Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Point (geometry)
Summary
In classical Euclidean geometry, a point is a primitive notion that models an exact location in space, and has no length, width, or thickness. In modern mathematics, a point refers more generally to an element of some set called a space.
Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy; for example, "there is exactly one line that passes through two different points".
Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". In the two-dimensional Euclidean plane, a point is represented by an ordered pair (x, y) of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z) with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, (a1, a2, ... , an) where n is the dimension of the space in which the point is located.
Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a line is an infinite set of points of the form
where c1 through cn and d are constants and n is the dimension of the space. Similar constructions exist that define the plane, line segment, and other related concepts. A line segment consisting of only a single point is called a degenerate line segment.
In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, that any two points can be connected by a straight line.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.