You cannot draw a circle with Beziers.

When I first read this some time ago I was a bit confused.
Why is this not a circle. It sure looks like one.

After learning a bit more about curve descriptions and calculations I kind of agree but the quote should better say:

You cannot draw a perfect circle with Beziers.

Getting the best tangent values.

We have a formula to calculate points on a Bezier Curve.

x = (1-t)**3*p1[0] + 3*(1-t)**2*t*t1[0] + 3*(1-t)*t**2*t2[0] + t**3*p2[0]
y = (1-t)**3*p1[1] + 3*(1-t)**2*t*t1[1] + 3*(1-t)*t**2*t2[1] + t**3*p2[1]

p1 and p2 are the two endpoints and t1 and t2 the two tangent points. t is the factor (0-1) at which we want the point on the curve.

On a bezier curve with the following coördinates:

p1 = (1, 0)
p2 = (0, 1)
t1 = (1, tangent_val)
t2 = (tangent_val, 1)

we can calculate the tangent_val if we assume that the point at t=0.5 (halfway the curve) is exactly on the circle. Halfway of a quartercircle is 45 degrees so x and y are both at 1/sqrt(2). If we plug these numbers into the formula we can calculate the tangent value:

1/sqrt(2) = (1-.5)**3*1 + 3*(1-.5)**2*.5*1 + 3*(1-.5)*.5**2*tangent_val + .5**3*0
...
tangent_val = 4*(sqrt(2)-1)/3 ≈ 0.5522847493

With this tangent_val we can calculate other points on the bezier curve. Of which we can calculate the angle in respect to the center point at (0, 0). And with this angle we can use sine and cosine to get the point that would be on a circle.

We can also calculate the distance of the bezier point and see where the offset is the biggest. As we can see the difference is really tiny. With a radius of 1 the biggest off value is 1.0002725295591013 (not even one permille). The difference is hardly visible so I added only the top part of a really long bar chart to make it show.
See the image below:

Code

# -------------------
#  settings 

pw, ph = 595, 842 #A4
margin = 70

tang = 4*(sqrt(2)-1)/3
r = pw - 2 * margin

density = 90 # amount of points 
dia_ = 1 # diameter to draw the points 

r_plot = 100000 # this is a magnifying factor to make the tiny difference visible. 

r_plot_y = margin - r_plot   #radius plotting 
c_plot_y = ph - 3 * margin - r  # circle plotting 

# -------------------
#  functions  

def get_bezier_p(p1, t1, t2, p2, t):
    ''' get the point on a bezier curve at t (should be between 0 and 1) '''
    x = (1 - t)**3 * p1[0] + 3*(1 - t)**2 * t * t1[0] + 3*(1 - t)* t**2 * t2[0] + t**3 * p2[0]
    y = (1 - t)**3 * p1[1] + 3*(1 - t)**2 * t * t1[1] + 3*(1 - t)* t**2 * t2[1] + t**3 * p2[1]
    return x, y

def get_dist(p1, p2): 
    ''' returns the distance between two points (x, y)'''
    return sqrt((p2[0]-p1[0])**2 + (p2[1]-p1[1])**2)

# -------------------
#  drawings  

newPage(pw, ph)
fill(1)
rect(0, 0, pw, ph)
translate(margin, margin)

# draw the tangent points 
rect(r - dia_, c_plot_y + tang*r - dia_, dia_*2, dia_*2)
rect(tang*r -dia_, c_plot_y + r - dia_, dia_*2, dia_*2)

max_off = 1

for i in range(density+1):

    # -------
    #  bezier 

    # calculate the factor f 
    f = i/density
    fill(0)
    if i == 0 or i == (density): dia = dia_ * 4
    else: dia = dia_
    # get the coordinates for the point at factor f 
    x_b, y_b = get_bezier_p( (1, 0), (1, tang), (tang, 1), (0, 1), f )
    # get the distance of the point from the center at (0, 0)
    r_b = get_dist((0, 0), (x_b, y_b))
    max_off = max(max_off, r_b)
    # get the angle of the point 
    angle = atan2(y_b, x_b)
    # draw the point and a rect for the bar chart     
    oval(x_b * r - dia/2, c_plot_y + y_b * r - dia/2, dia, dia)
    rect((angle/(pi/2)) * r - dia_/2, r_plot_y, dia_, r_plot * r_b)

