Cubic Equations in Python

Using Polynomial equations in Python

The equation of a cubic curve has the general form y = ax³ + bx² + cx + d;
where a, b, c, d are constants. The curve (graph) of any 3^rd degree polynomial has a maximum point and minimum point.

Generating Cubic Curves for Python

To generate a cubic equation, all we will need are the maximum and minimum points of the curve.

y = ax³ + bx² + cx + d ----- (eqn 0)

^dy/_dx = y^I = 3ax² + 2bx + c
At maximum point, y^I = 0
y^I|_{(x = x_max)} = 0
3ax_max² + 2bx_max + c = 0 ----- (eqn 1)
At minimum point, y^I = 0
y^I|_{(x = x_min)} = 0 ----- (eqn 2)
3ax_min² + 2bx_min + c = 0
Subtracting both derived equations
y^I|_{(x = x_max)} - y^I|_{(x = x_min)}
⇒ 3a(x_max² - x_min²) + 2b(x_max - x_min) = 0
2b(x_max - x_min) = -3a(x_max² - x_min²)

b =	-3a(x_max - x_min)(x_max + x_min)
	2(x_max - x_min)

b = -³/₂a(x_max + x_min)

Substituting b in (eqn 1)
3ax_max² + 2bx_max + c = 0
3ax_max² + 2(^-3a/₂)(x_max + x_min)x_max + c = 0
3ax_max² - 3ax_max(x_max + x_min) + c = 0
3ax_max² - 3ax_max² - 3ax_maxx_min + c = 0
c = 3ax_maxx_min

From the general equation(eqn 0)
y = ax³ + bx² + cx + d
y_max = ax_max³ + bx_max² + cx_max + d
Substituting for b & c
⇒ y_max = ax_max³ - ³/₂a(x_max + x_min)x_max² + 3ax_maxx_minx_max + d
y_max = ax_max³ - ³/₂ax_max³ - ³/₂ax_max²x_min + 3ax_max²x_min + d
y_max = ¹/₂[2ax_max³ - 3ax_max³ - 3ax_max²x_min + 6ax_max²x_min + 2d]
y_max = ¹/₂[ -ax_max³ + 3ax_max²x_min + 2d]
2y_max = -a(x_max - 3ax_min)x_max² + 2d
2d = 2y_max + a(x_max - 3ax_min)x_max²
d = y_max + ^a/₂(x_max - 3ax_min)x_max²

From the general equation(eqn 0)
y = ax³ + bx² + cx + d
y_max = ax_max³ + bx_max² + cx_max + d
y_min = ax_min³ + bx_min² + cx_min + d
Subtracting both derived equations
y_max - y_min = a(x_max³ - x_min³) + b(x_max² - x_min²) + c(x_max - x_min)
y_max - y_min = (x_max - x_min)[a(x_max² + x_maxx_min + x_min²) + b(x_max + x_min) + c]
Substituting for b & c
y_max - y_min = (x_max - x_min)[a(x_max² + x_maxx_min + x_min²) - ^3a/₂(x_max + x_min)² + 3ax_maxx_min]
y_max - y_min = a(x_max - x_min)[x_max² + x_maxx_min + x_min² - ³/₂(x_max² + 2x_maxx_min + x_min²) + 3x_maxx_min]
2(y_max - y_min) = a(x_max - x_min)[2x_max² + 2x_maxx_min + 2x_min² - 3(x_max² + 2x_maxx_min + x_min²) + 6x_maxx_min]
2(y_max - y_min) = a(x_max - x_min)(2x_max² + 2x_maxx_min + 2x_min² - 3x_max² - 6x_maxx_min - 6x_min² + 6x_maxx_min)
2(y_max - y_min) = a(x_max - x_min)(-x_max² + 2x_maxx_min - x_min²)
2(y_max - y_min) = -a(x_max - x_min)(x_max² - 2x_maxx_min + x_min²)
2(y_max - y_min) = -a(x_max - x_min)(x_max - x_min)²
2(y_max - y_min) = -a(x_max - x_min)³

Hence:

a =	-2(y_max - y_min)
	(x_max - x_min)³

b = -³/₂a(x_max + x_min)
c = 3ax_maxx_min
&
d = y_max + ^a/₂(x_max - 3ax_min)x_max²

Code to Make a Graphic Object Animate through a Cubic Curve in Python

To make a graphic (dot) travel by the equation of a cubic curve, continuously increment x by some interval, and use the equation to get the corresponding y value.
.

Create a new file; File, New File.
Call it and CubicPath.py.
Type out the adjoining Python / Turtle code for animating an image body through the path of a cubic / polynomial curve.

Code to Animate an Image Body through a Polynomial Curve in Python

Using Polynomial equations in Python

Generating Cubic Curves for Python

Code to Make a Graphic Object Animate through a Cubic Curve in Python

Python code for CubicPath file