/// <summary>
/// Returns circles that are tangent to another circle and line.
/// </summary>
/// <param name="circle">Circle that returned circles should be tangent to.</param>
/// <param name="line">Line that returned circles should be tangent to.</param>
/// <param name="radius">Radius of new circles.</param>
/// <returns>External tangent circles are listed first.</returns>
public static Circle2D[] FromTangentTangentRadius(Circle2D circle, LineSeg2D line, decimal radius)
{
List <Circle2D> orbits = new List <Circle2D>();
LineSeg2D[] parallels = new LineSeg2D[2];
List <Point2D> centerPoints = new List <Point2D>();
List <Circle2D> tangentCircles = new List <Circle2D>();
orbits.Add(new Circle2D(circle.X, circle.Y, circle.Radius + radius));
if (radius < circle.Radius)
{
orbits.Add(new Circle2D(circle.X, circle.Y, circle.Radius - radius));
}
parallels = line.GetParallels(radius);
foreach (Circle2D orbit in orbits)
{
for (int i = 0; i <= 1; i++)
{
centerPoints.AddRange(orbit.GetIntersect(parallels[i]));
}
}
for (int i = 0; i <= centerPoints.Count - 1; i++)
{
tangentCircles.Add(new Circle2D(centerPoints[i].X, centerPoints[i].Y, radius));
}
return(tangentCircles.ToArray());
}
/// <summary>
/// Returns circles that are tangent to two lines.
/// </summary>
/// <param name="lineA">First line.</param>
/// <param name="lineB">Second line.</param>
/// <param name="radius">Radius of the tangent circles.</param>
/// <returns>Will return no circles if the two lines are parallel or the same line.</returns>
public static Circle2D[] FromTangentTangentRadius(LineSeg2D lineA, LineSeg2D lineB, decimal radius)
{
List <LineSeg2D> parallels = new List <LineSeg2D>();
Nullable <Point2D> centerPoint = default(Nullable <Point2D>);
List <Point2D> centerPoints = new List <Point2D>();
List <Circle2D> tangentCircles = new List <Circle2D>();
parallels.AddRange(lineA.GetParallels(radius));
parallels.AddRange(lineB.GetParallels(radius));
for (int i = 0; i <= parallels.Count - 2; i++)
{
for (int j = i + 1; j <= parallels.Count - 1; j++)
{
centerPoint = parallels[i].GetIntersect(parallels[j], true);
if (centerPoint.HasValue)
{
centerPoints.Add(centerPoint.Value);
}
}
}
for (int i = 0; i <= centerPoints.Count - 1; i++)
{
tangentCircles.Add(new Circle2D(centerPoints[i].X, centerPoints[i].Y, radius));
}
return(tangentCircles.ToArray());
}
public static Arc2D FromLinesCircle(LineSeg2D line1, LineSeg2D line2, Circle2D c, int decimals = -1)
{
Point2D[] intersects1 = null;
Point2D[] intersects2 = null;
intersects1 = c.GetIntersect(line1, decimals);
intersects2 = c.GetIntersect(line2, decimals);
throw new NotImplementedException();
}
public static bool PointsAreColinear(decimal x1, decimal y1, decimal x2, decimal y2, decimal x3, decimal y3)
{
var l1 = new LineSeg2D(x1, y1, x2, y2);
var l2 = new LineSeg2D(x2, y2, x3, y3);
var intersect = l1.GetIntersect(l2, true);
// Points won't have an intersection if they are parallel. If they
// are parallel and share a point, then they are colinear.
return(!intersect.HasValue);
}
/// <summary>
/// Returns a line segment with the coordinates of a chord centered on
/// the given angle with the given length.
