//+-------------------------------------------------------------------------------------------------
//
// Function: ArcToBezier
//
// Synopsis: Compute the Bezier approximation of an arc
//
// Notes: This utilitycomputes the Bezier approximation for an elliptical arc as it is defined
// in the SVG arc spec. The ellipse from which the arc is carved is axis-aligned in its
// own coordinates, and defined there by its x and y radii. The rotation angle defines
// how the ellipse's axes are rotated relative to our x axis. The start and end points
// define one of 4 possible arcs; the sweep and large-arc flags determine which one of
// these arcs will be chosen. See SVG spec for details.
//
// Returning pieces = 0 indicates a line instead of an arc
// pieces = -1 indicates that the arc degenerates to a point
//
//--------------------------------------------------------------------------------------------------
public static PointCollection ArcToBezier(double xStart, double yStart, double xRadius, double yRadius, double rotationAngle,
bool isLargeArc, bool isClockwise, double xEnd, double yEnd, out int pieces)
{
double cosArcAngle, sinArcAngle, xCenter, yCenter, r, bezDist;
XVector vecToBez1, vecToBez2;
XMatrix matToEllipse;
double fuzz2 = FUZZ * FUZZ;
bool isZeroCenter = false;
pieces = -1;
// In the following, the line segment between between the arc's start and
// end points is referred to as "the chord".
// Transform 1: Shift the origin to the chord's midpoint
double x = (xEnd - xStart) / 2;
double y = (yEnd - yStart) / 2;
double halfChord2 = x * x + y * y; // (half chord length)^2
// Degenerate case: single point
if (halfChord2 < fuzz2)
{
// The chord degeneartes to a point, the arc will be ignored
return null;
}
// Degenerate case: straight line
if (!AcceptRadius(halfChord2, fuzz2, ref xRadius) || !AcceptRadius(halfChord2, fuzz2, ref yRadius))
{
// We have a zero radius, add a straight line segment instead of an arc
pieces = 0;
return null;
}
if (xRadius == 0 || yRadius == 0)
{
// We have a zero radius, add a straight line segment instead of an arc
pieces = 0;
return null;
}
// Transform 2: Rotate to the ellipse's coordinate system
rotationAngle = -rotationAngle * Calc.Deg2Rad;
double cos = Math.Cos(rotationAngle);
double sin = Math.Sin(rotationAngle);
r = x * cos - y * sin;
y = x * sin + y * cos;
x = r;
// Transform 3: Scale so that the ellipse will become a unit circle
x /= xRadius;
y /= yRadius;
// We get to the center of that circle along a verctor perpendicular to the chord
// from the origin, which is the chord's midpoint. By Pythagoras, the length of that
// vector is sqrt(1 - (half chord)^2).
halfChord2 = x * x + y * y; // now in the circle coordinates
if (halfChord2 > 1)
{
// The chord is longer than the circle's diameter; we scale the radii uniformly so
// that the chord will be a diameter. The center will then be the chord's midpoint,
// which is now the origin.
r = Math.Sqrt(halfChord2);
xRadius *= r;
yRadius *= r;
xCenter = yCenter = 0;
isZeroCenter = true;
// Adjust the unit-circle coordinates x and y
x /= r;
y /= r;
}
else
{
// The length of (-y,x) or (x,-y) is sqrt(rHalfChord2), and we want a vector
// of length sqrt(1 - rHalfChord2), so we'll multiply it by:
r = Math.Sqrt((1 - halfChord2) / halfChord2);
//if (isLargeArc != (eSweepDirection == SweepDirection.Clockwise))
if (isLargeArc != isClockwise)
// Going to the center from the origin=chord-midpoint
{
// in the direction of (-y, x)
xCenter = -r * y;
yCenter = r * x;
}
else
{
// in the direction of (y, -x)
xCenter = r * y;
yCenter = -r * x;
}
}
// Transformation 4: shift the origin to the center of the circle, which then becomes
// the unit circle. Since the chord's midpoint is the origin, the start point is (-x, -y)
// and the endpoint is (x, y).
