Class: PerfectShape::QuadraticBezierCurve

Inherits:
Shape
  • Object
show all
Includes:
MultiPoint
Defined in:
lib/perfect_shape/quadratic_bezier_curve.rb

Constant Summary collapse

OUTLINE_MINIMUM_DISTANCE_THRESHOLD = 
BigDecimal('0.001')

Instance Attribute Summary

Attributes included from MultiPoint

#points

Class Method Summary collapse

Instance Method Summary collapse

Methods included from MultiPoint

#initialize, #max_x, #max_y, #min_x, #min_y, normalize_point_array

Methods inherited from Shape

#==, #bounding_box, #center_point, #center_x, #center_y, #height, #max_x, #max_y, #min_x, #min_y, #width

Class Method Details

.point_crossings(x1, y1, xc, yc, x2, y2, px, py, level = 0) ⇒ Object

Calculates the number of times the quadratic bézier curve from (x1,y1) to (x2,y2) crosses the ray extending to the right from (x,y). If the point lies on a part of the curve, then no crossings are counted for that intersection. the level parameter should be 0 at the top-level call and will count up for each recursion level to prevent infinite recursion +1 is added for each crossing where the Y coordinate is increasing -1 is added for each crossing where the Y coordinate is decreasing



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# File 'lib/perfect_shape/quadratic_bezier_curve.rb', line 37

def point_crossings(x1, y1, xc, yc, x2, y2, px, py, level = 0)
  return 0 if (py <  y1 && py <  yc && py <  y2)
  return 0 if (py >= y1 && py >= yc && py >= y2)
  # Note y1 could equal y2...
  return 0 if (px >= x1 && px >= xc && px >= x2)
  if (px <  x1 && px <  xc && px <  x2)
    if (py >= y1)
      return 1 if (py < y2)
    else
      # py < y1
      return -1 if (py >= y2)
    end
    # py outside of y11 range, and/or y1==y2
    return 0
  end
  # double precision only has 52 bits of mantissa
  return PerfectShape::Line.point_crossings(x1, y1, x2, y2, px, py) if (level > 52)
  x1c = BigDecimal((x1 + xc).to_s) / 2
  y1c = BigDecimal((y1 + yc).to_s) / 2
  xc1 = BigDecimal((xc + x2).to_s) / 2
  yc1 = BigDecimal((yc + y2).to_s) / 2
  xc = BigDecimal((x1c + xc1).to_s) / 2
  yc = BigDecimal((y1c + yc1).to_s) / 2
  # [xy]c are NaN if any of [xy]0c or [xy]c1 are NaN
  # [xy]0c or [xy]c1 are NaN if any of [xy][0c1] are NaN
  # These values are also NaN if opposing infinities are added
  return 0 if (xc.nan? || yc.nan?)
  point_crossings(x1, y1, x1c, y1c, xc, yc, px, py, level+1) +
    point_crossings(xc, yc, xc1, yc1, x2, y2, px, py, level+1);
end

Instance Method Details

#contain?(x_or_point, y = nil, outline: false, distance_tolerance: 0) ⇒ @code true

Checks if quadratic bézier curve contains point (two-number Array or x, y args)

the quadratic bézier curve, false if the point lies outside of the quadratic bézier curve’s bounds.

Parameters:

  • x —

    The X coordinate of the point to test.

  • y (defaults to: nil) —

    The Y coordinate of the point to test.

Returns:

  • (@code true) —

    if the point lies within the bound of



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# File 'lib/perfect_shape/quadratic_bezier_curve.rb', line 82

def contain?(x_or_point, y = nil, outline: false, distance_tolerance: 0)
  x, y = Point.normalize_point(x_or_point, y)
  return unless x && y

  x1 = points[0][0]
  y1 = points[0][1]
  xc = points[1][0]
  yc = points[1][1]
  x2 = points[2][0]
  y2 = points[2][1]
  
