polynomial coordinates
example
spline functions using hermite polynomials can help generate smooth curves without any sharp corners for the body panels in 3D car models
answer
[p]B = [2, 0, -4, -8]
steps
1. We have a set of 4 special polynomials called the Hermite polynomials. These 4 polynomials are:
- 1st polynomial: 1
- 2nd polynomial: 2t
- 3rd polynomial: -2 + 4t^2
- 4th polynomial: -12t + 8t^3
2. These 4 polynomials form a basis B for the vector space P3. This means any 3rd degree polynomial can be written as a combination of these 4 basis polynomials.
3. We are given one 3rd degree polynomial:
p(t) = 2 - 4t^2 - 8t^3
4. To write p(t) in terms of the basis B, we need to find numbers c1, c2, c3, c4 such that:
p(t) = c1(1st polynomial) + c2(2nd polynomial) + c3(3rd polynomial) + c4(4th polynomial)
5. By matching coefficients, we find:
c1 = 2
c2 = 0
c3 = -4
c4 = -8
6. Therefore, the coordinate vector of p(t) relative to the basis B is:
[2, 0, -4, -8]
So in very simple terms, we wrote the given polynomial p(t) as a combination of the 4 special Hermite polynomials, and the numbers [2, 0, -4, -8] tell us how much to use of each polynomial.
- The basis B of P3 consisting of the Hermite polynomials is:
B = {1, 2t, -2 + 4t^2, -12t + 8t^3}
- The polynomial p(t) is given as:
p(t) = 2 - 4t^2 - 8t^3
To find the coordinate vector of p(t) relative to the basis B, we express p(t) as a linear combination of the basis polynomials:
p(t) = c1(1) + c2(2t) + c3(-2 + 4t^2) + c4(-12t + 8t^3)
Matching coefficients, we get:
c1 = 2
c2 = 0
c3 = -4
c4 = -8
Therefore, the coordinate vector of p(t) relative to the basis B is:
\[p\]B = [2, 0, -4, -8]
So the final answer is:
\[p\]B = [2, 0, -4, -8]
claudeAI