XPoint ptStart = new XPoint(-x - xCenter, -y - yCenter);
XPoint ptEnd = new XPoint(x - xCenter, y - yCenter);
// Set up the matrix that will take us back to our coordinate system. This matrix is
// the inverse of the combination of transformation 1 thru 4.
matToEllipse = new XMatrix(cos * xRadius, -sin * xRadius,
sin * yRadius, cos * yRadius,
(xEnd + xStart) / 2, (yEnd + yStart) / 2);
if (!isZeroCenter)
{
// Prepend the translation that will take the origin to the circle's center
matToEllipse.OffsetX += (matToEllipse.M11 * xCenter + matToEllipse.M21 * yCenter);
matToEllipse.OffsetY += (matToEllipse.M12 * xCenter + matToEllipse.M22 * yCenter);
}
// Get the sine & cosine of the angle that will generate the arc pieces
GetArcAngle(ptStart, ptEnd, isLargeArc, isClockwise, out cosArcAngle, out sinArcAngle, out pieces);
// Get the vector to the first Bezier control point
bezDist = GetBezierDistance(cosArcAngle, 1);
//if (eSweepDirection == SweepDirection.Counterclockwise)
if (!isClockwise)
bezDist = -bezDist;
vecToBez1 = new XVector(-bezDist * ptStart.Y, bezDist * ptStart.X);
PointCollection result = new PointCollection();
// Add the arc pieces, except for the last
for (int idx = 1; idx < pieces; idx++)
{
// Get the arc piece's endpoint
XPoint ptPieceEnd = new XPoint(ptStart.X * cosArcAngle - ptStart.Y * sinArcAngle, ptStart.X * sinArcAngle + ptStart.Y * cosArcAngle);
vecToBez2 = new XVector(-bezDist * ptPieceEnd.Y, bezDist * ptPieceEnd.X);
result.Add(matToEllipse.Transform(ptStart + vecToBez1));
result.Add(matToEllipse.Transform(ptPieceEnd - vecToBez2));
result.Add(matToEllipse.Transform(ptPieceEnd));
// Move on to the next arc
ptStart = ptPieceEnd;
vecToBez1 = vecToBez2;
}
// Last arc - we know the endpoint
vecToBez2 = new XVector(-bezDist * ptEnd.Y, bezDist * ptEnd.X);
result.Add(matToEllipse.Transform(ptStart + vecToBez1));
result.Add(matToEllipse.Transform(ptEnd - vecToBez2));
result.Add(new XPoint(xEnd, yEnd));
return result;
}
/// <summary>
/// Multiplies a point with a matrix.
/// </summary>
public static XPoint Multiply(XPoint point, XMatrix matrix)
{
return matrix.Transform(point);
}
/// <summary>
/// Appends a Bézier curve for an arc within a full quadrant.
/// </summary>
static void AppendPartialArcQuadrant(List<XPoint> points, double x, double y, double width, double height, double α, double β, PathStart pathStart, XMatrix matrix)
{
Debug.Assert(α >= 0 && α <= 360);
Debug.Assert(β >= 0);
if (β > 360)
β = β - Math.Floor(β / 360) * 360;
Debug.Assert(Math.Abs(α - β) <= 90);
// Scanling factor.
double δx = width / 2;
double δy = height / 2;
// Center of ellipse.
double x0 = x + δx;
double y0 = y + δy;
// We have the following quarters:
// |
// 2 | 3
// ----+-----
// 1 | 0
// |
// If the angles lie in quarter 2 or 3, their values are subtracted by 180 and the
// resulting curve is reflected at the center. This algorithm works as expected (simply tried out).
// There may be a mathematically more elegant solution...
bool reflect = false;
if (α >= 180 && β >= 180)
{
α -= 180;
β -= 180;
reflect = true;
}
double cosα, cosβ, sinα, sinβ;
if (width == height)
{
// Circular arc needs no correction.
