For RoboFont. Add a guideline for points (near selected points in the glyph) on the curve where the tangent is at a given angle.

guide_at_slantangle_tangent.py

# [email protected]
# April 2024
# version 1

# For robofont
# Dedicated to my github sponsors who encourate explorations like this.

# this can find the t for horizontals in a cubic bezier segment.
# this can find the tangent at any angle (by rotating the segment and finding horizontals)
# this adds guides to the current glyph at the font's italic angle
# .. near selected points. To prevent a mass of guidelines to be added.

# to do: find origins for the guides that will give the same line
# but don't have the guide label so near the tangent point.

# some of the reasoning:
# the function of the tangent is the derivative of the cubic.
# https://math.stackexchange.com/questions/477165/find-angle-at-point-on-bezier-curve
# 𝐏′(𝑡)=(1−𝑡)^2(𝐏1−𝐏0)+2𝑡(1−𝑡)(𝐏2−𝐏1)+𝑡^2(𝐏3−𝐏2)

# code should be similar for px = 0
# solve t for py = 0
# this is my solution, it seems to work.
#py = (1-t)**2 * a + 2*t*(1-t) * b + t**2 * c
#py = (1-t)(1-t) * a          +  2*t*(1-t) * b + t**2 * c
#py = (1 -2*t + t**2) * a     +  (2 * t -2 * t**2) * b        + c*t**2
#py = a - 2*a*t + a*t**2      +  2 * b * t - 2 * b * t**2     + c*t**2
#py =  a*t**2   -2*b*t**2  + c*t**2      - 2*a*t +  2 * b * t     + a 
#py = (a - 2*b + c)*t**2         +         (2*b - 2*a) * t         + a

import math
from math import degrees, atan2, sin, cos, pi, radians, degrees
from random import random
from fontTools.misc.bezierTools import cubicPointAtT, solveQuadratic


def tangentAtT(p0, p1, p2, p3, t):
    # not used in this script, but kept for reference
    # get the tangent point at t
    a = (1-t)**2
    b = 2*t*(1-t)
    c = t**2
    px = a * (p1[0]-p0[0]) + b*(p2[0]-p1[0]) + c*(p3[0]-p2[0]) 
    py = a * (p1[1]-p0[1]) + b*(p2[1]-p1[1]) + c*(p3[1]-p2[1])
    #return px + p0[0], py + p0[1]
    return px, py

def t_for_horizontal(p0, p1, p2, p3, horizontal=True):
    if horizontal:
        i = 1
    else:
        i = 0
    a = (p1[i]-p0[i])
    b = (p2[i]-p1[i])
    c = (p3[i]-p2[i])
    # solve the quadratic
    r = solveQuadratic((a - 2*b + c),  (2*b - 2*a),  a)
    return r

def add(p1, p2):
    return p1[0]+p2[0],p1[1]+p2[1]
    
def findHorizontals(glyph):
    results = []
    for contourIndex, c in enumerate(glyph.contours):
        bps = c.bPoints
        l = len(bps)
        for bPointIndex, bp1 in enumerate(bps):
            bp2 = bps[(bPointIndex+1)%l]
            if bp1.type != "curve" and bp2.type != "curve": continue
            # so the segment we're interested in will bp1.anchor, bp1.out, bp2.in, bp2.anchhor
            s = (bp1.anchor, add(bp1.anchor,bp1.bcpOut), add(bp2.anchor,bp2.bcpIn), bp2.anchor)        
            r = t_for_horizontal(s[0], s[1], s[2], s[3], 1)
            for value in r:
                # this does not catch everything --
                if value == 0:
                    results.append((contourIndex, bPointIndex, 0))
                elif value == 1:
                    results.append((contourIndex, bPointIndex, 1 ))
                elif 0 < value < 1:
                    cpt = cubicPointAtT(s[0], s[1], s[2], s[3], value)
                    results.append((contourIndex, bPointIndex, value))
                else:
                    pass
    return results

def findPointFromT(glyph, contourIndex, bPointIndex, value):
    bps = glyph.contours[contourIndex].bPoints
    bp1 = bps[bPointIndex]
    bp2 = bps[(bPointIndex + 1)%len( bps)]
    s = (bp1.anchor, add(bp1.anchor,bp1.bcpOut), add(bp2.anchor,bp2.bcpIn), bp2.anchor)        
    cpt = cubicPointAtT(s[0], s[1], s[2], s[3], value)
    return cpt

def findTangentInGlyph(glyph, angle):
    # pay attention, from robofont slant angle
    angle = -(angle - 90)
    g2 = glyph.copy()
    g2.rotate(angle)
    points = []
    results = findHorizontals(g2)
    for contourIndex, bPointIndex, value in results:
        pt = findPointFromT(g, contourIndex, bPointIndex, value)
        points.append(pt)
    return points

def nearSelected(glyph, pt, dst=100):
    # if the point is close to a currently selected point
    near = []
    if len(glyph.selectedPoints) == 0:
        return True
    for p in glyph.selectedPoints:
        if math.hypot(p.x-pt[0],p.y-pt[1]) <= dst:
            near.append(pt)
    if len(near) > 0:
        return True
    return False


# - - -
g = CurrentGlyph()
glyph = g.getLayer("foreground")

# this cleans guides with names that start with "angled_"
guideNamePrefix = "angled_"
remove = []
for guide in glyph.guidelines:
    if guide.name is not None:
        if guideNamePrefix in guide.name:
            remove.append(guide)
for guide in remove:
    glyph.removeGuideline(guide)

# this can be any other angle of course
# in case someone wants to write a UI for it.
angle = g.font.info.italicAngle

# add guides for candidate tangents near selected points in the glyph
# or all candidates if there is no selection
for p in findTangentInGlyph(g, angle):
    if nearSelected(glyph, p):
        glyph.appendGuideline(p, angle, color=(1,.5,0,1), name=f"{guideNamePrefix}{angle:3.3f}")
        glyph.appendGuideline(p, angle-90, color=(1,.25,0,1), name=f"{guideNamePrefix}_ortho_{angle:3.3f}")

# this adds guides on tangents +90 from the given angle. 
# maybe not what you need. Easy to remove.
for p in findTangentInGlyph(g, angle + 90):
    if nearSelected(glyph, p):
        glyph.appendGuideline(p, angle, color=(0,.5,1,1), name=f"angled_{angle:3.3f}")
        glyph.appendGuideline(p, angle-90, color=(1,.25,1,1), name=f"angled_ortho_{angle:3.3f}")