Getting curve control points from a quadratic

[M4573R]

Homework Statement

The first part is to find a quadratic parametric curve f(t) = (p1(t),p2(t)) which passes through the points (0,0) (1,1) and (2,1). (hint: find two interpolating polynomials, one for x and one for y)

Second part is to find the control points for the curve( ie. change basis to the berstein basis)

The Attempt at a Solution

I'm not exactly sure what the first part wants, but I have the equation:
f(t) = ( t, -1/2*t^2 + 3/2*t )
I got the y component from using the Newton form.

For the second part, I don't know how to get the control points. I would assume the first and second given points are control points, and the middle control point is the only one to find. We have been shown that you can get another control point by find the intersection of the tangent lines of the first and last control point. However I don't know if this point will result in a curve that goes exactly through the given middle point. I would change basis to the berstein basis, but I have no idea how to do that. That is the part would like to know more about.

 

[mighty2000]

Hello, thank you for posting your attempt at a solution. Let me try to clarify the first part for you. The problem is asking you to find a quadratic parametric curve (a curve defined by a set of parametric equations) that passes through the given points (0,0), (1,1), and (2,1). This means that for each value of t, the curve should go through these points. The hint is suggesting that you find two separate interpolating polynomials, one for the x-coordinate and one for the y-coordinate. This will give you two equations for x and y in terms of t, which you can then combine to get your parametric equations for f(t).

As for the second part, you are correct in assuming that the given points are control points. However, you are missing one more control point in order to fully define the curve. You can find this point by using the intersection of the tangent lines at the first and last control points, as you mentioned. This will ensure that the curve passes through the middle point as well.

To change basis to the Bernstein basis, you will need to use the Bernstein polynomials. These are a set of polynomials that are commonly used in Bezier curves (a type of parametric curve). You can find the Bernstein polynomials for a quadratic curve by doing a quick online search. Once you have these polynomials, you can use them to write your parametric equations in terms of the Bernstein basis instead of the standard basis. This will give you the control points that you need.

I hope this helps clarify the problem for you. Let me know if you have any further questions or need more help. it's important to always ask questions and seek out clarification when needed. Keep up the good work!