Quadratic vector extrapolation

[makc]

Hi, I need to extrapolate vector [tex]v_{-1}[/tex] from [tex]v_{0}[/tex], [tex]v_{1}[/tex] and [tex]v_{2}[/tex] (see attached pic), so that if [tex]v_{2}[/tex] is on the right/left (2D case for simplicity) of [tex]v_{1}-v_{0}[/tex], [tex]v_{-1}[/tex] would also be on the right/left.

My initial solution was like this:
[tex]v_{2} - v_{1} = v_{1} - v_{0} + dv[/tex],
[tex]v_{1} - v_{0} = v_{0} - v_{-1} + dv[/tex],
and from there
[tex]v_{-1} = 3(v_{0} - v_{1}) + v_{2}[/tex].

This, however, produces ugly results when abs ([tex]v_{2} - v_{1}[/tex]) < abs ([tex]v_{1} - v_{0}[/tex]) - point [tex]v_{-1}[/tex] is placed very far from [tex]v_{0}[/tex]. So I need a better formula for this.

Any help?

 

[makc]

okay, this forum is titled "physics and math help". so, once in a god knows how many month I post for help, and what do I receive? a warning? if this is how you intent to help, well... screw you and your warnings. you may ban me right away, I will get help at some other place :(

 

[m2003]

I would first suggest taking a step back and analyzing the problem at hand. What is the purpose of extrapolating vector v_{-1} from v_{0}, v_{1}, and v_{2}? Is there a specific application or scenario where this is needed? Understanding the context and purpose of the problem can often lead to a more effective solution.

With that being said, there are a few approaches you could take to improve upon your initial solution. One option could be to use a quadratic function to fit the three given vectors and then use that function to extrapolate v_{-1}. This would involve finding the coefficients of the quadratic function through a process such as least squares regression.

Another approach could be to use a weighted average of the three given vectors to calculate v_{-1}. This could help account for the scenario where v_{2} is far from v_{0} and v_{1}, as the weight of v_{2} in the calculation could be decreased. This approach would require determining appropriate weights for each vector, which could be based on factors such as the distance between the vectors or the confidence in their accuracy.

Ultimately, the best formula or method for extrapolating v_{-1} will depend on the specific circumstances and goals of your problem. It may also be helpful to consult with a colleague or expert in the field to discuss potential solutions and determine the most appropriate approach.