The non-negative real number x := ρ( O, X) is the absolute value of the x-coordinate of P. Given an arbitrary point P in the plane, it is possible to construct a half-line from P intersecting the x-axis perpendicularly at the point X: one says that X is the perpendicular projection of P on the x-axis. Briefly, one chooses an arbitrary point O (the origin) and let it be the point of intersection of two perpendicular straight lines, one line called the x-axis, the other line called the y-axis. This fact is expressed by the statement that a Euclidean plane is two-dimensional.Ī Cartesian coordinate frame can be erected anywhere in the plane, see this article for details and a figure. The maximum number of perpendicular lines that can intersect at a point of a Euclidean plane is equal for all points this maximum is two for a Euclidean plane. In particular, straight lines may be perpendicular, that is, their angle of intersection is 90°. Angles between intersecting lines can be measured and expressed in degrees. The point of intersection of two non-parallel straight lines is a single unique point. Two straight lines that do not intersect are called parallel. Hence, lines and half-lines in a Euclidean plane are of infinite length. If P also runs to infinity (away from Q), i.e., the straight segment is unbounded on both sides, the segment becomes a straight line. The straight line segment connecting P and Q may be extended to a half-line, which is a line segment terminated in P and unbounded on the other side. In a Euclidean plane no upper bound exists on the distance between two points: given a fixed point P and a "running" point Q, the distance ρ( P, Q) is unbounded, so that Q can "run to infinity" without leaving the plane. The length of the straight line segment connecting the points P and Q is by definition the Euclidean distance rho ( P, Q). Any two non-coinciding points P and Q in the plane lie on a unique straight line, that is to say, P and Q determine a unique straight line. Well-known subsets of the Euclidean plane are straight lines. In other words, the Euclidean plane is a metric space. ρ( P, Q) ≥ 0 and ρ( P, Q) = 0 if and only if P = Q.between which a distance ρ is defined, with the properties, The Euclidean plane is a collection of points P, Q, R. The plane and the geometry are named after the ancient-Greek mathematician Euclid. The Euclidean plane is the plane that is the object of study in Euclidean geometry (high-school geometry).
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