Initial Commit
This commit is contained in:
paoda
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# Assignment Name: Integral Approximator
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# Author: Rekai Nyangadzayi Musuka
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# Description: Approximates Any Given Function
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# Inputs: None
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# Outputs: A Graph in Turtle with the Approximated and Theorized Area, highlighted by drawing trapezoids or squares.
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# Version: 2019.0107.0
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#LINK TO GRAPH (DESMOS): https://www.desmos.com/calculator/igcdhrepy3
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import math
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import turtle
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CURSOR_SIZE = 0.1 # Makes the "Turtle" Invisible
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DRAW_SPEED = 0
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#Cosmetic
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FONT_FAMILY = "Consolas"
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NUM_FONT_SIZE = "8"
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LABEL_FONT_SIZE = "10"
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TEXT_SPACING = 15
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FIGURES = 5
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# Changes these to mess with the scale of the graph
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SCALE_X = 80 # Scale of the X Axis OG: 80
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SCALE_Y = 80 # Scale of the Y Axis OG: 80
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UNIT_LENGTH = 400 # Determines the maximum domain and Range of the cartesian plane (in this case (-400, 400) in both X and Y)
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# Best not to mess with these :)
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MIN_VALUE = -2 # The Domain of X is -2 <= x <= 2
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MAX_VALUE = 2
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TICK_STEP = 1 # Draw Ticks according to an interval of this variable
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PLANE_INCREMENT = 1
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# Change DRAW_STEP for a more or less accurately drawn function
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DRAW_STEP = 0.01 # What we Increment by when drawing the function
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# Change to shorten or lengthen the height of the tick
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TICK_HEIGHT = 10 # The Height of the Tick line
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# Changes How Detailed the Area Approximation (Trapezoid or Rectangle) is (higher is better)
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ITERATIONS = 100
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# Determines whether the program uses the rectangle rule
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# or trapezoid rule (trapezoid is more exact I think?)
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# TODO? Implement Simpson's Rule?
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APPROX_TYPE = "trapezoid" # or square
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# Initializing Turtle Variables
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cursor = turtle.Turtle()
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win = turtle.Screen()
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cursor.shape()
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cursor.hideturtle()
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cursor.speed(DRAW_SPEED)
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# Draws a Cursor Tick
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def draw_tick(num, cursor, height, x_axis = False):
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# Function draws the numerical value of the x or y value at any given tick
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def draw_num(angle, distance, num):
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cursor.penup()
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cursor.left(angle)
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cursor.forward(distance)
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cursor.write(num, align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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cursor.backward(distance)
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cursor.left(-angle)
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cursor.pendown()
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if x_axis:
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draw_num(-90, (TICK_HEIGHT * 2), num)
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else:
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draw_num(90, (TICK_HEIGHT * 2), num)
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for angle in [90, -180]:
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cursor.left(angle)
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cursor.forward(height / 2)
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cursor.backward(height / 2)
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cursor.left(90)
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# Draws the Cartesian Plane
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def create_plane():
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cursor.penup()
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cursor.goto(-UNIT_LENGTH, 0) # Goes to Fartheset left coordinate
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cursor.pendown()
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# Function will draw text nearby a given x value (used to draw axis labels).
