Analytic geometry

Builds on Cartesian coordinate system Euclidean space Euclidean distance

Definition

Let $E$ be a Euclidean space of dimension $n$ , and fix a Cartesian coordinate system on $E$ . It is a bijection $P \mapsto x(P) \in \R^n$ given by an origin and $n$ mutually perpendicular unit axes. Analytic geometry is the study of figures in $E$ through their coordinates:

a figure $F \subseteq E$ is described by equations and inequalities, as the set of points $P$ whose coordinates satisfy them;
the distance between $P$ and $Q$ is computed as $\abs{x(P) - x(Q)} = \sqrt{\sum_{i=1}^n (x_i(P) - x_i(Q))^2}$ , and angles through the dot product of coordinate vectors;
geometric statements are proved by algebraic computation on these equations.

Remarks

Example: in the plane, the line $y = x$ and the circle $x^2 + y^2 = 2$ meet where $2x^2 = 2$ , that is at $(1, 1)$ and $(-1, -1)$ . The distance between these points is $\sqrt{2^2 + 2^2} = 2\sqrt{2}$ , the diameter of the circle, as it should be for a line through the centre. Non-example: with oblique axes at an angle of $60^\circ$ and unit lengths on both axes, the formula of clause 2 fails. The points with coordinates $(1, 0)$ and $(0, 1)$ are then at distance $1$ , since with the origin they form an equilateral triangle, and not $\sqrt{2}$ . Affine notions such as lines, midpoints and parallelism can be studied with any affine coordinates; metric notions need Cartesian ones. In the plane, a line is $\set{(x, y) : ax + by = c}$ with $(a, b) \ne (0, 0)$ , and the circle with centre $(a, b)$ and radius $r$ is $\set{(x, y) : (x - a)^2 + (y - b)^2 = r^2}$ .

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Tag	`579PT4D349`
URI	`m2://579PT4D349`
Kind	definition
Writing	W0 stub
MSC	51N20, 51N10
Defines	Analytic geometry
Uses	Cartesian coordinate system, Euclidean space, Euclidean distance

Analytic geometry

Definition

Remarks

Reading path

Further reading

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