α = α * Calc.Deg2Rad;
β = β * Calc.Deg2Rad;
}
else
{
// Elliptic arc needs the angles to be adjusted such that the scaling transformation is compensated.
α = α * Calc.Deg2Rad;
sinα = Math.Sin(α);
if (Math.Abs(sinα) > 1E-10)
α = Math.PI / 2 - Math.Atan(δy * Math.Cos(α) / (δx * sinα));
β = β * Calc.Deg2Rad;
sinβ = Math.Sin(β);
if (Math.Abs(sinβ) > 1E-10)
β = Math.PI / 2 - Math.Atan(δy * Math.Cos(β) / (δx * sinβ));
}
double κ = 4 * (1 - Math.Cos((α - β) / 2)) / (3 * Math.Sin((β - α) / 2));
sinα = Math.Sin(α);
cosα = Math.Cos(α);
sinβ = Math.Sin(β);
cosβ = Math.Cos(β);
//XPoint pt1, pt2, pt3;
if (!reflect)
{
// Calculation for quarter 0 and 1.
switch (pathStart)
{
case PathStart.MoveTo1st:
points.Add(matrix.Transform(new XPoint(x0 + δx * cosα, y0 + δy * sinα)));
break;
case PathStart.LineTo1st:
points.Add(matrix.Transform(new XPoint(x0 + δx * cosα, y0 + δy * sinα)));
break;
case PathStart.Ignore1st:
break;
}
points.Add(matrix.Transform(new XPoint(x0 + δx * (cosα - κ * sinα), y0 + δy * (sinα + κ * cosα))));
points.Add(matrix.Transform(new XPoint(x0 + δx * (cosβ + κ * sinβ), y0 + δy * (sinβ - κ * cosβ))));
points.Add(matrix.Transform(new XPoint(x0 + δx * cosβ, y0 + δy * sinβ)));
}
else
{
// Calculation for quarter 2 and 3.
switch (pathStart)
{
case PathStart.MoveTo1st:
points.Add(matrix.Transform(new XPoint(x0 - δx * cosα, y0 - δy * sinα)));
break;
case PathStart.LineTo1st:
points.Add(matrix.Transform(new XPoint(x0 - δx * cosα, y0 - δy * sinα)));
break;
case PathStart.Ignore1st:
break;
}
points.Add(matrix.Transform(new XPoint(x0 - δx * (cosα - κ * sinα), y0 - δy * (sinα + κ * cosα))));
points.Add(matrix.Transform(new XPoint(x0 - δx * (cosβ + κ * sinβ), y0 - δy * (sinβ - κ * cosβ))));
points.Add(matrix.Transform(new XPoint(x0 - δx * cosβ, y0 - δy * sinβ)));
}
}
/// <summary>
/// Creates between 1 and 5 Béziers curves from parameters specified like in WPF.
/// </summary>
public static List<XPoint> BezierCurveFromArc(XPoint point1, XPoint point2, XSize size,
double rotationAngle, bool isLargeArc, bool clockwise, PathStart pathStart)
{
// See also http://www.charlespetzold.com/blog/blog.xml from January 2, 2008:
// http://www.charlespetzold.com/blog/2008/01/Mathematics-of-ArcSegment.html
double δx = size.Width;
double δy = size.Height;
Debug.Assert(δx * δy > 0);
double factor = δy / δx;
bool isCounterclockwise = !clockwise;
// Adjust for different radii and rotation angle.
XMatrix matrix = new XMatrix();
matrix.RotateAppend(-rotationAngle);
matrix.ScaleAppend(δy / δx, 1);
XPoint pt1 = matrix.Transform(point1);
XPoint pt2 = matrix.Transform(point2);
// Get info about chord that connects both points.
XPoint midPoint = new XPoint((pt1.X + pt2.X) / 2, (pt1.Y + pt2.Y) / 2);
XVector vect = pt2 - pt1;
double halfChord = vect.Length / 2;
// Get vector from chord to center.