/// </summary>
/// <param name="angleThroughCenter">The angle that passes through the
/// center of the chord, in degrees.</param>
/// <param name="chordLength">The length of the chord.</param>
public LineSeg2D GetChord(decimal angleThroughCenter, decimal chordLength)
{
LineSeg2D l = default(LineSeg2D);
decimal a = 0m;
// Create a right triangle where one side is half the chord length,
// the other side is from the center of the circle to the midpoint
// of the chord, and the hypotenuse is from the center of the circle
// to the chord start point
a = RightTriangle.GetAngleFromOppSideHyp(0.5m * chordLength, _radius);
l.Pt1 = PointAt(angleThroughCenter - a);
l.Pt2 = PointAt(angleThroughCenter + a);
return(l);
}
/// <summary>
/// Create a circle from two points.
/// </summary>
/// <param name="pt1">A point.</param>
/// <param name="pt2">A point.</param>
public Circle2D(Point2D pt1, Point2D pt2)
{
_center = default(Point2D);
_radius = 0m;
LineSeg2D l = new LineSeg2D(pt1, pt2);
if (l.IsZeroLength())
{
throw new Exception("Can't create circle from two points that are the same point!");
}
Center = l.MidPoint;
Radius = l.Length / 2m;
if (Radius == 0)
{
throw new Exception("Can't create circle from two points that are the same point!");
}
}
/// <summary>
/// Gets a line segment of length 1 starting at the point on the circle
/// and extending either clockwise or counterclockwise.
/// </summary>
/// <param name="degrees">An angle.</param>
public LineSeg2D TangentAt(decimal degrees, decimal length, bool clockwise)
{
var r = RightTriangleAbstract.FromTwoSides(_radius, length);
decimal rad = DecimalEx.ToRad(degrees + r.AngleA);
LineSeg2D ret = default(LineSeg2D);
Debug.Assert(r.AngleA == RightTriangle.GetAngleFromSides(length, _radius));
Debug.Assert(r.Hypotenuse == RightTriangle.GetHypFromSides(length, _radius));
ret.Pt1 = PointAt(degrees);
if (!clockwise)
{
ret.Pt2 = new Point2D(_center.X + r.Hypotenuse * DecimalEx.Cos(rad), _center.Y + r.Hypotenuse * DecimalEx.Sin(rad));
}
else
{
ret.Pt2 = new Point2D(ret.Pt1.X - (_center.X + r.Hypotenuse * DecimalEx.Cos(rad) - ret.Pt1.X), ret.Pt1.Y - (_center.Y + r.Hypotenuse * DecimalEx.Sin(rad) - ret.Pt1.Y));
}
return(ret);
}
public static Circle2D[] FromTwoPointsAndRadius(decimal x1, decimal y1, decimal x2, decimal y2, decimal radius)
{
LineSeg2D pointToPoint = default(LineSeg2D);
Point2D midPoint = default(Point2D);
Vector2D vPerpBisector = default(Vector2D);
if (x1 == x2 && y1 == y2)
{
return new Circle2D[] { }
}
;
pointToPoint = new LineSeg2D(x1, y1, x2, y2);
midPoint = pointToPoint.MidPoint;
vPerpBisector = pointToPoint.PerpendicularBisector().GetVectorP1toP2();
vPerpBisector.Magnitude = RightTriangle.GetSideFromSideHyp(pointToPoint.Length / 2, radius);
return(new Circle2D[] {
new Circle2D(midPoint + vPerpBisector, radius),
new Circle2D(midPoint - vPerpBisector, radius)
});
}
/// <summary>
/// Creates a circle from three points.
/// </summary>
/// <param name="x1">A number.</param>
/// <param name="y1">A number.</param>
/// <param name="x2">A number.</param>
/// <param name="y2">A number.</param>
/// <param name="x3">A number.</param>
/// <param name="y3">A number.</param>
/// <remarks>See http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=geometry2#circle</remarks>
public Circle2D(decimal x1, decimal y1, decimal x2, decimal y2, decimal x3, decimal y3)
{
_radius = 0m; // to satisfy compiler
_center = default(Point2D); // to satisfy compiler
if (Point2D.PointsAreColinear(x1, y1, x2, y2, x3, y3))
{
throw new Exception("Can't create circle! Points are colinear.");
}
LineSeg2D l1 = new LineSeg2D(x1, y1, x2, y2).PerpendicularBisector();
LineSeg2D l2 = new LineSeg2D(x2, y2, x3, y3).PerpendicularBisector();
Point2D? intersect = l1.GetIntersect(l2, true);
if (intersect == null)
{
// Should never get null here because lines won't be parallel so
// long as they aren't colinear which we checked for above.
