/// <summary>
/// Determines the minimum translation vector to be applied to the circle to
/// prevent overlap with the rectangle, when they are at their given positions.
/// </summary>
/// <param name="circle">The circle</param>
/// <param name="rect">The rectangle</param>
/// <param name="pos1">The top-left of the circles bounding box</param>
/// <param name="pos2">The rectangles origin</param>
/// <returns>MTV for circle at pos1 to prevent overlap with rect at pos2</returns>
public static Tuple <Vector2, float> IntersectMTV(Circle2 circle, Rect2 rect, Vector2 pos1, Vector2 pos2)
{
// Same as polygon rect, just converted to rects points
HashSet <Vector2> checkedAxis = new HashSet <Vector2>();
Vector2 bestAxis = Vector2.Zero;
float shortestOverlap = float.MaxValue;
Func <Vector2, bool> checkAxis = (axis) =>
{
var standard = Math2.MakeStandardNormal(axis);
if (!checkedAxis.Contains(standard))
{
checkedAxis.Add(standard);
var circleProj = Circle2.ProjectAlongAxis(circle, pos1, axis);
var rectProj = Rect2.ProjectAlongAxis(rect, pos2, axis);
var mtv = AxisAlignedLine2.IntersectMTV(circleProj, rectProj);
if (!mtv.HasValue)
{
return(false);
}
if (Math.Abs(mtv.Value) < Math.Abs(shortestOverlap))
{
bestAxis = axis;
shortestOverlap = mtv.Value;
}
}
return(true);
};
var circleCenter = new Vector2(pos1.X + circle.Radius, pos1.Y + circle.Radius);
int last = 4;
var lastVec = rect.UpperRight + pos2;
for (int curr = 0; curr < 4; curr++)
{
Vector2 currVec = Vector2.Zero;
switch (curr)
{
case 0:
currVec = rect.Min + pos2;
break;
case 1:
currVec = rect.LowerLeft + pos2;
break;
case 2:
currVec = rect.Max + pos2;
break;
case 3:
currVec = rect.UpperRight + pos2;
break;
}
// Test along circle center -> vector
if (!checkAxis(Vector2.Normalize(currVec - circleCenter)))
{
return(null);
}
// Test along line normal
if (!checkAxis(Vector2.Normalize(Math2.Perpendicular(currVec - lastVec))))
{
return(null);
}
last = curr;
lastVec = currVec;
}
return(Tuple.Create(bestAxis, shortestOverlap));
}
/// <summary>
/// Determines if the circle whose bounding boxs top left is at the first postion intersects the line
/// at the second position who is rotated the specified amount about the specified point.
/// </summary>
/// <param name="circle">The circle</param>
/// <param name="line">The line</param>
/// <param name="pos1">The top-left of the circles bounding box</param>
/// <param name="pos2">The origin of the line</param>
/// <param name="rot2">What rotation the line is under</param>
/// <param name="about2">What the line is rotated about</param>
/// <param name="strict">If overlap is required for intersection</param>
/// <returns>If the circle at pos1 intersects the line at pos2 rotated rot2 about about2</returns>
protected static bool CircleIntersectsLine(Circle2 circle, Line2 line, Vector2 pos1, Vector2 pos2, Rotation2 rot2, Vector2 about2, bool strict)
{
// Make more math friendly
var actualLine = new Line2(Math2.Rotate(line.Start, about2, rot2) + pos2, Math2.Rotate(line.End, about2, rot2) + pos2);
var circleCenter = new Vector2(pos1.X + circle.Radius, pos1.Y + circle.Radius);
// Check weird situations
if (actualLine.Horizontal)
{
return(CircleIntersectsHorizontalLine(circle, actualLine, circleCenter, strict));
}
if (actualLine.Vertical)
{
return(CircleIntersectsVerticalLine(circle, actualLine, circleCenter, strict));
}
// Goal:
// 1. Find closest distance, closestDistance, on the line to the circle (assuming the line was infinite)
// 1a Determine if closestPoint is intersects the circle according to strict
// - If it does not, we've shown there is no intersection.
// 2. Find closest point, closestPoint, on the line to the circle (assuming the line was infinite)
// 3. Determine if closestPoint is on the line (including edges)
// - If it is, we've shown there is intersection.
