How to check if line segment intersects a rectangle?

Question

If you have 2 points, (x1, y1) and (x2, y2), which represent two opposite corners of a rectangle, and 2 other points, (x3,y3) and (x4,y4), which represent 2 endpoints of a

Answer 1

To understand how to derive the formula for testing whether a line-segment intersects a rectangle, it's important to remember the properties of the vector dot product.

Represent the line-segment as a unit-vector and a distance between the line-segment's start-point and the origin. Here's some C# code to calculate that from the PointF variables a_ptStart and a_ptEnd, using a Vector:

Vector vecLine = new Vector(a_ptEnd.X - a_ptStart.X, a_ptEnd.Y - a_ptStart.Y);
double dLengthLine = vecLine.Length;
vecLine /= dLengthLine;
double dDistLine = Vector.Multiply(vecLine, new Vector(a_ptStart.X, a_ptStart.Y));

You'll also need to calculate the perpendicular vector, and its distance from the origin, for the line segment. Rotating a unit vector by 90° is easy.

Vector vecPerpLine = new Vector(-vecLine.Y, vecLine.X);
double dDistPerpLine = Vector.Multiply(vecPerpLine, new Vector(a_ptStart.X, a_ptStart.Y));

Assuming the four corners of the rectangle are in Vector variables called vecRect1, vecRect2, vecRect3, and vecRect4, calculate the distance between the line-segment and all four corners of the target's bounding-rectangle:

double dPerpLineDist1 = Vector.Multiply(vecPerpLine, vecRect1) - dDistPerpLine;
double dPerpLineDist2 = Vector.Multiply(vecPerpLine, vecRect2) - dDistPerpLine;
double dPerpLineDist3 = Vector.Multiply(vecPerpLine, vecRect3) - dDistPerpLine;
double dPerpLineDist4 = Vector.Multiply(vecPerpLine, vecRect4) - dDistPerpLine;
double dMinPerpLineDist = Math.Min(dPerpLineDist1, Math.Min(dPerpLineDist2,
    Math.Min(dPerpLineDist3, dPerpLineDist4)));
double dMaxPerpLineDist = Math.Max(dPerpLineDist1, Math.Max(dPerpLineDist2,
    Math.Max(dPerpLineDist3, dPerpLineDist4)));

If all distances are positive, or all distances are negative, then the rectangle is on one side of the line or the other, so there's no intersection. (Zero-extent rectangles are considered not to intersect with any line-segment.)

if (dMinPerpLineDist <= 0.0 && dMaxPerpLineDist <= 0.0
        || dMinPerpLineDist >= 0.0 && dMaxPerpLineDist >= 0.0)
    /* no intersection */;

Next, project all four corners of the target's bounding rectangle onto the line-segment. This gives us the distance between the line's origin and the projection of the rectangle-corner on that line.

double dDistLine1 = Vector.Multiply(vecLine, vecRect1) - dDistLine;
double dDistLine2 = Vector.Multiply(vecLine, vecRect2) - dDistLine;
double dDistLine3 = Vector.Multiply(vecLine, vecRect3) - dDistLine;
double dDistLine4 = Vector.Multiply(vecLine, vecRect4) - dDistLine;
double dMinLineDist = Math.Min(dDistLine1, Math.Min(dDistLine2,
    Math.Min(dDistLine3, dDistLine4)));
double dMaxLineDist = Math.Max(dDistLine1, Math.Max(dDistLine2,
    Math.Max(dDistLine3, dDistLine4)));

If the rectangle's points don't fall within the line segment's extent, then there's no intersection.

if (dMaxLineDist <= 0.0 || dMinLineDist >= dLengthLine)
    /* no intersection */;

I believe that's sufficient.