throw new Exception("Something bad happened. Should have found an intersect so long as points are not colinear.");
}
Center = intersect.Value;
Radius = intersect.Value.DistanceTo(x1, y1);
}
public decimal DistanceTo(LineSeg2D l, bool treatLineSegmentAsLine = false)
{
return(l.DistanceTo(this, treatLineSegmentAsLine));
}
/// <summary>
/// Gets the points of intersection between this circle and the line which
/// contains the given line segment.
/// </summary>
/// <param name="l">Line segment used to determine line equation.</param>
public Point2D[] GetIntersectFast(ref LineSeg2D l)
{
decimal[] x = new decimal[2];
decimal m = 0m;
decimal b = 0m;
decimal aCoeff = 0m;
decimal bCoeff = 0m;
decimal cCoeff = 0m;
decimal lineX = 0m;
decimal t = 0m;
decimal p = 0m;
decimal q = 0m;
Point2D[] pts = null;
if (!l.IsVertical)
{
// Circle (x - h) ^ 2 + (y - k) ^ 2 = r ^ 2
// Center: (h, k)
// Radius: r
// Line y = m * x + b
// (x - h) ^ 2 + (m * x + b - k) ^ 2 = r ^ 2
// (x - h) * (x - h) + (m * x + b - k) * (m * x + b - k) = r^2
// (x^2 - 2 * h * x + h^2) + (m^2 * x^2 + 2 * (b - k) * m * x + (b - k)^2 = r^2
// (m^2 + 1) * x^2 + (2 * (b - k) * m - 2 * h) * x + (h^2 + (b - k)^2 - r^2) = 0
m = l.Slope;
b = l.YIntersect;
aCoeff = DecimalEx.Pow(m, 2) + 1;
bCoeff = 2 * (b - _center.Y) * m - 2 * _center.X;
cCoeff = DecimalEx.Pow(_center.X, 2) + DecimalEx.Pow(b - _center.Y, 2) - DecimalEx.Pow(_radius, 2);
x = DecimalEx.SolveQuadratic(aCoeff, bCoeff, cCoeff);
if (x.Length == 0)
{
return new Point2D[] { }
}
;
pts = new Point2D[x.Length];
for (var i = 0; i <= x.Length - 1; i++)
{
pts[i] = new Point2D(x[i], m * x[i] + b);
}
}
else
{
// Circle (x - h) ^ 2 + (y - k) ^ 2 = r ^ 2
// Center: (h, k)
// Radius: r
// Line x = lineX
// Got the following from
// http://www.sonoma.edu/users/w/wilsonst/Papers/Geometry/circles/T1--2/T1-3-2.html
// http://www.sonoma.edu/users/w/wilsonst/Papers/Geometry/circles/default.html
lineX = l.Pt1.X;
t = _radius * _radius - (lineX - _center.X) * (lineX - _center.X);
if (t < 0)
{
return(new Point2D[] { });
}
else
{
p = _center.Y + DecimalEx.Sqrt(t);
q = _center.Y - DecimalEx.Sqrt(t);
pts = new Point2D[1];
pts[0] = new Point2D(lineX, p);
// NOTE that P=Q when t=0
if (p != q)
{
Array.Resize(ref pts, 2);
pts[1] = new Point2D(lineX, q);
}
}
}
return(pts);
}
/// <summary>
/// Gets the points of intersection between this circle and the line which
/// contains the given line segment.