  if outline
    distance_tolerance = BigDecimal(distance_tolerance.to_s)
    minimum_distance_threshold = OUTLINE_MINIMUM_DISTANCE_THRESHOLD + distance_tolerance
    point_distance(x, y, minimum_distance_threshold: minimum_distance_threshold) < minimum_distance_threshold
  else
    # We have a convex shape bounded by quad curve Pc(t)
    # and ine Pl(t).
    #
    #     P1 = (x1, y1) - start point of curve
    #     P2 = (x2, y2) - end point of curve
    #     Pc = (xc, yc) - control point
    #
    #     Pq(t) = P1*(1 - t)^2 + 2*Pc*t*(1 - t) + P2*t^2 =
    #           = (P1 - 2*Pc + P2)*t^2 + 2*(Pc - P1)*t + P1
    #     Pl(t) = P1*(1 - t) + P2*t
    #     t = [0:1]
    #
    #     P = (x, y) - point of interest
    #
    # Let's look at second derivative of quad curve equation:
    #
    #     Pq''(t) = 2 * (P1 - 2 * Pc + P2) = Pq''
    #     It's constant vector.
    #
    # Let's draw a line through P to be parallel to this
    # vector and find the intersection of the quad curve
    # and the line.
    #
    # Pq(t) is point of intersection if system of equations
    # below has the solution.
    #
    #     L(s) = P + Pq''*s == Pq(t)
    #     Pq''*s + (P - Pq(t)) == 0
    #
    #     | xq''*s + (x - xq(t)) == 0
    #     | yq''*s + (y - yq(t)) == 0
    #
    # This system has the solution if rank of its matrix equals to 1.
    # That is, determinant of the matrix should be zero.
    #
    #     (y - yq(t))*xq'' == (x - xq(t))*yq''
    #
    # Let's solve this equation with 't' variable.
    # Also let kx = x1 - 2*xc + x2
    #          ky = y1 - 2*yc + y2
    #
    #     t0q = (1/2)*((x - x1)*ky - (y - y1)*kx) /
    #                 ((xc - x1)*ky - (yc - y1)*kx)
    #
    # Let's do the same for our line Pl(t):
    #
    #     t0l = ((x - x1)*ky - (y - y1)*kx) /
    #           ((x2 - x1)*ky - (y2 - y1)*kx)
    #
    # It's easy to check that t0q == t0l. This fact means
    # we can compute t0 only one time.
    #
    # In case t0 < 0 or t0 > 1, we have an intersections outside
    # of shape bounds. So, P is definitely out of shape.
    #
    # In case t0 is inside [0:1], we should calculate Pq(t0)
    # and Pl(t0). We have three points for now, and all of them
    # lie on one line. So, we just need to detect, is our point
    # of interest between points of intersections or not.
    #
    # If the denominator in the t0q and t0l equations is
    # zero, then the points must be collinear and so the
    # curve is degenerate and encloses no area.  Thus the
    # result is false.
    kx = x1 - 2 * xc + x2;
    ky = y1 - 2 * yc + y2;
    dx = x - x1;
    dy = y - y1;
    dxl = x2 - x1;
    dyl = y2 - y1;
  
    t0 = (dx * ky - dy * kx) / (dxl * ky - dyl * kx)
    return false if (t0 < 0 || t0 > 1 || t0 != t0)
  
    xb = kx * t0 * t0 + 2 * (xc - x1) * t0 + x1;
    yb = ky * t0 * t0 + 2 * (yc - y1) * t0 + y1;
    xl = dxl * t0 + x1;
    yl = dyl * t0 + y1;
  
    (x >= xb && x < xl) ||
      (x >= xl && x < xb) ||
      (y >= yb && y < yl) ||
      (y >= yl && y < yb)
  end
end

#curve_center_point ⇒ Object

The center point on the outline of the curve in Array format as pair of (x, y) coordinates



