Given a line defined by two points A and B, an arbitrary point C, and a direction vector V, the goal is to find the point D where the ray from C in direction V intersects line AB. This is the "projected point" of C onto AB along V.

Approach — Parametric Line Equations

The parametric form of a line maps a scalar t to a point on the line, where t = 0 is the start point and t = 1 is the end point:

Line AB: x(t₁) = Xa + (Xb − Xa) · t₁
Line AB: y(t₁) = Ya + (Yb − Ya) · t₁
Projection ray from C along V: create point C′ = C + V, then define line CC′
Line CC′: x(t₂) = Xc + (Xc′ − Xc) · t₂
Line CC′: y(t₂) = Yc + (Yc′ − Yc) · t₂

At the intersection the x and y values match, so setting the two parametric equations equal and solving for t₁ gives the parameter of point D on line AB. Evaluating AB at t₁ yields the projected coordinates.

Implementation

csharp

public static double ParameterAtPoint(Line2D line2D, Point2D point2D)
{
    // Parametric coefficient A for the line direction
    double A = line2D.EndPoint.X - line2D.StartPoint.X;
    // t = (Xc - Xa) / A  →  position of point on the line (0 = start, 1 = end)
    double t = (point2D.X - line2D.StartPoint.X) / A;
    return t;
}

public static Point2D GetProjectedPointOnLine(
    Point2D toProject, Line2D line2D, Vector2D direction)
{
    // Create a second point along the projection direction from C
    Point2D otherPoint = new Point2D(
        toProject.X + direction.X,
        toProject.Y + direction.Y);

    // Parametric coefficients for line AB
    double Al = line2D.EndPoint.X - line2D.StartPoint.X;
    double Bl = line2D.EndPoint.Y - line2D.StartPoint.Y;

    // Parametric coefficients for the projection line CC'
    double Apl = otherPoint.X - toProject.X;
    double Bpl = otherPoint.Y - toProject.Y;

    // Solve the two parametric equations simultaneously.
    // t2 is the parameter on the projection line at the intersection:
    double t2 = ((toProject.Y - line2D.StartPoint.Y) * Al
               - (toProject.X - line2D.StartPoint.X) * Bl)
              / (Apl * Bl - Al * Bpl);

    // t1 is the parameter on line AB at the intersection — this is what we need:
    double t1 = ((line2D.StartPoint.Y - toProject.Y) * Apl
               - (line2D.StartPoint.X - toProject.X) * Bpl)
              / (Al * Bpl - Apl * Bl);

    // Evaluate line AB at t1 to get the projected point D
    return PointAtParameter(line2D, t1);
}

C# — Projecting a Point onto a Line by a Direction

Approach — Parametric Line Equations

Implementation