I have a parametric curve.
X = time, and y = position. If I know the path of the curve, lets say I want to find what the y value will be on the curve at time x.
If I know the value of x, how would I do this? Do I need to know the length of the curve? I have the derivatives down to the third, I am at a loss as far as how to go about this.
x = known
y = unknown
t = unknown
x(t) = 10t³-20t²+20t
y(t) = 25t³-99t²+25t
0<t<1
0<x<1
What is the value of y on the curve when x is at .7?
Is there a specific formula I use, or a term this is called? is it possible?
I asked a similar question on here but y was known, dont know if its ok to post another question. thanks
Broadly, you take your equation for $x$ in terms of $t$ and solve it for $t$ given $x$, then substitute that value in.
However, the way you would do so will depend on what that equation is, and in many cases it is not possible to do. You've given a cubic equation for $x(t)$, which while solvable is what mathematicians call "damn ugly". If your equation was a higher order polynomial, or not polynomial at all, then who knows whether it would be solvable?
If, however, you are more interested in getting results that aren't exact, but are accurate to a given precision, then you want some kind of root-finding algorithm. You take the equation $x = f(t)$, rewrite it as $x - f(t) = 0$, then use an appropriate algorithm to find the roots of $g(t; x) = x - f(t)$. For example, in your case, you'd try to find a root for $10t^3 - 20t^2 + 20t - 0.7 = 0$. Given that you know $t \in [0, 1]$, you could use a bracket-style method with 0 and 1 as your initial bounds, or you could try an open method starting at, say, $t = 0.5$.