/// </summary>
/// <param name="l">Line segment used to determine line equation.</param>
/// <param name="decimals">Determines rounding that should be performed when determining
/// if line intersection happens on zero, one, or two points. Default is -1 for
/// no rounding.</param>
public Point2D[] GetIntersect(LineSeg2D l, int decimals = -1)
{
LineSeg2D centToLine = default(LineSeg2D);
decimal centToLineLength = 0m;
decimal centToLineLengthForComp = 0m;
decimal radiusForComp = 0m;
if (l.IsHorizontal)
{
// The center to any horizontal line is a vertical line through
// the center of the circle.
centToLine = new LineSeg2D(this.Center, this.Center + new Vector2D(0, 1));
}
else
{
centToLine = LineSeg2D.FromPointSlope(this.Center, l.PerpendicularSlope);
}
centToLine.Pt2 = l.GetIntersect(centToLine, true).Value;
centToLineLength = centToLine.Length;
// Get numbers for comparison, rounding if necessary
centToLineLengthForComp = centToLineLength;
radiusForComp = _radius;
if (decimals >= 0)
{
centToLineLengthForComp = centToLineLengthForComp.RoundFromZero(decimals);
radiusForComp = radiusForComp.RoundFromZero(decimals);
}
// See if line is outside of circle
if (centToLineLengthForComp > radiusForComp)
{
return(new Point2D[] { });
}
// See if line is tangent to circle
if (centToLineLengthForComp == radiusForComp)
{
return(new Point2D[] { centToLine.Pt2 });
}
// Line must intersect in two places
Vector2D vCentToChord = default(Vector2D);
decimal halfChord = 0m;
// Get a vector from the center to the intersecting chord
vCentToChord = centToLine.GetVectorP1toP2();
if (vCentToChord.Magnitude == 0)
{
Vector2D offsetVector = default(Vector2D);
// Line goes through circle center
if (l.IsVertical)
{
// Slope undefined so just go up the length of the radius
offsetVector = new Vector2D(0, _radius);
}
else
{
offsetVector = new Vector2D(_radius * DecimalEx.Cos(DecimalEx.ATan(l.Slope)), _radius * DecimalEx.Sin(DecimalEx.ATan(l.Slope)));
}
return(new Point2D[] {
this.Center + offsetVector,
this.Center - offsetVector
});
}
else
{
Vector2D vChord = default(Vector2D);
// Get a vector along the chord
vChord = vCentToChord.GetPerpendicular();
// Determine the length of half the chord
halfChord = RightTriangle.GetSideFromSideHyp(centToLineLength, _radius);
// Set the magnitude of the vector along the chord
// to be half the chord length
vChord.Magnitude = halfChord;
// The two intersecting points are points translated
// from the center of the circle to the chord (+vCentToChord)
// and then translated to the ends of the chord (+-vChord)
return(new Point2D[] {
this.Center + vCentToChord + vChord,
this.Center + vCentToChord - vChord
});
}
}
/// <summary>
/// Returns circles that are tangent to another circle and line.
/// </summary>
/// <param name="line">Line that returned circles should be tangent to.</param>
/// <param name="circle">Circle that returned circles should be tangent to.</param>
/// <param name="radius">Radius of new circles.</param>
/// <returns>External tangent circles are listed first.</returns>
public static Circle2D[] FromTangentTangentRadius(LineSeg2D line, Circle2D circle, decimal radius)
{
return(FromTangentTangentRadius(circle, line, radius));
}
/// <summary>
/// Gets the points of intersection between this circle and another circle.
/// </summary>
/// <param name="other">The other circle.</param>
/// <param name="decimals">Determines rounding that should be performed when determining
/// if line intersection happens on zero, one, or two points. Default is -1 for
/// no rounding.</param>
/// <param name="impreciseResult">Which value to return for an imprecise result,
/// i.e. tangency determined by rounding. If positive, returns the point on the larger circle. If
/// negative, returns the point on the smaller circle. If 0, returns the midpoint between these
/// two points.</param>
public Point2D[] GetIntersect(Circle2D other, int decimals = -1, int impreciseResult = 0)
{
Vector2D meToOtherVector = default(Vector2D);
meToOtherVector = this.Center.GetVectorTo(other.Center);
if (decimals >= 0)
{
// The only intersection decision that can be made to a given precision is whether
// the two circles are tangent to each other, either internally or externally.