XVector vectRotated;
// (comparing two Booleans here!)
if (isLargeArc == isCounterclockwise)
vectRotated = new XVector(-vect.Y, vect.X);
else
vectRotated = new XVector(vect.Y, -vect.X);
vectRotated.Normalize();
// Distance from chord to center.
double centerDistance = Math.Sqrt(δy * δy - halfChord * halfChord);
if (double.IsNaN(centerDistance))
centerDistance = 0;
// Calculate center point.
XPoint center = midPoint + centerDistance * vectRotated;
// Get angles from center to the two points.
double α = Math.Atan2(pt1.Y - center.Y, pt1.X - center.X);
double β = Math.Atan2(pt2.Y - center.Y, pt2.X - center.X);
// (another comparison of two Booleans!)
if (isLargeArc == (Math.Abs(β - α) < Math.PI))
{
if (α < β)
α += 2 * Math.PI;
else
β += 2 * Math.PI;
}
// Invert matrix for final point calculation.
matrix.Invert();
double sweepAngle = β - α;
// Let the algorithm of GDI+ DrawArc to Bézier curves do the rest of the job
return BezierCurveFromArc(center.X - δx * factor, center.Y - δy, 2 * δx * factor, 2 * δy,
α / Calc.Deg2Rad, sweepAngle / Calc.Deg2Rad, pathStart, ref matrix);
}
internal static void TransformRect(ref XRect rect, ref XMatrix matrix)
{
if (!rect.IsEmpty)
{
XMatrixTypes types = matrix.type;
if (types != XMatrixTypes.Identity)
{
if ((types & XMatrixTypes.Scaling) != XMatrixTypes.Identity)
{
rect.x *= matrix.m11;
rect.y *= matrix.m22;
rect.width *= matrix.m11;
rect.height *= matrix.m22;
if (rect.width < 0)
{
rect.x += rect.width;
rect.width = -rect.width;
}
if (rect.height < 0)
{
rect.y += rect.height;
rect.height = -rect.height;
}
}
if ((types & XMatrixTypes.Translation) != XMatrixTypes.Identity)
{
rect.x += matrix.offsetX;
rect.y += matrix.offsetY;
}
if (types == XMatrixTypes.Unknown)
{
XPoint point = matrix.Transform(rect.TopLeft);
XPoint point2 = matrix.Transform(rect.TopRight);
XPoint point3 = matrix.Transform(rect.BottomRight);
XPoint point4 = matrix.Transform(rect.BottomLeft);
rect.x = Math.Min(Math.Min(point.X, point2.X), Math.Min(point3.X, point4.X));
rect.y = Math.Min(Math.Min(point.Y, point2.Y), Math.Min(point3.Y, point4.Y));
rect.width = Math.Max(Math.Max(point.X, point2.X), Math.Max(point3.X, point4.X)) - rect.x;
rect.height = Math.Max(Math.Max(point.Y, point2.Y), Math.Max(point3.Y, point4.Y)) - rect.y;
}
}
}
}
/// <summary>
///
/// </summary>
/// <param name="gfx"></param>
/// <param name="line"></param>
/// <param name="dx"></param>
/// <param name="dy"></param>
/// <param name="db"></param>
/// <param name="r"></param>
public void Draw(object gfx, Kaliber3D.Render.XLine line, double dx, double dy, ImmutableArray<Kaliber3D.Render.ShapeProperty> db, Kaliber3D.Render.Record r)
{
if (!line.IsStroked)
return;
var _gfx = gfx as XGraphics;
XPen strokeLine = ToXPen(line.Style, _scaleToPage);
XSolidBrush fillStartArrow = ToXSolidBrush(line.Style.StartArrowStyle.Fill);
XPen strokeStartArrow = ToXPen(line.Style.StartArrowStyle, _scaleToPage);