    # -------
    #  circle

    fill(1, 0, 0)

    # get the point on a circle 
    x_c, y_c = cos(angle), sin(angle)
    # draw the point and a rect for the bar chart     
    oval(x_c * r - dia/2, c_plot_y + y_c * r - dia/2, dia, dia)
    rect((angle/(pi/2)) * r - dia_/2, r_plot_y, dia_, r_plot)
        
    fill(.3)
    if i % 10 == 0:
        x_p, y_p = r * f, margin #* 1.5
        rect(x_p - .25, y_p, .5, 30)
        text('%d°' % (f * 90), (x_p-2, y_p * 1.5))

print (max_off)
text('%.6f' % max_off, (-50, max_off * r_plot - r_plot + margin - 2))
text('%.6f' % 1, (-50, margin - 2))
    

The maybe more interesting peculiarity of bezier curves is the difference between time and distance. We can compare the bezier distance at factor f (0-1) to the circle if we use the same factor to get the appropriate fraction of 90 degrees.

Code

# -------------------
#  settings 

pw, ph = 595, 842 #A4
margin = 70

tang = 4*(sqrt(2)-1)/3
r = pw - 2 * margin

density = 90 # amount of points 
dia_ = 1 # diameter to draw the points 

r_plot = 100000 # this is a magnifying factor to make the tiny difference visible. 

r_plot_y = margin - r_plot   #radius plotting 
c_plot_y = ph - 3 * margin - r  # circle plotting 

# -------------------
#  functions  

def get_bezier_p(p1, t1, t2, p2, t):
    ''' get the point on a bezier curve at t (should be between 0 and 1) '''
    x = (1 - t)**3 * p1[0] + 3*(1 - t)**2 * t * t1[0] + 3*(1 - t)* t**2 * t2[0] + t**3 * p2[0]
    y = (1 - t)**3 * p1[1] + 3*(1 - t)**2 * t * t1[1] + 3*(1 - t)* t**2 * t2[1] + t**3 * p2[1]
    return x, y

def get_dist(p1, p2): 
    ''' returns the distance between two points (x, y)'''
    return sqrt((p2[0]-p1[0])**2 + (p2[1]-p1[1])**2)

# -------------------
#  drawings  

newPage(pw, ph)
fill(1)
rect(0, 0, pw, ph)
translate(margin, margin)

# draw the tangent points 
rect(r - dia_, c_plot_y + tang*r - dia_, dia_*2, dia_*2)
rect(tang*r -dia_, c_plot_y + r - dia_, dia_*2, dia_*2)


for i in range(density+1):

    # -------
    #  bezier 

    # calculate the factor f 
    f = i/density
    fill(0)
    if i == 0 or i == (density): dia = dia_ * 4
    else: dia = dia_
    # get the coordinates for the point at factor f 
    x_b, y_b = get_bezier_p( (1, 0), (1, tang), (tang, 1), (0, 1), f )
    # get the distance of the point from the center at (0, 0)
    r_b = get_dist((0, 0), (x_b, y_b))
    # get the angle of the point 
    angle = atan2(y_b, x_b)
    # draw the point and a rect for the bar chart     
    oval(x_b * r - dia/2, c_plot_y + y_b * r - dia/2, dia, dia)
    rect((angle/(pi/2)) * r - dia_/2, r_plot_y, dia_, r_plot)

    # -------
    #  circle

    fill(1, 0, 0)

    # get the point on a circle 
    x_c, y_c = cos( pi/2*f ), sin( pi/2*f )
    # draw the point and a rect for the bar chart     
    oval(x_c * r - dia/2, c_plot_y + y_c * r - dia/2, dia, dia)
    rect(f * r - dia_/2, r_plot_y, dia_, r_plot)
        
    fill(.3)
    if i % 10 == 0:
        x_p, y_p = r * f, margin #* 1.5
        rect(x_p - .25, y_p, .5, 30)
        text('%d°' % (f * 90), (x_p-2, y_p * 1.5))

oh, yes!
that is of course a very nice solution.
will update some of the scripts in the next days.
thanks!