if (this.IsTangentTo(other, decimals))
{
// If the smaller circle is inside the other, then the smaller is
// considered internally tangent to the larger.
if ((this.Radius < other.Radius && this.IsInside(other, decimals)) || (other.Radius < this.Radius && other.IsInside(this, decimals)))
{
// Internal tangent
Point2D pointOnLargeCircle = default(Point2D);
Point2D pointOnSmallCircle = default(Point2D);
// Vectors to the two tangent points are both pointing in the same
// direction--from the center of the larger circle towards the
// center of the smaller circle. Their magnitude is the same as the
// radius of the circle whose center they start from.
if (this.Radius > other.Radius)
{
// Go from center of larger circle, Me, to smaller circle, other.
pointOnLargeCircle = this.Center + new Vector2D(meToOtherVector, this.Radius);
pointOnSmallCircle = other.Center + new Vector2D(meToOtherVector, other.Radius);
}
else
{
// Go from center of larger circle, other, to smaller circle, Me.
pointOnLargeCircle = other.Center - new Vector2D(meToOtherVector, other.Radius);
pointOnSmallCircle = this.Center - new Vector2D(meToOtherVector, this.Radius);
}
if (impreciseResult > 0)
{
// Point on larger
return(new Point2D[] { pointOnLargeCircle });
}
else if (impreciseResult < 0)
{
// Point on smaller
return(new Point2D[] { pointOnSmallCircle });
}
else
{
// Split difference
LineSeg2D l = new LineSeg2D(pointOnLargeCircle, pointOnSmallCircle);
return(new Point2D[] { l.MidPoint });
}
}
else
{
// External tangent
Point2D pointOnMe = default(Point2D);
Point2D pointOnOther = default(Point2D);
// Vectors to the two tangent points are simply pointing at the other
// circle's center with a magnitude of the radius of the circle whose
// center it originates in.
pointOnMe = this.Center + new Vector2D(meToOtherVector, this.Radius);
pointOnOther = other.Center - new Vector2D(meToOtherVector, other.Radius);
if (impreciseResult > 0)
{
// Point on larger
return(new Point2D[] { (this.Radius > other.Radius ? pointOnMe : pointOnOther) });
}
else if (impreciseResult < 0)
{
// Point on smaller
return(new Point2D[] { (this.Radius < other.Radius ? pointOnMe : pointOnOther) });
}
else
{
if (pointOnMe == pointOnOther)
{
return(new Point2D[] { pointOnMe });
}
else
{
// Split difference
return(new Point2D[] { new LineSeg2D(pointOnMe, pointOnOther).MidPoint });
}
}
}
}
}
// Detect situations where two points touch
decimal a = 0m;
decimal h = 0m;
decimal r2mina2 = 0m;
Vector2D vToIntersectMidPt = default(Vector2D);
Vector2D vToIntersect1 = default(Vector2D);
// The following two equations are from:
// http://paulbourke.net/geometry/2circle/
a = (DecimalEx.Pow(meToOtherVector.Magnitude, 2) - DecimalEx.Pow(other.Radius, 2) + DecimalEx.Pow(this.Radius, 2)) / (2 * meToOtherVector.Magnitude);
r2mina2 = DecimalEx.Pow(this.Radius, 2) - DecimalEx.Pow(a, 2);
// No intersection points -- one circle is inside or outside the other
if (r2mina2 < 0)
{
return new Point2D[] { }
}
;
vToIntersectMidPt = new Vector2D(this.Center, other.Center);
vToIntersectMidPt.Magnitude = a;
if (r2mina2 == 0)
{
// Only one intersection point
return(new Point2D[] { this.Center + vToIntersectMidPt });
}
h = DecimalEx.Sqrt(r2mina2);
vToIntersect1 = vToIntersectMidPt.GetPerpendicular();
vToIntersect1.Magnitude = h;
return(new Point2D[] {
this.Center + vToIntersectMidPt + vToIntersect1,
this.Center + vToIntersectMidPt - vToIntersect1
});
}
