Quadratic Region in Python

Checking the Boundaries of a Quadratic Curve in Python

Remember that any quadratic equation always have two roots for any value of y (except at it's maximum or minimum point).
All we need to do is use these two roots (x values) as boundaries for our check.
y = ax² + bx + c
ax² + bx + (c-y) = 0

x =	-b ± √(b² - 4a(c-y))
	2a

Our range will then be:

-b - √(b² - 4a(c-y))	≤ x ≤	-b + √(b² - 4a(c-y))
2a	≤ x ≤	2a

where a, b, and c are constants.

We will reuse the moving ball graphic and check for when it enters the region of our curve.

Code to Detect Entrance into Quadratic Region in Python

To check for when our ball enters the quadratic curve, we will continually check the x position of the ball against the x position gotten using the quadratic equation at the same y position as that of the ball.

We'll designate the coordinates of the ball as (x_b, y_b), and those of the curve as (x_q, y_q).

Create a new file; File, New File.
Call it QuadraticRegion.py.
Type out the adjoining Python / Turtle code for detecting the instance a travelling body crosses the boundary of a quadratic curve.

Python code for QuadraticRegion file

#!usr/bin/env python3

import math
from Facet import Template

def prep():
    global ball_colour, x_ball, y_ball, xq_lb, xq_ub, turtle_radius

    # bind fun function
    scene.button.onrelease(play)

    turtle.penup()
    # button text
    turtle.setposition(scene.button.xcor(), scene.button.ycor()-10)
    turtle.pendown()
    turtle.write("Move", align="center", font=("Arial",16,"bold"))

    # ball parameters
    diameter = 5
    ball_colour = "#ffff00"
    x_ball = 100 - scene.wnd_width/2; y_ball = 0

    # quadratic curve parameters
    xq_start = -200; yq_start = scene.wnd_height/3
    xq_min = 0; yq_min = -scene.wnd_height/3

    a = (yq_start - yq_min) / math.pow((xq_start - xq_min), 2)
    b = -2*a*xq_min
    c = yq_min + a*math.pow(xq_min, 2)

    # curve bounds
    discriminant = math.sqrt(b*b - 4*a*(c - y_ball))
    if a < 0:
        xq_lb = (-b + discriminant) / (2*a)
        xq_ub = (-b - discriminant) / (2*a)
    else:
        xq_lb = (-b - discriminant) / (2*a)
        xq_ub = (-b + discriminant) / (2*a)

    # draw quadratic curve
    xq = xq_start; yq = yq_start
    turtle.penup()
    turtle.setposition(xq, yq)
    turtle.pendown()
    while xq <= -xq_start:
        yq = a*xq*xq + b*xq + c
        turtle.setposition(xq, yq)
        xq += 1

    screen.delay(20)

    # transform turtle into a ball
    turtle.penup()
    turtle.setposition(x_ball, y_ball)
    turtle.setheading(0)
    turtle.shape("turtle")
    turtle.shapesize(diameter, diameter)
    turtle.color(ball_colour, ball_colour)

    turtle_radius = 10*turtle.shapesize()[1]

# fun function when button is clicked
# just moves turtle until it hits the right boundary
def play(x, y):
    global x_ball, y_ball, ball_colour, xq_lb, xq_ub, turtle_radius

    if x_ball < scene.wnd_width/2 - turtle_radius:
        # yellow outside the quadratic region
        ball_colour = "#ffff00"
        if (x_ball - turtle_radius <= xq_lb and x_ball + turtle_radius >= xq_lb) or\
           (x_ball - turtle_radius <= xq_ub and x_ball + turtle_radius >= xq_ub):
            # red on the quadratic outline
            ball_colour = "#ff0000"
        elif x_ball - turtle_radius >= xq_lb and x_ball + turtle_radius <= xq_ub:
            # green inside the quadratic region
            ball_colour = "#00ff00"
        turtle.color(ball_colour, ball_colour)

        # change ball's coordinates
        turtle.setposition(x_ball, y_ball)

        x_ball += 10
        screen.ontimer(play(0,0), 500)
    else:
        x_ball = 100 - scene.wnd_width/2; y_ball = 0
        screen.mainloop()

# Let fly
scene = Template();
turtle = scene.controlButtons()
screen = scene.screen

prep()

Important: When trying to click on the button to get things started, you might need to click away from the button text.

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