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Number line assumption. Every line is a set of points which can be put into a one-to-one correspondence with the real numbers. Any point can correspond with 0 (zero) and any other point can correspond with 1 (one). Dimension assumption. Given a line in a plane, there exists at least one point in the plane that is not on the line. Given a plane ...
Each line produces three possibilities per point: the point can be in one of the two open half-planes on either side of the line, or it can be on the line. Two points can be considered to be equivalent if they have the same classification with respect to all of the lines.
The process of logical synthesis begins with some arbitrary but definite starting point. This starting point is the introduction of primitive notions or primitives and axioms about these primitives: Primitives are the most basic ideas. Typically they include both objects and relationships.
If two points A, B of a line a lie in a plane α, then every point of a lies in α. In this case we say: “The line a lies in the plane α,” etc. If two planes α, β have a point A in common, then they have at least a second point B in common. There exist at least four points not lying in a plane. II. Order
A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic.
For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, [46] but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it. [47]