@frederik thank you!

There is one 'issue' that makes it difficult to abstract the code. And I was wondering if this could be implemented differently in drawbot.
when setting the fontVariations the axis tag cannot be a variable:
fontVariations( axisTag = someValue )

It would be quite convenient to do:

myAxis = 'wght' # or whatever
# ... do some stuff with myAxis
# ... and then just place the variable name
fontVariations( myAxis = someValue )

or to just iterate through axes without ever having to type the axistag.

After a request to add more axes to the previous simple variable fonts tutorial the follow up escalated a bit. Since there are several files I thought it might be better to put them on github.

@hanafolkwang hi, this is half a year too late and probably not even what you were looking for but I made a few more examples to generate .gifs of variable fonts.
Since there are several files I put them on github.

thanks @gferreira and @frederik!
great help. i managed to botch my rough attempt at type on a curve.

# --------------------------------
#  imports 

from fontTools.pens.basePen import BasePen

# --------------------------------
#  settings 

pw = ph = 400
r = 89

angle = pi * 3.1
baseline = ph/2 + r

txt = 'HELLO'

tang_f = .333 # tangent factor
center = pw/2, baseline
focus  = pw/2, baseline - r

# --------------------------------
#  functions 

def ip(a, b, f): return a + (b-a) * f

def conv_point(p):

    x, y = p
    f = (x - pw/2) / (pw/2)
    a = -angle/2 * f
    x_conv = center[0] + cos(a + pi/2) * (r + y - baseline)
    y_conv = center[1] + sin(a + pi/2) * (r + y - baseline) - r
    return x_conv, y_conv

class convert_path_pen(BasePen):
    def __init__(self, targetPen):
        self.targetPen = targetPen
    def _moveTo(self, pt):
        self.targetPen.moveTo( conv_point(pt) )
        self.prev_pt = pt
    def _lineTo(self, pt):
        if self.prev_pt[0] == pt[0]:
            self.targetPen.lineTo( conv_point(pt) )
        else:
            tang_1 = ip(self.prev_pt[0], pt[0], tang_f), ip(self.prev_pt[1], pt[1], tang_f)
            tang_2 = ip(self.prev_pt[0], pt[0], 1-tang_f), ip(self.prev_pt[1], pt[1], 1-tang_f)
            self.targetPen.curveTo(conv_point(tang_1), conv_point(tang_2), conv_point(pt))
        self.prev_pt = pt
    def _curveToOne(self, tang_1, tang_2, pt):
        self.targetPen.curveTo(conv_point(tang_1), conv_point(tang_2), conv_point(pt))
        self.prev_pt = pt
    def _closePath(self):
        self.targetPen.closePath()

# --------------------------------
#  drawings 

path = BezierPath()
path.text(txt, font="Helvetica", fontSize = 80, align = 'center', offset=(0, baseline))

newPage(pw, ph)
trans_path = BezierPath()
trans_pen = convert_path_pen( trans_path )

path.drawToPen( trans_pen )
drawPath(trans_path)

obligatory gifs

Is there some example code on how to feed a BezierPath into a custom pen (where I manipulate the coordinates with a function and change the _lineTo) and then draw that transformed path in my canvas?

thanks, j

ok, one more.
very similar to the previous one.