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def draw_axis_label(angle, distance, text):
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cursor.penup()
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cursor.left(angle)
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cursor.forward(distance)
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cursor.write(text, align="center", font=(FONT_FAMILY, LABEL_FONT_SIZE, "normal"))
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cursor.backward(distance)
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cursor.left(-angle)
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cursor.pendown()
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# This draws the x line of the cartesian plane
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while (cursor.pos()[0] <= UNIT_LENGTH):
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if (cursor.pos()[0] == -UNIT_LENGTH): draw_axis_label(-90, TICK_HEIGHT * 4, "X Axis") # If x is the furthermost left value, draw the x axis label below
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if (cursor.pos()[0] % (TICK_STEP * SCALE_X) == 0): draw_tick(cursor.pos()[0], cursor, TICK_HEIGHT, True) # Draw ticks whenever the condition is met
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cursor.forward(PLANE_INCREMENT)
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cursor.penup()
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cursor.goto(0, -UNIT_LENGTH) # Goes to the farthest bottom coordinate
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cursor.pendown()
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cursor.left(90)
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# This draws the y line of the cartesian plane
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while (cursor.pos()[1] <= UNIT_LENGTH):
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if (cursor.pos()[1] == -UNIT_LENGTH): draw_axis_label(90, TICK_HEIGHT * 4, "Y Axis") # If y is the furthermost bottom value, draw the y axis label there
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if (cursor.pos()[1] % (TICK_STEP * SCALE_Y) == 0): draw_tick(cursor.pos()[1], cursor, TICK_HEIGHT, True) # Draw ticks whenever the if condition is met
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cursor.forward(PLANE_INCREMENT)
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cursor.penup()
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# Used for Calculus Portion of Program
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def draw_rectangle(length, width):
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arr = [width, length, width, length]
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i = 0
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while (i < len(arr)):
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cursor.forward(arr[i])
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cursor.left(90)
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i += 1
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cursor.penup()
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cursor.forward(width)
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cursor.pendown()
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# Used for Calculus Portion of Program
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# Should Rewrite this sometime
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def draw_trapezoid(a, b, h):
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# Drawing Trapezoid given Area
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width_of_triangle = b - a
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c = math.sqrt(math.pow(h, 2) + math.pow(width_of_triangle, 2))
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angle = math.atan2(h, width_of_triangle) * (180 / math.pi)
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cursor.forward(h)
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cursor.left(90)
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cursor.forward(b)
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cursor.left(180 - angle)
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cursor.forward(c)
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cursor.left(angle)
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cursor.forward(a)
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cursor.left(90)
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cursor.penup()
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cursor.forward(h)
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cursor.pendown()
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# The function that draws the upper half of the Heart
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def f(x):
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return math.sqrt(1 - ((math.fabs(x) - 1) ** 2))
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# The function that draws the Lower Half the the Heart
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def g(x):
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return -3 * math.sqrt(1 - math.sqrt(math.fabs(x) / 2 ))
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# Calls create_plane() which draws and labels the cartesian plane.
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create_plane()
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# This will set x to -2 and this while loop will then draw f(x) at -2 to 2
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# Iterates between MIN_VALUE and MAX_VALUE and draws the calculated y value given the x value
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x = MIN_VALUE
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while (x <= MAX_VALUE):
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cursor.goto(SCALE_X * x, SCALE_Y * f(x)) # f(x) is the top half of the heart
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cursor.pendown()
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x += DRAW_STEP
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cursor.penup()
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# # This will set x to -2 and loop through g(x) until it reaches 2
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# # Iterates between MIN_VALUE and MAX_VALUE and draws the calculated y value given the x value
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x = MIN_VALUE
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while (x <= MAX_VALUE):
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cursor.goto(SCALE_X * x, SCALE_Y * g(x)) # g(x) is the bottom half of the heart
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cursor.pendown()
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x += DRAW_STEP
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# This code draws the functions used onto the screen.
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cursor.penup()
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cursor.goto(UNIT_LENGTH / 2, UNIT_LENGTH / 2) # We go to the middle of quadrant I
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cursor.right(180)
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cursor.write("Functions:", align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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cursor.forward(TEXT_SPACING)
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cursor.write("Top: y = math.sqrt(1 - ((math.abs(x) - 1) ** 2))", align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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cursor.forward(TEXT_SPACING)
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cursor.write("Bottom: y= math.sqrt(1 - math.sqrt(math.abs(x) / 2 ))", align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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# Intergral Stuff Goes Here
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cursor.left(90) # Settings Cursor Back to it's default position (heading right)
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# Allows us to import a reasonably acccurate estimate as to what the intergral of a given function will be
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# By accurate I mean more accurate than our estimate.
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import scipy.integrate as integrate
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# Calculates Percent Error (the one we learned in chemistry)
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def percent_error(exper, accept):
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return (math.fabs(exper - accept) / accept) * 100
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# Approximates the intergral of a function using the square rule.
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def approx_integral_square(func, a, b, n):
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# n is how many rectangles you want.
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# area is length * width
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area = 0.0
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width = (b - a) / n
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i = 1
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while (i <= n):
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height = func(a + (i - 1) * width)
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draw_rectangle(height * SCALE_Y, width * SCALE_Y)
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area += width * height
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i += 1
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return area
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# Approximates the integral of a function using the trapezoid rule.
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def approx_integral_trapezoid(func, a, b, n):
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# n is how many trapezoids you want
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width = (b - a) / n
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res = func(a) + func(b)
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i = 1
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prev_height = func(a)
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while (i < n):
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height = func(a + (i * width))
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draw_trapezoid(prev_height * SCALE_Y, height * SCALE_Y, width * SCALE_Y)
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res += 2 * height
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prev_height = height
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i += 1
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return (width / 2) * res
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# Exact Results
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upper_area = integrate.quad(lambda x: f(x), MIN_VALUE, MAX_VALUE)
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lower_area = integrate.quad(lambda x: g(x), MIN_VALUE, MAX_VALUE)
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# Don't really care for the margin of error at the moment
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# Absolute Value of the area because we don't care about signed area at the moment.math.
