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The concept of line is often considered in geometry as a primitive notion in axiomatic systems, [1]: 95 meaning it is not being defined by other concepts. [9] In those situations where a line is a defined concept, as in coordinate geometry , some other fundamental ideas are taken as primitives.
In constructive solid geometry, primitives are simple geometric shapes such as a cube, cylinder, sphere, cone, pyramid, torus. Modern 2D computer graphics systems may operate with primitives which are curves (segments of straight lines, circles and more complicated curves), as well as shapes (boxes, arbitrary polygons, circles).
Hilbert's axiom system is constructed with six primitive notions: three primitive terms: [5] point; line; plane; and three primitive relations: [6] Betweenness, a ternary relation linking points; Lies on (Containment), three binary relations, one linking points and straight lines, one linking points and planes, and one linking straight lines ...
Euclidean geometry is a mathematical system ... of geometric objects such as points and lines, ... and distance become primitive concepts. [55 ...
Thus, a line segment AB defined as the points A and B and all the points between A and B in absolute geometry, needs to be reformulated. A line segment in this new geometry is determined by three collinear points A, B and C and consists of those three points and all the points not separated from B by A and C. There are further consequences.
Because points are the only primitive objects, and because Tarski's system is a first-order theory, it is not even possible to define lines as sets of points. The only primitive relations are "betweenness" and "congruence" among points. Tarski's axiomatization is shorter than its rivals, in a sense Tarski and Givant (1999) make explicit.
In classical Euclidean geometry, a point is a primitive notion, defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, called axioms, that they must satisfy; for example, "there is exactly one straight line that passes through two distinct points".
In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons , the cells of the arrangement, line segments and rays , the edges of the arrangement, and points where two or more lines cross, the vertices of the arrangement.
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