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Home / Geometry / Coordinate Geometry

Coordinate Geometry

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Descartes' revolution — merging algebra and geometry into a single powerful framework.

The Coordinate Plane

The Cartesian coordinate system, invented by René Descartes, assigns every point in the plane a unique pair (x, y). This seemingly simple idea is one of the most important in all of mathematics — it lets us use algebraic equations to describe geometric shapes.

Distance & Midpoint

Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

The distance formula is a direct consequence of the Pythagorean theorem. It generalizes to n dimensions in linear algebra: d = ‖v₁ − v₂‖.

Example: Distance between (1, 2) and (4, 6)

d = √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5

Lines & Slope

Slope: m = (y₂ − y₁)/(x₂ − x₁) = Δy/Δx
Slope-intercept form: y = mx + b
Point-slope form: y − y₁ = m(x − x₁)
Standard form: Ax + By = C

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: m₁ · m₂ = −1.

The concept of slope is the geometric precursor to the derivative. In calculus, we ask: what is the slope of a curved line at a single point?

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Conic Sections

The four curves obtained by cutting a cone with a plane — each has a standard equation on the coordinate plane:

Circle: (x − h)² + (y − k)² = r²
Ellipse: (x − h)²/a² + (y − k)²/b² = 1
Parabola: y = a(x − h)² + k  or  x = a(y − k)² + h
Hyperbola: (x − h)²/a² − (y − k)²/b² = 1

Parabolas arise in quadratic equations and physics (projectile motion). Ellipses describe planetary orbits (Kepler's first law). Hyperbolas appear in navigation systems and special relativity.

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All conic sections are unified by the focus-directrix property: a conic is the set of points where the ratio of distance-to-focus / distance-to-directrix equals the eccentricity e. Circle: e = 0, Ellipse: 0 < e < 1, Parabola: e = 1, Hyperbola: e > 1. In linear algebra, conics are classified by the eigenvalues of their associated matrix.

Transformations

Geometric transformations can be expressed algebraically using coordinates:

  • Translation by (a, b): (x, y) → (x + a, y + b)
  • Reflection over x-axis: (x, y) → (x, −y)
  • Reflection over y-axis: (x, y) → (−x, y)
  • Rotation by θ: (x, y) → (x cos θ − y sin θ, x sin θ + y cos θ) — uses trigonometry
  • Dilation by factor k: (x, y) → (kx, ky)

The rotation formula uses sine and cosine. In linear algebra, all these transformations are represented as matrix multiplication — an incredibly powerful unification.

Coordinate geometry is the birthplace of calculus. Newton and Leibniz asked: how do curves defined by equations change locally? The answer — derivatives and integrals — launched modern mathematics.

Related Topics

Equations The algebra that makes coordinate geometry possible. Linear Transformations Geometric transformations as matrix operations — the modern view. Derivatives Find tangent lines, normals, and curvature of coordinate curves. The Unit Circle The connection between circles and trigonometric functions.