// 4. Determine which edge, edgeClosest, is closest to closestPoint
// 5. Determine if edgeClosest intersects the circle according to strict
// - If it does, we've shown there is intersection
// - If it does not, we've shown there is no intersection
// Step 1
// We're trying to find closestDistance
// Recall that the shortest line from a line to a point will be normal to the line
// Thus, the shortest distance from a line to a point can be found by projecting
// the line onto it's own normal vector and projecting the point onto the lines
// normal vector; the distance between those points is the shortest distance from
// the two points.
// The projection of a line onto its normal will be a single point, and will be same
// for any point on that line. So we pick a point that's convienent (the start or end).
var lineProjectedOntoItsNormal = Vector2.Dot(actualLine.Start, actualLine.Normal);
var centerOfCircleProjectedOntoNormalOfLine = Vector2.Dot(circleCenter, actualLine.Normal);
var closestDistance = Math.Abs(centerOfCircleProjectedOntoNormalOfLine - lineProjectedOntoItsNormal);
// Step 1a
if (strict)
{
if (closestDistance >= circle.Radius)
{
return(false);
}
}
else
{
if (closestDistance > circle.Radius)
{
return(false);
}
}
// Step 2
// We're trying to find closestPoint
// We can just walk the vector from the center to the closest point, which we know is on
// the normal axis and the distance closestDistance. However it's helpful to get the signed
// version End - Start to walk.
var signedDistanceCircleCenterToLine = lineProjectedOntoItsNormal - centerOfCircleProjectedOntoNormalOfLine;
var closestPoint = circleCenter - actualLine.Normal * signedDistanceCircleCenterToLine;
// Step 3
// Determine if closestPoint is on the line (including edges)
// We're going to accomplish this by projecting the line onto it's own axis and the closestPoint onto the lines
// axis. Then we have a 1D comparison.
var lineStartProjectedOntoLineAxis = Vector2.Dot(actualLine.Start, actualLine.Axis);
var lineEndProjectedOntoLineAxis = Vector2.Dot(actualLine.End, actualLine.Axis);
var closestPointProjectedOntoLineAxis = Vector2.Dot(closestPoint, actualLine.Axis);
if (AxisAlignedLine2.Contains(lineStartProjectedOntoLineAxis, lineEndProjectedOntoLineAxis, closestPointProjectedOntoLineAxis, false, true))
{
return(true);
}
// Step 4
// We're trying to find edgeClosest.
//
// We're going to reuse those projections from step 3.
//
// (for each "point" in the next paragraph I mean "point projected on the lines axis" but that's wordy)
//
// We know that the start is closest iff EITHER the start is less than the end and the
// closest point is less than the start, OR the start is greater than the end and
// closest point is greater than the end.
var closestEdge = Vector2.Zero;
if (lineStartProjectedOntoLineAxis < lineEndProjectedOntoLineAxis)
{
closestEdge = (closestPointProjectedOntoLineAxis <= lineStartProjectedOntoLineAxis) ? actualLine.Start : actualLine.End;
}
else
{
closestEdge = (closestPointProjectedOntoLineAxis >= lineEndProjectedOntoLineAxis) ? actualLine.Start : actualLine.End;
}
// Step 5
// Circle->Point intersection for closestEdge
var distToCircleFromClosestEdgeSq = (circleCenter - closestEdge).LengthSquared();
if (strict)
{
return(distToCircleFromClosestEdgeSq < (circle.Radius * circle.Radius));
}
else
{
return(distToCircleFromClosestEdgeSq <= (circle.Radius * circle.Radius));
}
// If you had trouble following, see the horizontal and vertical cases which are the same process but the projections
// are simpler
}
/// <summary>
/// Determines if the specified circle, at the given position, intersects the specified polygon,
/// at the given position and rotation.
/// </summary>
/// <param name="circle">The circle</param>
/// <param name="poly">The polygon</param>
/// <param name="pos1">The top-left of the circles bounding box</param>
/// <param name="pos2">The origin of the polygon</param>
/// <param name="rot2">The rotation of the polygon</param>
/// <param name="strict">If overlap is required for intersection</param>
/// <returns>If circle at pos1 intersects poly at pos2 with rotation rot2</returns>
public static bool Intersects(Circle2 circle, Polygon2 poly, Vector2 pos1, Vector2 pos2, Rotation2 rot2, bool strict)
{
return(Intersects(poly, circle, pos2, pos1, rot2, strict));
}
/// <summary>
/// Determines if the specified rectangle and circle intersect at their given positions.
