深入探讨：从CSS到晶体管

if (allBordersSame &&
      mCompositeColors[0] == nullptr &&
      mBorderStyles[0] == NS_STYLE_BORDER_STYLE_SOLID &&
      !mAvoidStroke &&
      !mNoBorderRadius)
  {
    // Relatively simple case.
    gfxRect outerRect = ThebesRect(mOuterRect);
    RoundedRect borderInnerRect(outerRect, mBorderRadii);
    borderInnerRect.Deflate(mBorderWidths[eSideTop],
                            mBorderWidths[eSideBottom],
                            mBorderWidths[eSideLeft],
                            mBorderWidths[eSideRight]);

    // Instead of stroking we just use two paths: an inner and an outer.
    // This allows us to draw borders that we couldn't when stroking. For example,
    // borders with a border width >= the border radius. (i.e. when there are
    // square corners on the inside)
    //
    // Further, this approach can be more efficient because the backend
    // doesn't need to compute an offset curve to stroke the path. We know that
    // the rounded parts are elipses we can offset exactly and can just compute
    // a new cubic approximation.
    RefPtr<PathBuilder> builder = mDrawTarget->CreatePathBuilder();
    AppendRoundedRectToPath(builder, mOuterRect, mBorderRadii, true);
    AppendRoundedRectToPath(builder, ToRect(borderInnerRect.rect), borderInnerRect.corners, false);
    RefPtr<Path> path = builder->Finish();
    mDrawTarget->Fill(path, color);
    return;
  }

    //      radius.width
    // |<----------------->|
    // |                   |
    // |             ___---+-------------
    // |         __--     #|#       ###
    // |       _-        ##|##     #####
    // |     /           ##+##     ##+##
    // |   /             # P #     #####
    // |  |               #|#       ###
    // | |             __--+-------------
    // ||            _-    ^
    // ||          /       |
    // |         /        first dot is filled
    // |        |
    // |       |
    // |      |
    // |      |
    // |      |
    // +------+
    // |##  ##|
    // |##  ##|
    // |##  ##|

void
AppendRoundedRectToPath(PathBuilder* aPathBuilder,
                        const Rect& aRect,
                        const RectCornerRadii& aRadii,
                        bool aDrawClockwise)
{
  // For CW drawing, this looks like:
  //
  //  ...******0**      1    C
  //              ****
  //                  ***    2
  //                     **
  //                       *
  //                        *
  //                         3
  //                         *
  //                         *
  //
  // Where 0, 1, 2, 3 are the control points of the Bezier curve for
  // the corner, and C is the actual corner point.
  //
  // At the start of the loop, the current point is assumed to be
  // the point adjacent to the top left corner on the top
  // horizontal.  Note that corner indices start at the top left and
  // continue clockwise, whereas in our loop i = 0 refers to the top
  // right corner.
  //
  // When going CCW, the control points are swapped, and the first
  // corner that's drawn is the top left (along with the top segment).
  //
  // There is considerable latitude in how one chooses the four
  // control points for a Bezier curve approximation to an ellipse.
  // For the overall path to be continuous and show no corner at the
  // endpoints of the arc, points 0 and 3 must be at the ends of the
  // straight segments of the rectangle; points 0, 1, and C must be
  // collinear; and points 3, 2, and C must also be collinear.  This
  // leaves only two free parameters: the ratio of the line segments
  // 01 and 0C, and the ratio of the line segments 32 and 3C.  See
  // the following papers for extensive discussion of how to choose
  // these ratios:
  //
  //   Dokken, Tor, et al. "Good approximation of circles by
  //      curvature-continuous Bezier curves."  Computer-Aided
  //      Geometric Design 7(1990) 33--41.
  //   Goldapp, Michael. "Approximation of circular arcs by cubic
  //      polynomials." Computer-Aided Geometric Design 8(1991) 227--238.
  //   Maisonobe, Luc. "Drawing an elliptical arc using polylines,
  //      quadratic, or cubic Bezier curves."
  //      http://www.spaceroots.org/documents/ellipse/elliptical-arc.pdf
  //
  // We follow the approach in section 2 of Goldapp (least-error,
  // Hermite-type approximation) and make both ratios equal to
  //
  //          2   2 + n - sqrt(2n + 28)
  //  alpha = - * ---------------------
  //          3           n - 4
  //
  // where n = 3( cbrt(sqrt(2)+1) - cbrt(sqrt(2)-1) ).
  //
  // This is the result of Goldapp's equation (10b) when the angle
  // swept out by the arc is pi/2, and the parameter "a-bar" is the
  // expression given immediately below equation (21).
  //
  // Using this value, the maximum radial error for a circle, as a
  // fraction of the radius, is on the order of 0.2 x 10^-3.
  // Neither Dokken nor Goldapp discusses error for a general
  // ellipse; Maisonobe does, but his choice of control points
  // follows different constraints, and Goldapp's expression for
  // 'alpha' gives much smaller radial error, even for very flat
  // ellipses, than Maisonobe's equivalent.
  //
  // For the various corners and for each axis, the sign of this
  // constant changes, or it might be 0 -- it's multiplied by the
  // appropriate multiplier from the list before using.

