/// <summary>
/// Gets hypotenuse from a known side and the angle adjacent to that side.
/// </summary>
/// <param name="side">Length of the known side.</param>
/// <param name="angleAdjacentToSide">The angle adjacent to the known side in degrees.</param>
public static decimal GetHypFromSideAdjAngle(decimal side, decimal angleAdjacentToSide)
{
// cos(a) = s / h
// h * cos(a) = s
// h = s / cos(a)
return(side / DecimalEx.Cos(DecimalEx.ToRad(angleAdjacentToSide)));
}
/// <summary>
/// Gets the Y values for which the equation of the circle yields the given X.
/// </summary>
/// <param name="x">The X value.</param>
public decimal[] SolveForY(ref decimal x)
{
// Alternative calculations. May be faster/more accurate, but need to be tested.
//Dim absDiff = Math.Abs(x - _center.X)
//Select Case absDiff
// Case Is > _radius
// Return New Decimal() {}
// Case _radius
// Return New Decimal() {_center.Y}
// Case 0
// Return New Decimal() {_center.Y + _radius, _center.Y - _radius}
// Case Else
// Dim height = RightTriangle.GetSideFromSideHyp(absDiff, _radius)
// Return New Decimal() {_center.Y + height, _center.Y - height}
//End Select
decimal aCoeff = 0m;
decimal bCoeff = 0m;
decimal cCoeff = 0m;
// Solve for y
// (x - h)^2 + (y - k)^2 = r^2
// simplifies to
// y^2 - 2 * k * y + k^2 + (x - h)^2 - r^2 = 0
aCoeff = 1;
bCoeff = -2 * _center.Y;
cCoeff = _center.Y * _center.Y + (x - _center.X) * (x - _center.X) - _radius * _radius;
return(DecimalEx.SolveQuadratic(aCoeff, bCoeff, cCoeff));
}
public decimal DistanceTo(decimal x, decimal y)
{
var xDist = this.X - x;
var yDist = this.Y - y;
return(DecimalEx.Sqrt(xDist * xDist + yDist * yDist));
}
/// <summary>
/// Gets the angle in degrees for a ray that extends from the center
/// of the circle through the given point. An exception will occur,
/// however, if the point specified is the same point as the center
/// of the circle.
/// </summary>
/// <param name="pt">The point.</param>
public decimal AngleThroughPoint(Point2D pt)
{
decimal a = 0m;
// Shift point so it's relative to the origin.
pt += new Vector2D(-Center.X, -Center.Y);
// Do check here after we've translated the point
// to account for small precision rounding.
if (pt == Point2D.Origin)
{
throw new Exception("No angle through given point since point is at center of circle!");
}
// Special case for 0 and 180 where we would get
// a divide by zero below.
if (pt.Y == 0)
{
if (pt.X > 0)
{
return(0);
}
else
{
return(180);
}
}
// Special case for 90 and 270 where we can't
// determine quadrant below.
if (pt.X == 0)
{
if (pt.Y > 0)
{
return(90);
}
else
{
return(270);
}
}
a = DecimalEx.ToDeg(DecimalEx.ATan(pt.X / pt.Y));
switch (pt.Quadrant)
{
case 1:
case 2:
a = 90 - a;
break;
case 3:
case 4:
a = 270 - a;
break;
}
return(a);
}
/// <summary>
/// Gets the angle in degrees between this vector and another.
/// </summary>
/// <param name="other">The other vector to get the angle to.</param>
/// <remarks>See http://en.wikipedia.org/wiki/Dot_product</remarks>
public decimal AngleTo(Vector2D other)
{
if (this.IsNull || other.IsNull)
{
throw new Exception("Can't find angle when one or both vectors are null vectors!");
}
// Cos(theta) = DotProduct(v1,v2) / (length(v1) * length(v2))
// aka theta = acos(v.normalize.dot(other.normalize)), however, the equation
// used gives us better precision
return(DecimalEx.ToDeg(DecimalEx.ACos(this.Dot(other) / (this.Magnitude * other.Magnitude))));
}
/// <summary>
/// Gets a side from a known side and the hypotenuse.