/// </summary>
/// <param name="rect">The rectangle</param>
/// <param name="circle">The circle</param>
/// <param name="pos1">The origin of the rectangle</param>
/// <param name="pos2">The top-left of the circles bounding box</param>
/// <param name="strict">If overlap is required for intersection</param>
/// <returns></returns>
public static bool Intersects(Rect2 rect, Circle2 circle, Vector2 pos1, Vector2 pos2, bool strict)
{
return(Intersects(circle, rect, pos2, pos1, strict));
}
/// <summary>
/// Determines if the specified polygon at the specified position and rotation
/// intersects the specified circle at it's respective position.
/// </summary>
/// <param name="poly">The polygon</param>
/// <param name="circle">The circle</param>
/// <param name="pos1">The origin for the polygon</param>
/// <param name="pos2">The top-left of the circles bounding box</param>
/// <param name="rot1">The rotation of the polygon</param>
/// <param name="strict">If overlap is required for intersection</param>
/// <returns>If poly at pos1 with rotation rot1 intersects the circle at pos2</returns>
public static bool Intersects(Polygon2 poly, Circle2 circle, Vector2 pos1, Vector2 pos2, Rotation2 rot1, bool strict)
{
// look at pictures of https://stackoverflow.com/questions/401847/circle-rectangle-collision-detection-intersection if you don't
// believe this is true
return(poly.Lines.Any((l) => CircleIntersectsLine(circle, l, pos2, pos1, rot1, poly.Center, strict)) || Polygon2.Contains(poly, pos1, rot1, new Vector2(pos2.X + circle.Radius, pos2.Y + circle.Radius), strict));
}
/// <summary>
/// Determines the minimum translation that must be applied the specified polygon (at the given position
/// and rotation) to prevent intersection with the circle (at its given rotation). If the two are not overlapping,
/// returns null.
///
/// Returns a tuple of the axis to move the polygon in (unit vector) and the distance to move the polygon.
/// </summary>
/// <param name="poly">The polygon</param>
/// <param name="circle">The circle</param>
/// <param name="pos1">The origin of the polygon</param>
/// <param name="pos2">The top-left of the circles bounding box</param>
/// <param name="rot1">The rotation of the polygon</param>
/// <returns></returns>
public static Tuple <Vector2, float> IntersectMTV(Polygon2 poly, Circle2 circle, Vector2 pos1, Vector2 pos2, Rotation2 rot1)
{
// We have two situations, either the circle is not strictly intersecting the polygon, or
// there exists at least one shortest line that you could push the polygon to prevent
// intersection with the circle.
// That line will either go from a vertix of the polygon to a point on the edge of the circle,
// or it will go from a point on a line of the polygon to the edge of the circle.
// If the line comes from a vertix of the polygon, the MTV will be along the line produced
// by going from the center of the circle to the vertix, and the distance can be found by
// projecting the cirle on that axis and the polygon on that axis and doing 1D overlap.
// If the line comes from a point on the edge of the polygon, the MTV will be along the
// normal of that line, and the distance can be found by projecting the circle on that axis
// and the polygon on that axis and doing 1D overlap.
// As with all SAT, if we find any axis that the circle and polygon do not overlap, we've
// proven they do not intersect.
// The worst case performance is related to 2x the number of vertices of the polygon, the same speed
// as for 2 polygons of equal number of vertices.