XSolidBrush fillEndArrow = ToXSolidBrush(line.Style.EndArrowStyle.Fill);
XPen strokeEndArrow = ToXPen(line.Style.EndArrowStyle, _scaleToPage);
double _x1 = line.Start.X + dx;
double _y1 = line.Start.Y + dy;
double _x2 = line.End.X + dx;
double _y2 = line.End.Y + dy;
Kaliber3D.Render.XLine.SetMaxLength(line, ref _x1, ref _y1, ref _x2, ref _y2);
double x1 = _scaleToPage(_x1);
double y1 = _scaleToPage(_y1);
double x2 = _scaleToPage(_x2);
double y2 = _scaleToPage(_y2);
var sas = line.Style.StartArrowStyle;
var eas = line.Style.EndArrowStyle;
double a1 = Math.Atan2(y1 - y2, x1 - x2) * 180.0 / Math.PI;
double a2 = Math.Atan2(y2 - y1, x2 - x1) * 180.0 / Math.PI;
var t1 = new XMatrix();
var c1 = new XPoint(x1, y1);
t1.RotateAtPrepend(a1, c1);
var t2 = new XMatrix();
var c2 = new XPoint(x2, y2);
t2.RotateAtPrepend(a2, c2);
XPoint pt1;
XPoint pt2;
double radiusX1 = sas.RadiusX;
double radiusY1 = sas.RadiusY;
double sizeX1 = 2.0 * radiusX1;
double sizeY1 = 2.0 * radiusY1;
switch (sas.ArrowType)
{
default:
case Kaliber3D.Render.ArrowType.None:
{
pt1 = new XPoint(x1, y1);
}
break;
case Kaliber3D.Render.ArrowType.Rectangle:
{
pt1 = t1.Transform(new XPoint(x1 - sizeX1, y1));
var rect = new XRect(x1 - sizeX1, y1 - radiusY1, sizeX1, sizeY1);
_gfx.Save();
_gfx.RotateAtTransform(a1, c1);
DrawRectangleInternal(_gfx, fillStartArrow, strokeStartArrow, sas.IsStroked, sas.IsFilled, ref rect);
_gfx.Restore();
}
break;
case Kaliber3D.Render.ArrowType.Ellipse:
{
pt1 = t1.Transform(new XPoint(x1 - sizeX1, y1));
_gfx.Save();
_gfx.RotateAtTransform(a1, c1);
var rect = new XRect(x1 - sizeX1, y1 - radiusY1, sizeX1, sizeY1);
DrawEllipseInternal(_gfx, fillStartArrow, strokeStartArrow, sas.IsStroked, sas.IsFilled, ref rect);
_gfx.Restore();
}
break;
case Kaliber3D.Render.ArrowType.Arrow:
{
pt1 = t1.Transform(new XPoint(x1, y1));
var p11 = t1.Transform(new XPoint(x1 - sizeX1, y1 + sizeY1));
var p21 = t1.Transform(new XPoint(x1, y1));
var p12 = t1.Transform(new XPoint(x1 - sizeX1, y1 - sizeY1));
var p22 = t1.Transform(new XPoint(x1, y1));
DrawLineInternal(_gfx, strokeStartArrow, sas.IsStroked, ref p11, ref p21);
DrawLineInternal(_gfx, strokeStartArrow, sas.IsStroked, ref p12, ref p22);
}
break;
}
double radiusX2 = eas.RadiusX;
double radiusY2 = eas.RadiusY;
double sizeX2 = 2.0 * radiusX2;
double sizeY2 = 2.0 * radiusY2;
switch (eas.ArrowType)
{
default:
case Kaliber3D.Render.ArrowType.None:
{
pt2 = new XPoint(x2, y2);
}
break;
case Kaliber3D.Render.ArrowType.Rectangle:
{
pt2 = t2.Transform(new XPoint(x2 - sizeX2, y2));
var rect = new XRect(x2 - sizeX2, y2 - radiusY2, sizeX2, sizeY2);
_gfx.Save();
_gfx.RotateAtTransform(a2, c2);
DrawRectangleInternal(_gfx, fillEndArrow, strokeEndArrow, eas.IsStroked, eas.IsFilled, ref rect);
_gfx.Restore();
}
break;
case Kaliber3D.Render.ArrowType.Ellipse:
{
pt2 = t2.Transform(new XPoint(x2 - sizeX2, y2));