  const Float alpha = Float(0.55191497064665766025);

  typedef struct { Float a, b; } twoFloats;

  twoFloats cwCornerMults[4] = { { -1,  0 },    // cc == clockwise
                                 {  0, -1 },
                                 { +1,  0 },
                                 {  0, +1 } };
  twoFloats ccwCornerMults[4] = { { +1,  0 },   // ccw == counter-clockwise
                                  {  0, -1 },
                                  { -1,  0 },
                                  {  0, +1 } };

  twoFloats *cornerMults = aDrawClockwise ? cwCornerMults : ccwCornerMults;

  Point cornerCoords[] = { aRect.TopLeft(), aRect.TopRight(),
                           aRect.BottomRight(), aRect.BottomLeft() };

  Point pc, p0, p1, p2, p3;

  if (aDrawClockwise) {
    aPathBuilder->MoveTo(Point(aRect.X() + aRadii[RectCorner::TopLeft].width,
                               aRect.Y()));
  } else {
    aPathBuilder->MoveTo(Point(aRect.X() + aRect.Width() - aRadii[RectCorner::TopRight].width,
                               aRect.Y()));
  }

  for (int i = 0; i < 4; ++i) {
    // the corner index -- either 1 2 3 0 (cw) or 0 3 2 1 (ccw)
    int c = aDrawClockwise ? ((i+1) % 4) : ((4-i) % 4);

    // i+2 and i+3 respectively.  These are used to index into the corner
    // multiplier table, and were deduced by calculating out the long form
    // of each corner and finding a pattern in the signs and values.
    int i2 = (i+2) % 4;
    int i3 = (i+3) % 4;

    pc = cornerCoords[c];

    if (aRadii[c].width > 0.0 && aRadii[c].height > 0.0) {
      p0.x = pc.x + cornerMults[i].a * aRadii[c].width;
      p0.y = pc.y + cornerMults[i].b * aRadii[c].height;

      p3.x = pc.x + cornerMults[i3].a * aRadii[c].width;
      p3.y = pc.y + cornerMults[i3].b * aRadii[c].height;

      p1.x = p0.x + alpha * cornerMults[i2].a * aRadii[c].width;
      p1.y = p0.y + alpha * cornerMults[i2].b * aRadii[c].height;

      p2.x = p3.x - alpha * cornerMults[i3].a * aRadii[c].width;
      p2.y = p3.y - alpha * cornerMults[i3].b * aRadii[c].height;

      aPathBuilder->LineTo(p0);
      aPathBuilder->BezierTo(p1, p2, p3);
    } else {
      aPathBuilder->LineTo(pc);
    }
  }

  aPathBuilder->Close();
}