[pens.areaPen] Clean up a bit

Based on discussion in
https://github.com/behdad/fonttools/pull/618#issuecomment-226105824

Lib/fontTools/pens/areaPen.py

@@ -5,29 +5,6 @@ from fontTools.misc.py23 import *
 from fontTools.pens.basePen import BasePen
 
 
-def polygon_area(p0, p1):
-	return -(p1[0] - p0[0]) * (p1[1] + p0[1]) * 0.5
-
-
-def quadratic_curve_area(p0, p1, p2):
-	x0, y0 = p0[0], p0[1]
-	x1, y1 = p1[0] - x0, p1[1] - y0
-	x2, y2 = p2[0] - x0, p2[1] - y0
-	return (x1*y2 - x2*y1) / 3
-
-
-def cubic_curve_area(p0, p1, p2, p3):
-	x0, y0 = p0[0], p0[1]
-	x1, y1 = p1[0] - x0, p1[1] - y0
-	x2, y2 = p2[0] - x0, p2[1] - y0
-	x3, y3 = p3[0] - x0, p3[1] - y0
-	return -(
-		x1 * (   -   y2 -   y3) +
-		x2 * (y1        - 2*y3) +
-		x3 * (y1 + 2*y2       )
-	) * 0.15
-
-
 class AreaPen(BasePen):
 
 	def __init__(self, glyphset=None):
@@ -35,23 +12,56 @@ class AreaPen(BasePen):
 		self.value = 0
 
 	def _moveTo(self, p0):
-		self.__startPoint = p0
+		"""Remember the first point in this contour, in case it's closed. Also
+		set the initial value for p0 in this contour, which will always refer to
+		the most recent point.
+		"""
+		self._p0 = self._startPoint = p0
 
 	def _lineTo(self, p1):
-		p0 = self._getCurrentPoint()
-		self.value += polygon_area(p0, p1)
-
-	def _curveToOne(self, p1, p2, p3):
-		p0 = self._getCurrentPoint()
-		self.value += cubic_curve_area(p0, p1, p2, p3)
-		self.value += polygon_area(p0, p3)
+		"""Add the signed area beneath the line from the latest point to this
+		one. Signed areas cancel each other based on the horizontal direction of
+		the line.
+		"""
+		x0, y0 = self._p0
+		x1, y1 = p1
+		self.value -= (x1 - x0) * (y1 + y0) * .5
+		self._p0 = p1
 
 	def _qCurveToOne(self, p1, p2):
-		p0 = self._getCurrentPoint()
-		self.value += quadratic_curve_area(p0, p1, p2)
-		self.value += polygon_area(p0, p2)
+		"""Add the signed area of this quadratic curve.
+		https://github.com/Pomax/bezierinfo/issues/44
+		"""
+		p0 = self._p0
+		x0, y0 = p0[0], p0[1]
+		x1, y1 = p1[0] - x0, p1[1] - y0
+		x2, y2 = p2[0] - x0, p2[1] - y0
+		self.value -= (x2 * y1 - x1 * y2) / 3
+		self._lineTo(p2)
+		self._p0 = p2
+
+	def _curveToOne(self, p1, p2, p3):
+		"""Add the signed area of this cubic curve.
+		https://github.com/Pomax/bezierinfo/issues/44
+		"""
+		p0 = self._p0
+		x0, y0 = p0[0], p0[1]
+		x1, y1 = p1[0] - x0, p1[1] - y0
+		x2, y2 = p2[0] - x0, p2[1] - y0
+		x3, y3 = p3[0] - x0, p3[1] - y0
+		self.value -= (
+				x1 * (   -   y2 -   y3) +
+				x2 * (y1        - 2*y3) +
+				x3 * (y1 + 2*y2       )
+			      ) * 0.15
+		self._lineTo(p3)
+		self._p0 = p3
 
 	def _closePath(self):
-		p0 = self._getCurrentPoint()
-		if p0 != self.__startPoint:
-			self.value += polygon_area(p0, self.__startPoint)
+		"""Add the area beneath this contour's closing line."""
+		self._lineTo(self._startPoint)
+		del self._p0, self._startPoint
+
+	def _endPath(self):
+		"""Area is not defined for open contours."""
+		raise NotImplementedError