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A necessary condition for two lines to intersect is that they are in the same plane—that is, are not skew lines. Satisfaction of this condition is equivalent to the tetrahedron with vertices at two of the points on one line and two of the points on the other line being degenerate in the sense of having zero volume.
The Möller–Trumbore ray-triangle intersection algorithm, named after its inventors Tomas Möller and Ben Trumbore, is a fast method for calculating the intersection of a ray and a triangle in three dimensions without needing precomputation of the plane equation of the plane containing the triangle. [1]
For the example in the question, the above code gives xi = 3 and yi = 3, as expected. If you want to check that the intersection point lies between the endpoints of the lines (i.e. they are finite line segments), you just have to check that the values ua and ub both lie between 0 and 1: isInSegment = all(([ua ub] >= 0) & ([ua ub] <= 1));
To check if a given point lies inside or outside a polygon: Draw a horizontal line to the right of each point and extend it to infinity. Count the number of times the line intersects with polygon edges. A point is inside the polygon if either count of intersections is odd or point lies on an edge of the polygon.
Green line has two intersections. Yellow line lies tangent to the cylinder, so has infinitely many points of intersection. Line-cylinder intersection is the calculation of any points of intersection, given an analytic geometry description of a line and a cylinder in 3d space. An arbitrary line and cylinder may have no intersection at all.
The Shamos–Hoey algorithm [1] applies this principle to solve the line segment intersection detection problem, as stated above, of determining whether or not a set of line segments has an intersection; the Bentley–Ottmann algorithm works by the same principle to list all intersections in logarithmic time per intersection.
The three possible line-sphere intersections: 1. No intersection. 2. Point intersection. 3. Two point intersection. In analytic geometry, a line and a sphere can intersect in three ways: No intersection at all; Intersection in exactly one point; Intersection in two points.
The ideal case of epipolar geometry. A 3D point x is projected onto two camera images through lines (green) which intersect with each camera's focal point, O 1 and O 2. The resulting image points are y 1 and y 2. The green lines intersect at x. In practice, the image points y 1 and y 2 cannot be measured with arbitrary accuracy.