Drawing circles, cylinders and cones is a good start for an image synthesis software, but one have to admit it is not fully satisfying. . . This project is the continuation of the previous one, and should allow you to draw more complex forms, such as a torus, which do not emerge from 2nd degree equations, but from superior degree equations (4th degree in the torus case).
The objective of this project is to solve a 4th degree equation: a4x⁴ + a3x³ + a2x² +a1x1 +a0 = 0. A direct resolution method does exist (Ferrari’s method), but does not generalize to higher degrees. Thus, we will rather compare 3 iterative algorithms:
• The bisection method,
• Newton’s method,
• The secant method.
Equations to be solved here will all have one and only one solution, in the [0, 1] interval.
This is the solution we are looking for. The initial value for Newton’s method will be 0.5,
those for the 2 other methods will be 0 and 1.
Just in case you would need it, the derivative of the polynomial function x → a4x⁴ +
a3x³ + a2x² + a1x + a0 is the function x → 4a4x³ + 3a3x² + 2a2x + a1