HashSet <Vector2> checkedAxis = new HashSet <Vector2>();
Vector2 bestAxis = Vector2.Zero;
float shortestOverlap = float.MaxValue;
Func <Vector2, bool> checkAxis = (axis) =>
{
var standard = Math2.MakeStandardNormal(axis);
if (!checkedAxis.Contains(standard))
{
checkedAxis.Add(standard);
var polyProj = Polygon2.ProjectAlongAxis(poly, pos1, rot1, axis);
var circleProj = Circle2.ProjectAlongAxis(circle, pos2, axis);
var mtv = AxisAlignedLine2.IntersectMTV(polyProj, circleProj);
if (!mtv.HasValue)
{
return(false);
}
if (Math.Abs(mtv.Value) < Math.Abs(shortestOverlap))
{
bestAxis = axis;
shortestOverlap = mtv.Value;
}
}
return(true);
};
var circleCenter = new Vector2(pos2.X + circle.Radius, pos2.Y + circle.Radius);
int last = poly.Vertices.Length - 1;
var lastVec = Math2.Rotate(poly.Vertices[last], poly.Center, rot1) + pos1;
for (int curr = 0; curr < poly.Vertices.Length; curr++)
{
var currVec = Math2.Rotate(poly.Vertices[curr], poly.Center, rot1) + pos1;
// Test along circle center -> vector
if (!checkAxis(Vector2.Normalize(currVec - circleCenter)))
{
return(null);
}
// Test along line normal
if (!checkAxis(Vector2.Normalize(Math2.Perpendicular(currVec - lastVec))))
{
return(null);
}
last = curr;
lastVec = currVec;
}
return(Tuple.Create(bestAxis, shortestOverlap));
}
/// <summary>
/// Projects the specified circle with the upper-left at the specified position onto
/// the specified axis.
/// </summary>
/// <param name="circle">The circle</param>
/// <param name="pos">The position of the circle</param>
/// <param name="axis">the axis to project along</param>
/// <returns>Projects circle at pos along axis</returns>
public static AxisAlignedLine2 ProjectAlongAxis(Circle2 circle, Vector2 pos, Vector2 axis)
{
return(ProjectAlongAxis(circle.Radius, pos, axis));
}
/// <summary>
/// Determines the shortest axis and overlap for which the first circle at the specified position
/// overlaps the second circle at the specified position. If the circles do not overlap, returns null.
/// </summary>
/// <param name="circle1">First circle</param>
/// <param name="circle2">Second circle</param>
/// <param name="pos1">Top-left of the first circles bounding box</param>
/// <param name="pos2">Top-left of the second circles bounding box</param>
/// <returns></returns>
public static Tuple <Vector2, float> IntersectMTV(Circle2 circle1, Circle2 circle2, Vector2 pos1, Vector2 pos2)
{
return(IntersectMTV(circle1.Radius, circle2.Radius, pos1, pos2));
}
/// <summary>
/// Determines if the first circle at the specified position intersects the second circle
/// at the specified position.
/// </summary>
/// <param name="circle1">First circle</param>
/// <param name="circle2">Second circle</param>
/// <param name="pos1">Top-left of the bounding box of the first circle</param>
/// <param name="pos2">Top-left of the bounding box of the second circle</param>
/// <param name="strict">If overlap is required for intersection</param>
/// <returns>If circle1 at pos1 intersects circle2 at pos2</returns>
public static bool Intersects(Circle2 circle1, Circle2 circle2, Vector2 pos1, Vector2 pos2, bool strict)
{
return(Intersects(circle1.Radius, circle2.Radius, pos1, pos2, strict));
}
/// <summary>
/// Determines the minimum translation vector to be applied to the circle to prevent
/// intersection with the specified polyogn, when they are at the given positions.
/// </summary>
/// <param name="circle">The circle</param>
/// <param name="poly">The polygon</param>
/// <param name="pos1">The top-left of the circles bounding box</param>
/// <param name="pos2">The origin of the polygon</param>
/// <returns></returns>
public static Tuple <Vector2, float> IntersectMTV(Circle2 circle, Polygon2 poly, Vector2 pos1, Vector2 pos2)
{
return(IntersectMTV(circle, poly, pos1, pos2, Rotation2.Zero));
}
/// <summary>
/// Determines if the circle and polygon intersect when at the given positions.
/// </summary>
/// <param name="circle">The circle</param>
/// <param name="poly">The polygon</param>
/// <param name="pos1">The top-left of the circles bounding box</param>
/// <param name="pos2">The origin of the polygon</param>
/// <param name="strict">If overlap is required for intersection</param>
/// <returns>If circle at pos1 intersects poly at pos2</returns>
public static bool Intersects(Circle2 circle, Polygon2 poly, Vector2 pos1, Vector2 pos2, bool strict)
{
return(Intersects(circle, poly, pos1, pos2, Rotation2.Zero, strict));
}