_gfx.Save();
_gfx.RotateAtTransform(a2, c2);
var rect = new XRect(x2 - sizeX2, y2 - radiusY2, sizeX2, sizeY2);
DrawEllipseInternal(_gfx, fillEndArrow, strokeEndArrow, eas.IsStroked, eas.IsFilled, ref rect);
_gfx.Restore();
}
break;
case Kaliber3D.Render.ArrowType.Arrow:
{
pt2 = t2.Transform(new XPoint(x2, y2));
var p11 = t2.Transform(new XPoint(x2 - sizeX2, y2 + sizeY2));
var p21 = t2.Transform(new XPoint(x2, y2));
var p12 = t2.Transform(new XPoint(x2 - sizeX2, y2 - sizeY2));
var p22 = t2.Transform(new XPoint(x2, y2));
DrawLineInternal(_gfx, strokeEndArrow, eas.IsStroked, ref p11, ref p21);
DrawLineInternal(_gfx, strokeEndArrow, eas.IsStroked, ref p12, ref p22);
}
break;
}
_gfx.DrawLine(strokeLine, pt1, pt2);
}
public static XVector Multiply(XVector vector, XMatrix matrix)
{
return matrix.Transform(vector);
}
private static XPoint DrawLineArrowInternal(XGraphics gfx, XPen pen, XSolidBrush brush, double x, double y, double angle, Core2D.Style.ArrowStyle style)
{
XPoint pt;
var rt = new XMatrix();
var c = new XPoint(x, y);
rt.RotateAtPrepend(angle, c);
double rx = style.RadiusX;
double ry = style.RadiusY;
double sx = 2.0 * rx;
double sy = 2.0 * ry;
switch (style.ArrowType)
{
default:
case Core2D.Style.ArrowType.None:
{
pt = new XPoint(x, y);
}
break;
case Core2D.Style.ArrowType.Rectangle:
{
pt = rt.Transform(new XPoint(x - sx, y));
var rect = new XRect(x - sx, y - ry, sx, sy);
gfx.Save();
gfx.RotateAtTransform(angle, c);
DrawRectangleInternal(gfx, brush, pen, style.IsStroked, style.IsFilled, ref rect);
gfx.Restore();
}
break;
case Core2D.Style.ArrowType.Ellipse:
{
pt = rt.Transform(new XPoint(x - sx, y));
gfx.Save();
gfx.RotateAtTransform(angle, c);
var rect = new XRect(x - sx, y - ry, sx, sy);
DrawEllipseInternal(gfx, brush, pen, style.IsStroked, style.IsFilled, ref rect);
gfx.Restore();
}
break;
case Core2D.Style.ArrowType.Arrow:
{
pt = rt.Transform(new XPoint(x, y));
var p11 = rt.Transform(new XPoint(x - sx, y + sy));
var p21 = rt.Transform(new XPoint(x, y));
var p12 = rt.Transform(new XPoint(x - sx, y - sy));
var p22 = rt.Transform(new XPoint(x, y));
DrawLineInternal(gfx, pen, style.IsStroked, ref p11, ref p21);
DrawLineInternal(gfx, pen, style.IsStroked, ref p12, ref p22);
}
break;
}
return pt;
}
void AddCapToPath(Path path, PathFigure figure, double length, double lineWidthHalf, LineCap lineCap, XMatrix matrix)
{
// sketch:
// 1. create Transform that make a horizontal line with start in 0,0
// 2. create a Polygon with the shape of the line including its caps
// 3. render the shape with the brush of the pen
//PolyLineSegment seg;
switch (lineCap)
{
case LineCap.Flat:
matrix.Transform(new XPoint(length + lineWidthHalf, -lineWidthHalf));
break;
case LineCap.Square:
matrix.Transform(new XPoint(length + lineWidthHalf, -lineWidthHalf));
break;
case LineCap.Round:
break;
case LineCap.Triangle:
break;
}
}