words, I would like to find $t$ which, if the Bezier curve would be evaluated at it, would result in given $x, y$.
I found similar question already posted here but mine is a bit different:
- the accepted answer refers to the notes of Tom Sederberg. I tried to look it up but there seems to be an error: the equation at the very bottom of the page 203:
$984100t^9_2 - 458200t^8_2 + 8868537t^7_2 - 9420593t^6_2 + 5949408t^5_2 - 2282850t^4_2 + 522890t^3_2 - 67572t^2_2 + 4401t_2 - 109 = 0$
seemingly cannot result in the very last column of the table at the p. 204:
Am I correctly assuming that the column should contain the roots of the equation? When I plug the coefficients into a online solver:
9 984100 -458200 8868537 -9420593 5949408 -2282850 522890 -67572 4401 -109
I get several complex roots instead.
- Even if I could make this "inversion technique" work, what would I do if the point would lie on a self-intersection (there: "double point")? I mean, if the given point should result in two or more $t$-s, how to find all of them? Sederberg just leaves a comment: "At a double point, an inversion equation will always give 0/0."
Thank you in advance!