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upper_area = math.fabs(upper_area[0])
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lower_area = math.fabs(lower_area[0])
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upper_area_approx = 0
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lower_area_approx = 0
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if (APPROX_TYPE == "square"):
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cursor.goto(MIN_VALUE * SCALE_X, 0)
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upper_area_approx = math.fabs(approx_integral_square(lambda x: f(x), MIN_VALUE, MAX_VALUE, ITERATIONS))
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cursor.goto(MIN_VALUE * SCALE_X, 0)
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lower_area_approx = math.fabs(approx_integral_square(lambda x: g(x), MIN_VALUE, MAX_VALUE, ITERATIONS))
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elif (APPROX_TYPE == "trapezoid"):
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cursor.goto(MIN_VALUE * SCALE_X, 0)
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upper_area_approx = math.fabs(approx_integral_trapezoid(lambda x: f(x), MIN_VALUE, MAX_VALUE, ITERATIONS))
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cursor.goto(MIN_VALUE * SCALE_X, 0)
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lower_area_approx = math.fabs(approx_integral_trapezoid(lambda x: g(x), MIN_VALUE, MAX_VALUE, ITERATIONS))
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# Probably Want to Display the Results or smth here.
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cursor.penup()
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cursor.goto(-(UNIT_LENGTH / 2), UNIT_LENGTH / 2) # We go to the middle of quadrant II
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cursor.right(90)
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if (APPROX_TYPE == "square"):
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cursor.write("Using the Square Rule:", align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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else:
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cursor.write("Using the Trapezoid Rule:", align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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cursor.forward(TEXT_SPACING)
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cursor.forward(TEXT_SPACING)
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cursor.write(("Approximated Top Half Area: " + str(round(upper_area_approx, FIGURES)) + " | Theorized: " + str(round(upper_area, FIGURES))), align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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cursor.forward(TEXT_SPACING)
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cursor.write(("Percent Error: " + str(round(percent_error(upper_area_approx, upper_area), FIGURES)) + "%"), align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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cursor.forward(TEXT_SPACING)
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cursor.forward(TEXT_SPACING)
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cursor.write(("Approximated Bottom Half Area: " + str(round(lower_area_approx, FIGURES)) + " | Theorized: " + str(round(lower_area, FIGURES))), align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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cursor.forward(TEXT_SPACING)
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cursor.write(("Percent Error: " + str(round(percent_error(lower_area_approx, lower_area), FIGURES)) + "%"), align="center", font=(FONT_FAMILY, NUM_FONT_SIZE, "normal"))
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win.exitonclick()
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@ -0,0 +1,147 @@
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import math
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import turtle
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import scipy.integrate as integrate
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MIN_VALUE = -2
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MAX_VALUE = 2
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ITERATIONS = 100
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TURTLE_SCALE = 100
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#### TURTLE STUFF
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cursor = turtle.Turtle()
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win = turtle.Screen()
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cursor.shape()
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cursor.hideturtle()
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cursor.speed(0)
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cursor.goto(-2, 0)
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def draw_rectangle(length, width):
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arr = [width, length, width, length]
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i = 0
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while (i < len(arr)):
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cursor.forward(arr[i])
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cursor.left(90)
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i += 1
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cursor.penup()
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cursor.forward(width)
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cursor.pendown()
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def draw_trapezoid(a, b, h):
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# Drawing Trapezoid given Area
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width_of_triangle = b - a
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c = math.sqrt(math.pow(h, 2) + math.pow(width_of_triangle, 2))
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angle = math.atan2(h, width_of_triangle) * (180 / math.pi)
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cursor.forward(h)
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cursor.left(90)
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cursor.forward(b)
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cursor.left(180 - angle)
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cursor.forward(c)
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cursor.left(angle)
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cursor.forward(a)
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cursor.left(90)
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cursor.penup()
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cursor.forward(h)
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cursor.pendown()
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###
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# The function that draws the upper half of the Heart
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def f(x):
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return math.sqrt(1 - ((math.fabs(x) - 1) ** 2))
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# The function that draws the Lower Half the the Heart
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def g(x):
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return -3 * math.sqrt(1 - math.sqrt(math.fabs(x) / 2 ))
|
||||||
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def percent_error(exper, accept):
|
||||||
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return (math.fabs(exper - accept) / accept) * 100
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def approx_integral_square(func, a, b, n):
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# n is how many rectangles you want.
