Intersection (geometry)

In geometry, an intersection is a point, line, or curve that two or more objects share. This can happen between lines, curves, planes, and surfaces. The simplest example is when two lines cross each other. They might meet at one point, which is sometimes called a vertex, or they might never meet if they are parallel.

There are many types of intersections, such as where a line meets a plane, a line meets a sphere, or where a line crosses a polyhedron. Finding these intersections can be done using tools from linear algebra, which involves solving systems of linear equations. Sometimes, more complex shapes like circles or spheres are involved, leading to quadratic or quartic equations that can be solved with special methods.

The idea of intersection in geometry has also been used in set theory, where it describes how groups of things share common elements. This concept helps us understand how different shapes and sets relate to each other.

On a plane

Further information: Plane (geometry) and Two-dimensional space

Two lines

Main article: Line–line intersection

When finding where two lines that aren't parallel cross, we can use simple steps to get the spot where they meet. If the lines are parallel, they never meet.

Two line segments

See also: Multiple line segment intersection and Line–line intersection § Given two points on each line segment

When we look at two lines that aren't parallel, made from points like (x1, y1) to (x2, y2) and (x3, y3) to (x4, y4), they might not actually cross if we only look at the pieces of the lines. We check this by seeing if the meeting point lands on both pieces.

A line and a circle

Further information: Line–sphere intersection

To find where a line meets a circle, we solve the line's equation together with the circle's. This can give us up to two points where they cross.

Two circles

Two conic sections

Solving where shapes like ellipses, hyperbolas, or parabolas meet another shape can be tricky. Sometimes we use special steps to find the answers.

Two smooth curves

When two smooth curves cross, they meet at points where they share a common spot. We can use steps to find these points, depending on how the curves are described.

Two polygons

To find where two shapes made of many straight lines meet, we check each pair of lines from both shapes. This can take time, so we use tricks to make it faster.

In space (three dimensions)

Further information: three-dimensional space

In three-dimensional space, we can find points where curves and surfaces meet. We'll look at cases where things cross each other in a straight way.

A line and a plane

Main article: Line–plane intersection

When a straight line crosses a flat surface in space, it usually meets at exactly one point.

If we describe the line using equations and put them into the equation of the plane, we can find this meeting point. Sometimes, the line might lie entirely on the plane or run parallel to it without ever touching.

Three planes

If we have three flat surfaces, we can find where they all meet by solving their equations together. When the surfaces are arranged just right, they meet at a single point.

A curve and a surface

A curved path can also meet a flat or curved surface. This needs solving more complex equations, but it works in a similar way.

A line and a polyhedron

Main article: Intersection of a polyhedron with a line

Two surfaces

Main article: Intersection curve

When two curved surfaces cross each other, they usually form a curve. The simplest example is when two flat surfaces cross, making a straight line.

A sphere and a plane

Two spheres

When two spheres meet, they usually form a circle where they overlap. This circle lies in a special flat surface between the two spheres.