Arc, tangents & Bezier studies

from sketch-a-day project, sketch_2020_09_26a

This repository contains the product of my studies trying to work with arcs, bezier approximations of arcs and tangents. Most functions were made initially on Processing Python mode and now work with py5. Some will work with pyp5js too, and I have ported b_arc for Processing Java and p5js. You are welcome to help porting more stuff!

I have added very little new ideas here, you can look at the findings & previous studies I moved into the PRIOR_ART folder. I tried to attribute stuff with links.

Please note that the most recent (and maybe unstable?) version of the Python functions shown here are kept at this other repository: github.com/villares/villares. You will probably want to get arcs.py and line_geometry.py it depends on.

Bezier approximation of an arc

Processing PShape insfrastructure (and Py5hape that depends on it) does not contain a fuction for embeding an arc in a larger polyline shape. The b_arc function provided here can be used inside beginShape()/endShape() as a kind of "arcVertex" (which doesn't exist). It follows mostly the Processing arc signature, but does not include PIE and CHORD modes. You can find demos at:

• b_arc Processing Python mode (also works with pyp5js and py5)

• b_arc Processing Java

• b_arc p5.js

  

  x, y, w, h, start_angle, end_angle = 75, 100, 100, 100, 0, PI + QUARTER_PI
  # Standalone arc replacement
  b_arc(x, y, w, h, start_angle, end_angle)
  
  # mode=2 for use inside beginShape/endShape
  x += 125
  beginShape()
  b_arc(x, y, w, h, start_angle, end_angle, mode=2) 
  endShape(CLOSE)
  x += 125
  beginShape()
  b_arc(x, y, w, h, start_angle, end_angle, mode=2)
  vertex(x, y)
  endShape(CLOSE)

More arcs and the first tangents

Other functions based on b_arc, and a polygonal approximation called p_arc, are kept at arcs.py.

• The var_bar and bar functions draw "two connected circles". They can be used with p_arc(a polygonal approximation of an arc) instead of the default b_arc.

• The var_bar_pts function, based on arc_pts, returns the points that var_bar would draw with the same arguments (except the internal feature).

• The arc_pts function returns a list of points (as tuples), that p_arc would draw, but does not draw them. p_arc now uses arc_pts internally.

  

  …
  # default var_bar & bar with b_arc
  var_bar(50, 165, 350, 315, 40, 0) # by default arc_func=b_arc
  bar(50, 55, 350, 255, thickness=60, shorter=mouseX)
  var_bar(50, 255, 50 + mouseX * 0.6, 255 + mouseX * 0.25, 20, 40)
  … 
  # var_bar & bar with p_arc
  var_bar(50, 165, 350, 315, 40, 0, arc_func=p_arc, num_points=6)
  bar(50, 55, 350, 255, thickness=60,
      shorter=mouseX, arc_func=p_arc, num_points=3)
  var_bar(50, 255,
          50 + mouseX * 0.6, 255 + mouseX * 0.25, 20, 40,
          arc_func=p_arc, num_points=8)
  … 
  # var_bar_pts
  pts1 = var_bar_pts(50, 165, 350, 315, 40, 0, num_points=6)
  pts2 = var_bar_pts(50, 55, 350, 255, 30, 30,
                     shorter=mouseX, num_points=3)
  pts3 = var_bar_pts(50, 255,
                     50 + mouseX * 0.6, 255 + mouseX * 0.25, 20, 40,
                     num_points=8)
  strokeWeight(5)
  for px, py in pts1 + pts2 + pts3:
      point(px, py)
  …

  The full code of the Processing Python mode example above is at villares_arcs_and_bars.

• The same interactive var_bar & var_bar_pts example above, but running on the browser using pyp5js!

• There is a similar, simpler, interactive var_bar_pts example using py5 imported mode at var_bar_pts_py5_example.

Filleted polygons and nice shapes that can wrap circles

Perhaps the bigest motivation for starting the studies in this repository, next, we have some functions that povide continous poly-based shapes with tangent arcs.

• Rounding polygons "in", filleted polygons arc_filleted_poly, takes a sequence of points and radii and uses the arc_cornerfunction to draw. Notice it may need to make a radius smaller to fit sometimes. A recently added radius keyword can be supplied instead of the radius values list.

  

  p_list = [(30, 160), (250, 50), (350, 150), (200, 100)]
  r_list = [20, 30, 40, 30]
  …
  arc_filleted_poly(p_list,r_list)  # arc_func=b_arc by default

  

  p_list = [(30, 160), (250, 50), (350, 150), (mouseX, mouseY)]
  r_list = [20, 30, 40, 30]
  …
  arc_filleted_poly(p_list,r_list)  # arc_func=b_arc by default

• Rounding polygons "outside" with arc_augmented_poly, takes a sequence of points and radii and calculates geometry with the circ_circ_tangent function. If two points are too close it will reduce the radii.

  

  p_list = [(30, 160), (250, 50), (350, 150), (200, 100)]
  r_list = [20, 30, 40, 30]
  …
  arc_augmented_poly(p_list,r_list)  # arc_func=b_arc by default

  

  p_list = [(30, 160), (250, 50), (350, 150), (mouseX, mouseY)]
  r_list = [20, 30, 40, 30]
  …
  arc_augmented_poly(p_list,r_list)  # arc_func=b_arc by default

  • TO DO: I should document an "ugly" feature of arc_augmented_poly that checks for self intersections calculating a polygonal approximation, without drawing the shapes.

  A few other silly arc helpers

• The circle_arc function tries to create a simpler interface for Processing's arc, asking for x, y, radius, start_angle, and sweep (radius instead of width, height and sweep instead of end_angle). It also allows drawing with b_arc or p_arc.

• The half_circle and quarter_circle are similar, very silly arc (or b_arc/p_arc) wrappers using a mix of Processing constants to define rotation.

  

  x, y, radius, start_angle, sweep = 75, 105, 50, 0, PI + QUARTER_PI
  circle_arc(x, y, radius, start_angle, sweep)    # default 'arc' wrapper mode
  circle_arc(x, y, radius, -QUARTER_PI / 2, -HALF_PI, arc_func=p_arc, num_points=4)
  
  x, y1, y2 = 190, 95, 105
  half_circle(x, y1, radius, TOP, CHORD)  # default 'arc' wrapper mode
  half_circle(x, y2, radius, BOTTOM, arc_func=b_arc)
  
  x1, x2, y1, y2 = 300, 310, 95, 105
  quarter_circle(x1, y1, radius, TOP + LEFT, CHORD)  # default 'arc' wrapper mode
  quarter_circle(x1, y2, radius, BOTTOM + LEFT, PIE)
  quarter_circle(x2, y1, radius, TOP + RIGHT)
  quarter_circle(x2, y2, radius, BOTTOM + RIGHT, arc_func=b_arc)

• circle_arc_pts will return a list of points like the ones p_arc would draw.

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Please open an issue if you have a question. Consider supporting the author, Alexandre B A Villares, at gumroad.com/villares.