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Basics of Turtle Programming
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Computer Science 111
Fundamentals of Programming IAdvanced Turtle Graphics
Recursive Patterns in Art• The 20th century Dutch artist Piet Mondrian painted a series
of pictures that displayed abstract, rectangular patterns of color
• Start with a single colored rectangle
• Subdivide the rectangle into two unequal parts (say, 1/3 and 2/3) and paint these in different colors
• Repeat this process until an aesthetically appropriate “moment” is reached
Level 1: A Single Filled Rectangle
Level 2: Split at the Aesthetically
Appropriate Spot
Level 3: Continue the Same Process with Each Part
Level 4
Level 5
Level 6
Level 7
Level 8
Level 9
Design a Recursive Function
• The function expects a Turtle, the corner points of a rectangle, and the current level as arguments
• If the level is greater than 0
– Draw a filled rectangle with the given corner points
– Calculate the corner points of two new rectangles within the current one and decrement the level by 1
– Call the function recursively to draw these two rectangles
from turtle import Turtleimport random
def drawRectangle(t, x1, y1, x2, y2): red = random.randint(0, 255) green = random.randint(0, 255) blue = random.randint(0, 255) t.pencolor(red, green, blue) # Code for drawing goes here
# Definition of the recursive mondrian function goes here
t = Turtle()x = 50y = 50mondrian(t, -x, y, x, -y, 3)
Program Structure
def mondrian(t, x1, y1, x2, y2, level): if level > 0: drawRectangle(t, x1, y1, x2, y2)
vertical = random.randint(1, 2) if vertical == 1: # Vertical split mondrian(t, x1, y1, (x2 - x1) // 3 + x1, y2, level - 1) mondrian(t, (x2 - x1) // 3 + x1, y1, x2, y2, level - 1)
else: # Horizontal split
mondrian(t, x1, y1, x2, (y2 - y1) // 3 + y1, level - 1) mondrian(t, x1, (y2 - y1) // 3 + y1, x2, y2, level - 1)
The mondrian Function
Recursive Patterns in Nature
• A fractal is a mathematical object that exhibits the same pattern when it is examined in greater detail
• Many natural phenomena, such as coastlines and mountain ranges, exhibit fractal patterns
The C-curve
• A C-curve is a fractal pattern
• A level 0 C-curve is a vertical line segment
• A level 1 C-curve is obtained by bisecting a level 0 C-curve and joining the sections at right angles
• A level N C-curve is obtained by joining two level N - 1 C-curves at right angles
Level 0 and Level 1
(50,50)
(50,-50)
(0,0)
(50,-50)
(50,50)
drawLine(50, -50, 50, 50)
drawLine(50, -50, 0, 0)drawLine(0, 0, 50, 50)
Bisecting and Joining
(50,50)
(50,-50)
(0,0)
(50,-50)
(50,50)
0 = (50 + 50 + -50 - 50) // 20 = (50 + -50 + 50 - 50) // 2drawLine(50, -50, 0, 0)drawLine(0, 0, 50, 50)
drawLine(50, -50, 50, 50)
Generalizing
(50,50)
(50,-50)
(0,0)
(50,-50)
(50,50)
drawLine(x1, y1, x2, y2) xm = (x1 + x2 + y1 - y2) // 2ym = (x2 + y1 + y2 - x1) // 2drawLine(x1, y1, xm, ym)drawLine(xm, ym, x2, y2)
Recursing
(50,50)
(50,-50)
(0,0)
(50,-50)
(50,50)
drawLine(x1, y1, x2, y2) xm = (x1 + x2 + y1 - y2) // 2ym = (x2 + y1 + y2 - x1) // 2cCurve(x1, y1, xm, ym)CCurve(xm, ym, x2, y2)
Base case Recursive step
def cCurve(t, x1, y1, x2, y2, level): if level == 0: drawLine(t, x1, y1, x2, y2) else: xm = (x1 + x2 + y1 - y2) // 2 ym = (x2 + y1 + y2 - x1) // 2 cCurve(t, x1, y1, xm, ym, level - 1) cCurve(t, xm, ym, x2, y2, level - 1)
Note that recursive calls occur before any C-curve is drawn when level > 0
The cCurve Function
from turtle import Turtle
def drawLine(t, x1, y1, x2, y2): """Draws a line segment between the endpoints.""" t.up() t.goto(x1, y1) t.down() t.goto(x2, y2)
# Definition of the recursive cCurve function goes here
for level in range(0, 10): t = Turtle() cCurve(t, 50, -50, 50, 50, level)
Program Structure
Draws 10 C-curves of increasing levels of detail
ccurve
A call tree diagram shows the number of calls of a function for a given argument value
Call Tree for ccurve(0)
ccurve(0) uses one call, the top-level one
ccurve
Call Tree for ccurve(1)
ccurve(1) uses three calls, a top-level one and two recursive calls
ccurve ccurve
ccurve
Call Tree for ccurve(2)ccurve(2) uses 7 calls, a top-level one and 6 recursive calls
ccurve ccurve
ccurve
ccurve ccurve
ccurve
ccurve
Call Tree for ccurve(n)ccurve(n) uses 2n+1 - 1 calls, a top-level one and 2n+1 - 2 recursive calls
ccurve ccurve
ccurve
ccurve ccurve
ccurve
ccurve
Call Tree for ccurve(2)The number of line segments drawn equals the number of calls on the frontier of the tree (2n)
ccurve ccurve
ccurve
ccurve ccurve
ccurve
Summary• A recursive algorithm passes the buck repeatedly to the
same function
• Recursive algorithms are well-suited for solving problems in domains that exhibit recursive patterns
• Recursive strategies can be used to simplify complex solutions to difficult problems