Added Chris's attempt at an ellipse function

chris_ellipse_exploration/ellipse.py

@@ -0,0 +1,102 @@
+from math import pi, sqrt, sin, cos, atan
+from turtle import penup, pendown, forward, left, radians, degrees
+
+π = pi
+
+def ellipse(rx, ry, num_points, points_to_draw=None):
+    """
+    Draws an ellipse with x-radius rx and y-radius ry, 
+    with num_points control points. Optionally specify
+    points_to_draw if you want only a partial ellipse.
+    Assumes the turtle starts at the center, pointed in 
+    the +x direction, and ends in the same position and heading.
+    """
+    radians()
+    fly(rx)
+    left(π/2)
+    for i in range(points_to_draw or num_points):
+        φi = get_angle(rx, ry, num_points, i)
+        di = get_distance(rx, ry, num_points, i)
+        left(φi)
+        forward(di)
+    left(-π/2)
+    fly(-rx)
+    degrees()
+
+def fly(distance):
+    """Like forward, but with the pen up.
+    """
+    penup()
+    forward(distance)
+    pendown()
+
+def get_angle(rx, ry, n, i):
+    """Returns the angle φi, or how far left the turtle should
+    turn at point i of n. 
+    """
+    coords_prev = get_coordinates(rx, ry, n, i - 1)
+    coords      = get_coordinates(rx, ry, n, i)
+    coords_next = get_coordinates(rx, ry, n, i + 1)
+    vec_prev = get_vector(coords_prev, coords)
+    vec_next = get_vector(coords, coords_next)
+    vec_parallel, vec_perpendicular = project_onto_basis(vec_next, vec_prev)
+    φi = atan(mag(vec_perpendicular) / mag(vec_parallel))
+    return φi/3
+
+def get_distance(rx, ry, n, i):
+    """
+    Returns the distance from point i to point i + 1.
+    """
+    coords      = get_coordinates(rx, ry, n, i)
+    coords_next = get_coordinates(rx, ry, n, i + 1)
+    vec_next = get_vector(coords, coords_next)
+    return mag(vec_next)
+
+def get_coordinates(rx, ry, n, i):
+    """Returns the (x, y) coordinates of point i for the ellipse
+    specified by rx, ry, and n.
+
+    Normally, we can get the coordinates of a point on the circle
+    at angle θ as (r * cos(θ), r * sin(θ)). An ellipse is a circle
+    scaled differently along the x- and y- axes, so we need to
+    apply different scaling factors rx and ry. 
+    """
+    θi = 2*π*i/n
+    xi = rx * cos(θi)
+    yi = ry * sin(θi)
+    return xi, yi
+
+def get_vector(start, end):
+    """Returns a vector from the start point to the end point. 
+    """
+    x0, y0 = start
+    x1, y1 = end
+    return (x1 - x0, y1 - y0)
+
+def project_onto_basis(vec, basis):
+    """Projects a vector onto a basis vector, returning two vectors:
+    the component which is parallel to basis (vec_a) and the component which 
+    is perpendicular (vec_b). 
+    """
+    v0, v1 = vec
+    b0, b1 = basis
+    len_a = dot(vec, basis)
+    vec_a = (len_a * v0, len_a * v1)
+    vec_b = get_vector(vec_a, vec)
+    return vec_a, vec_b
+
+def dot(vec0, vec1):
+    """Computes the dot product of vec0 and vec1. 
+    The result is the magnitude of vec0 projected onto vec1.
+    """
+    x0, y0 = vec0
+    x1, y1 = vec1
+    return x0 * x1 + y0 * y1
+
+def mag(vec):
+    """Returns the magnitude, or length, of a vector.
+    We can simplify the familiar formula sqrt(x*x + y*y)
+    using the dot product. 
+    """
+    return sqrt(dot(vec, vec))
+

chris_ellipse_exploration/ellipse_test.py

@@ -0,0 +1,5 @@
+from ellipse import ellipse
+
+ellipse(100, 200, 1024)
+
+input()

turtle.py

@@ -1,7 +0,0 @@
-def draw_turtle(frame):
-    # ...
-
-
-for frame in animate(100):
-    draw_turtle(frame)
-    draw_bubbles(frame)