page_s = 500
cell_amount = 12

rhythm = [1, 0, 0, 1, 0, 1, 1, 0]

cell_s = page_s / cell_amount
corner_s = cell_s/4

newPage(page_s, page_s)
translate(-cell_s/2, -cell_s/2)

def lozenge(pos, s):
    x, y = pos
    polygon((x, y), (x + s/2, y + s/2), (x + s, y), (x + s/2, y - s/2))

for x in range(cell_amount+1):
    for y in range(cell_amount+1):
        pos_x, pos_y = x * cell_s, y * cell_s
        fill(0.5) if (x+y) % 2 == 0 else fill(1)
        rect(pos_x, pos_y, cell_s, cell_s)

        fill(1)
        lozenge((pos_x - corner_s, pos_y), corner_s*2)

        fill(0)
        shift = (x + y) % len(rhythm)

        if rhythm[ shift ]:
            lozenge((pos_x - corner_s, pos_y), corner_s)
            lozenge((pos_x, pos_y), corner_s)
        else:
            lozenge((pos_x - corner_s/2, pos_y + corner_s/2), corner_s)
            lozenge((pos_x - corner_s/2, pos_y - corner_s/2), corner_s)

nice, ones!
here is one more:

grid = 12
page_s = 600
frames = 48

tile_s = page_s/grid

rhythm = [1, 0, 0, 1, 0, 1, 1, 0]

def cross(pos, s):
    x, y = pos
    polygon((x - s/2, y),
            (x - s/2 + s/8, y - s/8),
            (x - s/8, y - s/8),
            (x - s/8, y - s/2 + s/8),
            (x, y - s/2),
            (x + s/8, y - s/2 + s/8),
            (x + s/8, y - s/8),
            (x + s/2 - s/8, y - s/8),
            (x + s/2, y),
            (x + s/2 - s/8, y + s/8),
            (x + s/8, y + s/8),
            (x + s/8, y + s/2 - s/8),
            (x, y + s/2),
            (x - s/8, y + s/2 - s/8),
            (x - s/8, y + s/8),
            (x - s/2 + s/8, y + s/8)
        )

for f in range(frames):

    newPage(page_s, page_s)
    frameDuration(.1)
    # linear 
    alpha = f / frames if f < (frames) else 2 - (f / frames)
    # sinus wave 
    alpha = .5 + .5 * sin(f/frames * 2 * pi)

    for y in range(grid):
        for x in range(grid):

            fill(.8) if (x+y) % 2 == 0 else fill(.7)
            rect(x * tile_s, y * tile_s, tile_s, tile_s)
            shift = (x + y) % len(rhythm)
            fill(1, alpha) if rhythm[ shift ] else fill(.5, alpha)
            cross((x * tile_s, y * tile_s), tile_s/3 )
            
# saveImage('wavy_checkerboard.gif')

nautil.us has a nice blogpost of optical illusions. I could not resist the attempt to build one of them in drawbot.

pw = ph = 400 # define variables to set width and height of the canvas
cols = 14 # define how many columns there should be 
rows = 4 # define how many rows there should be 
pages = 30 # define how many frames the gif should have. more pages mean slower movement

angle_diff = 2 * pi/pages # we want the loop to make one full circle. 2 times pi is 360 degrees. Divided by the amount of pages this gives us the angle difference between pages. 

gap = pw/(cols+1) # calculate the gap between the lines. 
l   = (ph - 2 * gap)/rows # calculate the length of the line. 

for p in range(pages): 
    newPage(pw, ph)
    translate(gap, gap)
    fill(None)
    strokeWidth(2)
    for x in range(cols):
        y_off = cos( 3 * pi * x/cols + angle_diff * p ) * gap/2 # calculate how high or low the arrowhead should be 
        for y in range(rows+1):
            stroke(1,0 ,0) if y % 2 == 0 else stroke(0, 0, 1)
            if y < rows:
                line((x*gap, y*l), (x*gap, y*l+l))
            stroke(0)
            y_off *= -1 # switch to change direction 
            line( (x*gap, y*l), (x*gap - gap/2, y*l - y_off) )
            line( (x*gap, y*l), (x*gap + gap/2, y*l - y_off) )

# saveImage('~/Desktop/opti_illu.gif')

this should generate something like this: