math.txt

# from https://de.wikipedia.org/wiki/Trigonometrische_Funktion
#
# from https://de.wikipedia.org/wiki/Schnittwinkel_(Geometrie)
# wikipedia has an abs() in the formula, which extracts the smaller of the two angles.
# we don't want that. We need to distinguish betwenn spitzwingklig and stumpfwinklig.

# cut-away length 
# where r is the radious of the desired circle, and alpha is the angle (>0, <pi) under which the lines meet.

l = r / tan(alpha/2)


# given two vectors a = (xa, ya) and b = (xb, yb), they meet under angle alpha:

alpha = acos( (xa*xb+ya*yb) / ( sqrt(xa*xa+ya*ya) * sqrt(xb*xb+yb*yb) ) )


# https://stackoverflow.com/questions/1734745/how-to-create-circle-with-b%C3%A9zier-curves
  [To cover a full circle] for Bezier curve with n segments the optimal distance to the control 
  points, in the sense that the middle of the curve lies on the circle itself, is (4/3)*tan(pi/(2n)).

# https://stackoverflow.com/questions/734076/how-to-best-approximate-a-geometrical-arc-with-a-bezier-curve
  Per the corrections listed in this blog post, given the start and end points of
  the arc ([x1, y1] and [x4, y4], respectively) and the the center of the circle
  ([xc, yc]), one can derive the control points for a cubic bezier curve ([x2,
  y2] and [x3, y3]) as follows:
  
  ax = x1 – xc
  ay = y1 – yc
  bx = x4 – xc
  by = y4 – yc
  q1 = ax * ax + ay * ay
  q2 = q1 + ax * bx + ay * by
  k2 = 4/3 * (√(2 * q1 * q2) – q2) / (ax * by – ay * bx)
  
  
  x2 = xc + ax – k2 * ay
  y2 = yc + ay + k2 * ax
  x3 = xc + bx + k2 * by                                 
  y3 = yc + by – k2 * bx

# https://hansmuller-flex.blogspot.com/2011/10/more-about-approximating-circular-arcs.html?showComment=1498749617507#c2109832351939371205