/// </summary>
/// <param name="side">Length of the known side.</param>
/// <param name="hypotenuse">Length of the hypotenuse.</param>
public static decimal GetSideFromSideHyp(decimal side, decimal hypotenuse)
{
// a^2 + b^2 = c^2
// a^2 = c^2 - b^2
// a = sqrt(c^2 - b^2)
decimal sideSquared = hypotenuse * hypotenuse - side * side;
if (sideSquared < 0)
{
throw new Exception("Error finding side from other side and hypotenuse! Side and hypotenuse swapped or invalid right triangle.");
}
return(DecimalEx.Sqrt(sideSquared));
}
/// <summary>
/// Determines whether or not the given angle lies on the arc.
/// </summary>
/// <param name="angle">The angle in question.</param>
public bool IsAngleOnArc(decimal angle)
{
angle = DecimalEx.NormalizeAngleDeg(angle);
if (_startAngle <= _endAngle)
{
return((_startAngle <= angle) && (angle <= _endAngle));
}
else
{
return((angle > _startAngle) || (angle < _endAngle));
}
}
/// <summary>
/// Gets the X values for which the equation of the circle yields the given Y.
/// </summary>
/// <param name="y">The Y value.</param>
public decimal[] SolveForX(decimal y)
{
decimal aCoeff = 0m;
decimal bCoeff = 0m;
decimal cCoeff = 0m;
// Solve for x
// (x - h)^2 + (y - k)^2 = r^2
// simplifies to
// x^2 - 2 * h * x + h^2 + (y - k)^2 - r^2 = 0
aCoeff = 1;
bCoeff = -2 * _center.X;
cCoeff = _center.X * _center.X + (y - _center.Y) * (y - _center.Y) - _radius * _radius;
return(DecimalEx.SolveQuadratic(aCoeff, bCoeff, cCoeff));
}
/// <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);
}
/// <summary>
/// Gets a side from the opposite angle and the length of the hypotenuse.
/// </summary>
/// <param name="oppositeAngle">Angle opposite to the side to calculate in degrees.</param>
/// <param name="hypotenuse">Length of the hypotenuse.</param>
public static decimal GetSideFromOppAngleHyp(decimal oppositeAngle, decimal hypotenuse)
{
// sin(oppositeAngle) = x / hypotenuse
// x = hypotenuse * sin(oppositeAngle)
return(hypotenuse * Convert.ToDecimal(DecimalEx.Sin(DecimalEx.ToRad(oppositeAngle))));
}
/// <summary>
/// Gets angle in degrees from the opposite side and the hypotenuse.
/// </summary>
/// <param name="oppositeSide">Length of the known side opposite the angle.</param>
/// <param name="hypotenuse">Length of the hypotenuse.</param>
public static decimal GetAngleFromOppSideHyp(decimal oppositeSide, decimal hypotenuse)
{
// sin(a) = oppositeSide / hypotenuse
// a = asin(oppositeSide / hypotenuse)
return(DecimalEx.ToDeg(DecimalEx.ASin(oppositeSide / hypotenuse)));
}
/// <summary>
/// Gets the angle of the vector in degrees from the X+ axis. Will return result in the range
/// -180 to +180. Will return 0 for null vector.
/// </summary>
public decimal Angle()
{
return(DecimalEx.ToDeg(DecimalEx.ATan2(Y, X)));
}
/// <summary>
/// Gets angle in degrees from the adjacent side and the hypotenuse.
/// </summary>
/// <param name="adjacentSide">Length of the known side adjacent to the angle.</param>
/// <param name="hypotenuse">Length of the hypotenuse.</param>
public static decimal GetAngleFromAdjSideHyp(decimal adjacentSide, decimal hypotenuse)
{
// cos(a) = adjacentSide / hypotenuse
// a = acos(adjacentSide / hypotenuse)
return(DecimalEx.ToDeg(DecimalEx.ACos(adjacentSide / hypotenuse)));
}
/// <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>
/// Gets the hypotenuse from the two other sides.