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# File 'lib/perfect_shape/quadratic_bezier_curve.rb', line 200

def curve_center_point
  subdivisions.last.points[0]
end

#curve_center_x ⇒ Object

The center point x on the outline of the curve



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# File 'lib/perfect_shape/quadratic_bezier_curve.rb', line 205

def curve_center_x
  subdivisions.last.points[0][0]
end

#curve_center_y ⇒ Object

The center point y on the outline of the curve



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# File 'lib/perfect_shape/quadratic_bezier_curve.rb', line 210

def curve_center_y
  subdivisions.last.points[0][1]
end

#point_crossings(x_or_point, y = nil, level = 0) ⇒ Object

Calculates the number of times the quad crosses the ray extending to the right from (x,y). If the point lies on a part of the curve, then no crossings are counted for that intersection. the level parameter should be 0 at the top-level call and will count up for each recursion level to prevent infinite recursion +1 is added for each crossing where the Y coordinate is increasing -1 is added for each crossing where the Y coordinate is decreasing



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# File 'lib/perfect_shape/quadratic_bezier_curve.rb', line 192

def point_crossings(x_or_point, y = nil, level = 0)
  x, y = Point.normalize_point(x_or_point, y)
  return unless x && y
  QuadraticBezierCurve.point_crossings(points[0][0], points[0][1], points[1][0], points[1][1], points[2][0], points[2][1], x, y, level)
end

#point_distance(x_or_point, y = nil, minimum_distance_threshold: OUTLINE_MINIMUM_DISTANCE_THRESHOLD) ⇒ Object 



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# File 'lib/perfect_shape/quadratic_bezier_curve.rb', line 249

def point_distance(x_or_point, y = nil, minimum_distance_threshold: OUTLINE_MINIMUM_DISTANCE_THRESHOLD)
  x, y = Point.normalize_point(x_or_point, y)
  return unless x && y
  
  point = Point.new(x, y)
  current_curve = self
  minimum_distance = point.point_distance(curve_center_point)
  last_minimum_distance = minimum_distance + 1 # start bigger to ensure going through loop once at least
  while minimum_distance >= minimum_distance_threshold && minimum_distance < last_minimum_distance
    curve1, curve2 = current_curve.subdivisions
    distance1 = point.point_distance(curve1.curve_center_point)
    distance2 = point.point_distance(curve2.curve_center_point)
    last_minimum_distance = minimum_distance
    if distance1 < distance2
      minimum_distance = distance1
      current_curve = curve1
    else
      minimum_distance = distance2
      current_curve = curve2
    end
  end
  if minimum_distance < minimum_distance_threshold
    minimum_distance
  else
    last_minimum_distance
  end
end

#subdivisions(level = 1) ⇒ Object

Subdivides QuadraticBezierCurve exactly at its curve center returning 2 QuadraticBezierCurve’s as a two-element Array by default

Optional ‘level` parameter specifies the level of recursions to perform to get more subdivisions. The number of resulting subdivisions is 2 to the power of `level` (e.g. 2 subdivisions for level=1, 4 subdivisions for level=2, and 8 subdivisions for level=3)



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# File 'lib/perfect_shape/quadratic_bezier_curve.rb', line 221

def subdivisions(level = 1)
  level -= 1 # consume 1 level
  
  x1 = points[0][0]
  y1 = points[0][1]
  ctrlx = points[1][0]
  ctrly = points[1][1]
  x2 = points[2][0]
  y2 = points[2][1]
  ctrlx1 = BigDecimal((x1 + ctrlx).to_s) / 2
  ctrly1 = BigDecimal((y1 + ctrly).to_s) / 2
  ctrlx2 = BigDecimal((x2 + ctrlx).to_s) / 2
  ctrly2 = BigDecimal((y2 + ctrly).to_s) / 2
  centerx = BigDecimal((ctrlx1 + ctrlx2).to_s) / 2
  centery = BigDecimal((ctrly1 + ctrly2).to_s) / 2
  
  default_subdivisions = [
    QuadraticBezierCurve.new(points: [x1, y1, ctrlx1, ctrly1, centerx, centery]),
    QuadraticBezierCurve.new(points: [centerx, centery, ctrlx2, ctrly2, x2, y2])
  ]
  
  if level == 0
    default_subdivisions
  else
    default_subdivisions.map { |curve| curve.subdivisions(level) }.flatten
  end
end