geometry

Coordinate (System)

A coordinate system assigns numbers to points in space. Cartesian (x, y) is most common in 2D; polar (r, θ) is used for circular symmetry.

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A coordinate system assigns numerical labels to each point in space, enabling algebraic methods to solve geometric problems.

Common 2D systems:

  • Cartesian: (x,y)(x, y). Distance: (x2−x1)2+(y2−y1)2\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.
  • Polar: (r,θ)(r, \theta). Conversion: x=rcos⁡θx = r\cos\theta, y=rsin⁡θy = r\sin\theta.

3D extensions:

  • Cartesian: (x,y,z)(x, y, z).
  • Cylindrical: (r,θ,z)(r, \theta, z).
  • Spherical: (ρ,θ,ϕ)(\rho, \theta, \phi).

Choice of system affects problem difficulty. A circle is awkward in Cartesian (x2+y2=r2x^2 + y^2 = r^2) but trivial in polar (r=r = const). Physics with circular / spherical symmetry → polar / spherical.

Foundation of analytic geometry, computer graphics, and geographic coordinates (latitude / longitude).