Line–line intersection

In geometry, an intersection occurs when two lines share common points. This concept is vital for modern technology. Computer graphics use intersections to render images. Motion planning uses them to map paths. Collision detection uses them to stop objects from overlapping.

In Euclidean geometry, which describes flat space, the result of an intersection can be three different things. The lines might coincide, meaning they are the same line and share infinitely many points. They might be parallel, meaning they have the same direction but share no points. Finally, they might intersect at a single point, known as a singleton set.

To understand these outcomes, we must look at how lines exist in space. In a two-dimensional plane, lines are either parallel, coincident, or intersecting. However, in three-dimensional space, a new possibility arises. Lines can be non-coplanar, meaning they do not lie on the same flat surface. These are called skew lines. Skew lines do not intersect and are not parallel.

A mathematical way to check if lines are coplanar is to look at a tetrahedron. If you pick two points on each line, the resulting tetrahedron must have zero volume to be coplanar.

Mathematicians use several methods to find these intersection points. In 2D space, a line is often defined by two distinct points. We can use determinants to calculate where these lines meet. If the lines are parallel or coincident, the denominator in these formulas becomes zero. When working with line segments instead of infinite lines, we use Bézier parameters. These parameters are real numbers that describe a position along a segment. By testing specific inequalities, computers can quickly determine if segments intersect without performing slow division.

Algebraic equations also provide a clear way to find intersections. We can represent non-vertical lines using their slopes and y-intercepts. The slope is the gradient, and the intercept is where the line crosses the vertical axis. At the exact point of intersection, the x and y coordinates for both lines are identical. By setting the equations equal to each other, we can solve for the unknown values. Another advanced method uses homogeneous coordinates. This approach represents 2D points as projections of 3D points. By using ordered triples, we can solve 2D intersection problems with great efficiency.

Geometry changes significantly when we move away from flat Euclidean space.

In spherical geometry, lines are represented by great circles. These are formed where a plane cuts through the center of a sphere. Because of this shape, every pair of lines must intersect. They actually meet at two antipodal points on opposite sides of the sphere.

In elliptic geometry, we treat these antipodal points as the same point. This means every pair of lines in elliptic geometry intersects at exactly one point. In these curved spaces, parallel lines do not exist.

Hyperbolic geometry offers a different perspective. In this space, there is constant negative Gaussian curvature. This curvature causes lines to behave in ways that defy Euclidean logic. Given a single line and a point not on that line, there are infinitely many lines through that point that will never intersect the first line. Lines in hyperbolic space can be intersecting, asymptotically parallel, or ultraparallel. Ultraparallel lines are disjoint but share a common perpendicular.

Projective geometry acts as a unifying framework for all these different behaviors. It extends the idea of intersection by adding ideal points, also called points at infinity. This allows mathematicians to describe parallel lines as meeting at a single projective point. In projective geometry, any two distinct lines intersect in exactly one point. This framework lets us study Euclidean, elliptic, and hyperbolic geometries under one system. It focuses on incidence relations rather than measuring distance or angles.