This project draws a loopy curve, that mathematicians call an epitrochoid, and everyone else calls a spirograph pattern. It introduces the idea of an object and its methods.
Turtles, turtles, turtles¶
Start the project by making an empty file
Right-click and open it with IDLE.
Type in the editor:
# Fun with epitrochoids from turtle import *
In the introduction we used only one turtle,
and so your commands like
went to that unnamed turtle.
Here you’ll create two other turtles,
and send your commands to each of them by name.
Let’s see how this works.
Save and run your program,
import happens and the shell opens.
This is how you make a new turtle,
and a variable
ted to refer to it.
Try this in the shell window:
>>> ted = Turtle()
Now we can give
ted the turtle some things to do:
>>> ted.color("blue") >>> ted.left(60) >>> ted.forward(100)
These instructions are just like function calls.
A function addressed to a particular object is called a method.
. is how you address a method to an object in Python.
It has to be something the object knows how to do, so:
raises an error.
Setting up the guide turtles¶
We need two “guide” turtles, each of which draws a circle. Each has to be set up a certain distance from home, so add this function to your code:
def new_guide(r, c): "Return a new guide turtle of radius r and colour c" t = Turtle() t.speed("fastest") t.color(c) # Go to (r,0) without drawing t.penup() t.setposition(r,0) t.left(90) t.pendown() return t
Notice this function returns a turtle. Save and run, then try this in the shell:
>>> g = new_guide(100, "blue") >>> p = g.position() >>> g.circle(50, 90) >>> q = g.position() >>> >>> mid = (1/2) * (p+q) >>> setposition(mid)
You should see this:
The unnamed turtle (black) has moved halfway between start and end of the blue arc.
Moving the guides¶
The first job is to make the guides move in their orbits. Add this to your program at the end:
def epitrochoid(a, b, L, M=1): ta = new_guide(a, "blue") tb = new_guide(b, "red") # N little steps s make one circle N = 500 s = 360/N for i in range(N): # ta will go L times round ta.circle(a, L*s) # tb will go M times round tb.circle(b, M*s) # Test epitrochoid(90, 100, 3, 2)
Save and run. You should see blue and red circles drawn.
The blue turtle goes round L=3 times, and the red turtle M=2 times.
You can see how this works in the code.
N just has to be a big enough number to make the final curve smooth.
N steps of size
s make 360 degrees, exactly one circle.
N steps of
M steps make
M full circles.
M=1 on the first line says that,
if you don’t give it a value in the call to
M will be equal to 1.
Compute the shape¶
The shape we are looking for is drawn by keeping our pen mid-way between the two guides.
epitrochoid function to add these lines:
def epitrochoid(a, b, L, M=1): "Epitrochoid: a, b are guide radii; L, M the number of orbits." ta = new_guide(a, "blue") tb = new_guide(b, "red") # Local function for midpoint between the guides def midpoint(): return (1/2) * (ta.position() + tb.position()) # Set start position for unnamed turtle penup() setposition(midpoint()) pendown() # N little steps s make one circle N = 500 s = 360/N for i in range(N): # ta will go L times round ta.circle(a, L*s) # tb will go M times round tb.circle(b, M*s) # unnamed will be half-way between them setposition(midpoint())
Save and run. You should see this:
It would be nice if the guide circles were not on the final drawing.
Add this tidy-up code at the end of
and style the unnamed turtle to your liking:
def epitrochoid(a, b, L, M=1): ta = new_guide(a, "blue") tb = new_guide(b, "red") . . . # Erase guides guide_erase(ta) guide_erase(tb) def guide_erase(t): "Erase what turtle t drew" t.hideturtle() t.clear() speed("fastest") width(5) color("lime green") # Test epitrochoid(90, 100, 3, 2) hideturtle()
Try changing the numbers in the call to
epitrochoid like this:
a, b = 100, 300 epitrochoid(a, b, 4) color("goldenrod") epitrochoid(a, b, 5) color("sienna") epitrochoid(a, b, 6)
(Remember, M=1 if you don’t give a fourth argument.) Suppose you change just one line now:
a, b = -100, 300
and run again. When the loops point outwards, the shape is called a hypotrochoid.
What’s happening here?
a, b = 250, 300 epitrochoid(a, b, 4) color("goldenrod") epitrochoid(a, b, 5) color("sienna") epitrochoid(a, b, 6)
And what about here?
L = 6 a, x = 50, 80 epitrochoid(a, L*a, L) color("goldenrod") epitrochoid(a, L*a + x, L) color("sienna") epitrochoid(a, L*a - x, L)
Find other interesting shapes of your own.
Some advanced questions¶
If you like investigating mathematical patterns, this code project poses some interesting questions.
- What determines the number of loops?
- What values for
bmake the curve pass through (0,0)? (Hint: where would the guide turtles be at that moment?)
A shape in this family, where the curve passes through zero, is called a “rose”.
- When do the loops become points?
- Both curves below have 3 loops: what is the difference between them?