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Polynomial Parametric Curves

import sympy as sp
sp.init_printing(order='grevlex')

t = sp.symbols('t')

The coefficients are written as bold letters, because they are elements of a vector space (e.g. \(\mathbb{R}^3\)).

We are using bold symbols because apart from simple scalars (for one-dimensional functions), these symbols can also represent vectors in two- or three-dimensional space.

coefficients = sp.Matrix(sp.symbols('a:dbm')[::-1])
coefficients

$\displaystyle \left[\begin{matrix}\boldsymbol{d}\\\boldsymbol{c}\\\boldsymbol{b}\\\boldsymbol{a}\end{matrix}\right]$

Monomial basis functions:

b_monomial = sp.Matrix([t**3, t**2, t, 1]).T
b_monomial

$\displaystyle \left[\begin{matrix}t^{3} & t^{2} & t & 1\end{matrix}\right]$

b_monomial.dot(coefficients)

$\displaystyle \boldsymbol{d} t^{3} + \boldsymbol{c} t^{2} + \boldsymbol{b} t + \boldsymbol{a}$

This is a cubic polynomial in its canonical form (monomial basis).

Monomial basis functions:

from helper import plot_basis

plot_basis(*b_monomial, labels=b_monomial)

It doesn’t look like much, but every conceivable cubic polynomial can be formulated as exactly one linear combination of those basis functions.

Example:

example_polynomial = (2 * t - 1)**3 + (t + 1)**2 - 6 * t + 1
example_polynomial

$\displaystyle \left(2 t - 1\right)^{3} + \left(t + 1\right)^{2} - 6 t + 1$

sp.plot(example_polynomial, (t, 0, 1));

<sympy.plotting.plot.Plot at 0x7ff97a689a90>

Can be re-written with monomial basis functions:

example_polynomial.expand()

$\displaystyle 8 t^{3} - 11 t^{2} + 2 t + 1$