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||||||
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# area is length * width
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area = 0.0
|
||||||
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width = (b - a) / n
|
||||||
|
i = 1
|
||||||
|
|
||||||
|
while (i <= n):
|
||||||
|
height = func(a + (i - 1) * width)
|
||||||
|
|
||||||
|
# draw_rectangle(height * TURTLE_SCALE, width * TURTLE_SCALE)
|
||||||
|
|
||||||
|
area += width * height
|
||||||
|
i += 1
|
||||||
|
|
||||||
|
return area
|
||||||
|
|
||||||
|
def approx_integral_trapezoid(func, a, b, n):
|
||||||
|
# n is how many trapezoids you want
|
||||||
|
width = (b - a) / n
|
||||||
|
res = func(a) + func(b)
|
||||||
|
i = 1
|
||||||
|
|
||||||
|
prev_height = func(a)
|
||||||
|
|
||||||
|
while (i < n):
|
||||||
|
height = func(a + (i * width))
|
||||||
|
draw_trapezoid(prev_height * TURTLE_SCALE, height * TURTLE_SCALE, width * TURTLE_SCALE)
|
||||||
|
res += 2 * height
|
||||||
|
|
||||||
|
prev_height = height
|
||||||
|
i += 1
|
||||||
|
|
||||||
|
return (width / 2) * res
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
# Compare Results
|
||||||
|
upper_area = integrate.quad(lambda x: f(x), MIN_VALUE, MAX_VALUE)
|
||||||
|
lower_area = integrate.quad(lambda x: g(x), MIN_VALUE, MAX_VALUE)
|
||||||
|
|
||||||
|
# Don't really care for the margin of error at the moment
|
||||||
|
# Absolute Value of the area because we don't care about signed area at the moment.math.
|
||||||
|
upper_area = math.fabs(upper_area[0])
|
||||||
|
lower_area = math.fabs(lower_area[0])
|
||||||
|
|
||||||
|
|
||||||
|
print("Using the Square Method:")
|
||||||
|
|
||||||
|
upper_area_approx = math.fabs(approx_integral_square(lambda x: f(x), MIN_VALUE, MAX_VALUE, ITERATIONS))
|
||||||
|
# cursor.goto(-2, 0)
|
||||||
|
lower_area_approx = math.fabs(approx_integral_square(lambda x: g(x), MIN_VALUE, MAX_VALUE, ITERATIONS))
|
||||||
|
|
||||||
|
print("\nApproximated Top Half Area:", upper_area_approx, "Original:", upper_area)
|
||||||
|
print("Percent Error: ", percent_error(upper_area_approx, upper_area), "%", sep="")
|
||||||
|
|
||||||
|
print("\nApproximated Bottom Half Area:", lower_area_approx, "Original:", lower_area)
|
||||||
|
print("Percent Error: ", percent_error(lower_area_approx, lower_area), "%\n", sep="")
|
||||||
|
|
||||||
|
print("------")
|
||||||
|
print("\nUsing the Trapezoid Method:")
|
||||||
|
|
||||||
|
upper_area_approx = math.fabs(approx_integral_trapezoid(lambda x: f(x), MIN_VALUE, MAX_VALUE, ITERATIONS))
|
||||||
|
cursor.goto(-2, 0)
|
||||||
|
lower_area_approx = math.fabs(approx_integral_trapezoid(lambda x: g(x), MIN_VALUE, MAX_VALUE, ITERATIONS))
|
||||||
|
|
||||||
|
|
||||||
|
print("\nApproximated Top Half Area:", upper_area_approx, "Original:", upper_area)
|
||||||
|
print("Percent Error: ", percent_error(upper_area_approx, upper_area), "%", sep="")
|
||||||
|
|
||||||
|
print("\nApproximated Bottom Half Area:", lower_area_approx, "Original:", lower_area)
|
||||||
|
print("Percent Error: ", percent_error(lower_area_approx, lower_area), "%", sep="")
|
||||||
|
|
||||||
|
|
||||||
|
# Contratulations!
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
# Figuring out How to draw a Trapezoid in Turtle.
|
||||||
|
|
||||||
|
win.exitonclick()
|
||||||
Reference in New Issue