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In mathematics, intersection is a relationship between two geometric figures. The intersection of two graphs means that they have a common part, or the point sets belonging to both graphs are not empty sets. If there is only one intersection point between two geometries, it is called tangency rather than intersection. If two figures are completely coincident, it is generally not called intersection. In set theory, the intersection of two sets means that their intersection is not an empty set.

2. Intersection of straight lines

On the Euclidean plane, two straight lines are either parallel, intersect or coincide. This is the inference of Euclid's fifth postulate. Two intersecting straight lines have exactly one intersection point. In non-Euclidean geometry, it can be divided into two categories according to geometric characteristics (curvature).

In Luo Barczewski geometry, two straight lines are either parallel or intersect, but there are more than one parallel line. In Riemannian geometry, two straight lines always intersect. In three-dimensional space or higher-dimensional space, when two straight lines intersect, they must be * * * planes.

3. Intersection of circles

In Euclidean geometry, there are four relationships between two circles on the same plane: separation, tangency, compatibility and intersection. Separation means that two circles do not intersect, and neither circle is in the other circle. Tangency means that two circles have only one intersection.

Intersection means that two circles have more than one intersection. Compatibility means that two circles do not intersect and one circle is inside the other. Two circles intersect if and only if the distance between the centers of the two circles is strictly smaller than the sum of the radii of the two circles and strictly larger than the difference between the radii of the two circles.

4. Straight line introduction

A straight line consists of countless points, and the points move into a line. A straight line is a part of a surface and then constitutes a body. There is no end point, extending to both ends indefinitely, and the length cannot be measured. A straight line is an axisymmetric figure.

It has countless symmetry axes, all of which are straight lines perpendicular to it (there are countless). There is only one straight line between two non-overlapping points on the plane, that is, two non-overlapping points determine a straight line. On the sphere, countless similar straight lines can be made after two points.

The most basic element of geometry. In the axiomatic system of Euclidean geometry established by D Hilbert, points, lines and surfaces belong to basic concepts, which are defined by their relationships and five groups of axioms.