/// <summary>
/// Enumerates the times and places, if any, where a moving line crosses a point during a time step.
/// The line's endpoints move at a constant rate along their linear paths.
/// </summary>
public static IEnumerable<Sweep> WhenLineSweepsPoint(LineSegment pathOfLineStartPoint,
LineSegment pathOfLineEndPoint,
Point point)
{
var a = point - pathOfLineStartPoint.Start;
var b = -pathOfLineStartPoint.Delta;
var c = pathOfLineEndPoint.Start - pathOfLineStartPoint.Start;
var d = pathOfLineEndPoint.Delta - pathOfLineStartPoint.Delta;
return from t in QuadraticRoots(b.Cross(d), a.Cross(d) + b.Cross(c), a.Cross(c))
where t >= 0 && t <= 1
let start = pathOfLineStartPoint.LerpAcross(t)
let end = pathOfLineEndPoint.LerpAcross(t)
let s = point.LerpProjectOnto(new LineSegment(start, end))
where s >= 0 && s <= 1
orderby t
select new Sweep(timeProportion: t, acrossProportion: s);
}
///<summary>
///The proportion that, when lerped across the given line, results in the given point.
///If the point is not on the line segment, the result is the closest point on the extended line.
///</summary>
public static double LerpProjectOnto(this Point point, LineSegment line)
{
var b = point - line.Start;
var d = line.Delta;
return (b * d) / (d * d);
}