/// </summary>
/// <param name="sideA">Length of one side.</param>
/// <param name="sideB">Length of other side.</param>
public static decimal GetHypFromSides(decimal sideA, decimal sideB)
{
// a^2 + b^2 = c^2
// c = sqrt(a^2 + b^2)
return(DecimalEx.Sqrt(sideA * sideA + sideB * sideB));
}
/// <summary>
/// Computes the radius of a circle given a chord length and the chord's sagitta.
/// </summary>
/// <param name="chordLength">Length of the chord.</param>
/// <param name="sagitta">Sagitta.</param>
/// <remarks> http://www.physicsforums.com/showpost.php?p=2069646&postcount=2 </remarks>
public static decimal GetRadiusFromChordLengthSagitta(decimal chordLength, decimal sagitta)
{
return((DecimalEx.Pow(sagitta, 2) + 0.25m * DecimalEx.Pow(chordLength, 2)) / (2 * sagitta));
}
/// <summary>
/// Gets hypotenuse from a known side and the angle opposite to it.
/// </summary>
/// <param name="side">Length of the known side.</param>
/// <param name="angleOppositeToSide">Angle opposite to the known side in degrees.</param>
public static decimal GetHypFromSideOppAngle(decimal side, decimal angleOppositeToSide)
{
// sin(a) = s / h
// h = s / sin(a)
return(side / DecimalEx.Sin(DecimalEx.ToRad(angleOppositeToSide)));
}
/// <summary>
/// Gets the coordinates of the point on the circle at the given angle.
/// </summary>
/// <param name="degrees">The angle in degrees.</param>
public Point2D PointAt(decimal degrees)
{
decimal rad = DecimalEx.ToRad(degrees);
return(new Point2D(_center.X + _radius * DecimalEx.Cos(rad), _center.Y + _radius * DecimalEx.Sin(rad)));
}
/// <summary>
/// Gets angle in degrees from the two known sides.
/// </summary>
/// <param name="oppositeSide">Length of the known side opposite the angle.</param>
/// <param name="adjacentSide">Length of the known side adjacent to the angle.</param>
public static decimal GetAngleFromSides(decimal oppositeSide, decimal adjacentSide)
{
// tan(a) = opposideSide / adjacentSide
// a = atan(opposideSide / adjacentSide)
return(DecimalEx.ToDeg(DecimalEx.ATan(oppositeSide / adjacentSide)));
}
/// <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
});
}
/// <summary>
/// Gets a side from the adjacent angle and the length of the hypotenuse.
/// </summary>
/// <param name="adjacentAngle">Angle adjacent to the side to calculate in degrees.</param>
/// <param name="hypotenuse">Length of the hypotenuse.</param>
public static decimal GetSideFromAdjAngleHyp(decimal adjacentAngle, decimal hypotenuse)
{
// cos(adjacentAngle) = x / hypotenuse
// x = hypotenuse * cos(adjacentAngle)
return(hypotenuse * Convert.ToDecimal(DecimalEx.Cos(DecimalEx.ToRad(adjacentAngle))));
}
/// <summary>
/// Gets a side from the opposite angle and the length of the opposite side.
/// </summary>
/// <param name="oppositeAngle">Angle opposite the side to calculate in degrees.</param>
/// <param name="oppositeSide">Length of the opposite side.</param>
public static decimal GetSideFromOppAngleOppSide(decimal oppositeAngle, decimal oppositeSide)
{
// tan(oppositeAngle) = x / oppositeSide
// x = oppositeSide * tan(oppositeAngle)
return(oppositeSide * Convert.ToDecimal(DecimalEx.Tan(DecimalEx.ToRad(oppositeAngle))));
}
/// <summary>
/// Gets a side from the adjacent angle and the length of the opposite side.
/// </summary>
/// <param name="adjacentAngle">Angle adjacent to the side to calculate in degrees.</param>
/// <param name="oppositeSide">Length of the opposite side.</param>
public static decimal GetSideFromAdjAngleOppSide(decimal adjacentAngle, decimal oppositeSide)
{
// tan(adjacentAngle) = oppositeSide / x
// x = oppositeSide / tan(adjacentAngle)
return(oppositeSide / Convert.ToDecimal(DecimalEx.Tan(DecimalEx.ToRad(adjacentAngle))));
}
