Find Bezier 4 control points from parametric eqn

[atrus_ovis]

Homework Statement

Given a parametric eqn for the x,y coordinates , x(t),y(t) , how can i find the control points?

Homework Equations

de Casteljau's algorithm
is used to compute the value for a given t given the control points.

The Attempt at a Solution

Control Points 0,3 are given for values t=0,t=1.
Maybe for an arbitrary t, you can backstep the de Casteljau algorithm, and end up in a relation of the other two dependent on the first and last control points' coordinates?
But i tried this and get no definite solution.

 

[chiro]

Hey atrus_ovis.

For this problem you have four unknowns so you will need four equations.

You already have found two control points using t=0 and t=1 so now you have two unknowns left (remember each point is independent).

So the next thing is to use the definition of the Bezier curve given the control points, and substitute in the start and end-point to get an equation in terms of t for points 2 and 3.

Now you already have an equation in t for x(t) and y(t), so the next step comes equating the two together in terms of your parameter. Finally you can simplify your answer by comparing the two equations and choosing a value of t (twice) that gives the simplest system of equations for your two unknown points. Then solve for this using substitution (just like you would do for any two equations where you substitute one into the other).

The above will depend on the equation you have: one t-value may give a simpler expression in comparison to using another value.

 

[atrus_ovis]

Got it!

Thanks for your reply Chiro.