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In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines , which either is one point (sometimes called a vertex ) or does not exist (if the lines are parallel ).
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems ) from these.
In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer graphics , motion planning , and collision detection .
The geometric definition of a closed line segment: the intersection of all points at or to the right of A with all points at or to ... Beyond Euclidean geometry, ...
Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space. It simply means the overlapping area of two or more objects or geometries. Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point is the most common in a plane geometry.
In the Euclidean plane, using the geometric definition, a degenerate case arises when the cutting plane passes through the apex of the cone. The degenerate conic is either: a point , when the plane intersects the cone only at the apex; a straight line , when the plane is tangent to the cone (it contains exactly one generator of the cone); or a ...
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.
In Geometric Algebra, Emil Artin has proved that all these definitions of a Euclidean space are equivalent. [9] It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is the ...