Coordinate Geometry — Bench

The Coordinate Plane

René Descartes’ key insight: every point in a plane can be described by two numbers — its horizontal position (x) and vertical position (y). This bridges geometry and algebra completely. Every geometric shape becomes an equation; every equation becomes a shape.

A point is written as an ordered pair (x, y).

Distance and Midpoint

Distance between two points (x₁, y₁) and (x₂, y₂):

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Just Pythagoras applied to the horizontal and vertical separations.

Midpoint:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Average the coordinates. The midpoint is literally the mean of the two positions.

Lines

A straight line is the set of all points satisfying a linear equation.

Slope

The slope m measures steepness — rise over run:

m = (y₂ − y₁) / (x₂ − x₁)

Positive slope: rises left to right
Negative slope: falls left to right
Zero slope: horizontal line
Undefined slope: vertical line (x₂ = x₁, division by zero)

Forms of a line equation

Slope-intercept form: y = mx + b

m = slope, b = y-intercept (where line crosses y-axis)
Most useful for graphing and reading off slope directly

Point-slope form: y − y₁ = m(x − x₁)

Most useful when you know a point and the slope

Standard form: ax + by = c (integers, a > 0)

Most useful for systems of equations

Converting between forms:

y − 3 = 2(x − 1)      (point-slope, point (1,3), slope 2)
y − 3 = 2x − 2
y = 2x + 1             (slope-intercept)
2x − y = −1            (standard)

Parallel and perpendicular lines

Parallel lines: same slope, m₁ = m₂ (never intersect)
Perpendicular lines: slopes are negative reciprocals, m₁ × m₂ = −1

Line with slope 3 → perpendicular slope = −1/3
Line with slope 2/3 → perpendicular slope = −3/2

Intersection of Lines

Two lines intersect at the point satisfying both equations simultaneously — solve the system:

y = 2x + 1
y = −x + 7
2x + 1 = −x + 7
3x = 6
x = 2, y = 5
Intersection: (2, 5)

Parallel lines: no intersection (same slope, different intercepts). Same line: infinitely many intersections (same slope, same intercept).

Circles

A circle is the set of all points at distance r from a centre (h, k):

(x − h)² + (y − k)² = r²

Standard form — centre (h, k), radius r. Read them off directly.

Expanded form (general form):

x² + y² + Dx + Ey + F = 0

To convert from general to standard, complete the square for x and y:

x² + y² − 4x + 6y − 3 = 0
(x² − 4x + 4) + (y² + 6y + 9) = 3 + 4 + 9
(x − 2)² + (y + 3)² = 16
Centre: (2, −3), radius: 4

Distance from a Point to a Line

The shortest distance from point (x₀, y₀) to line ax + by + c = 0:

d = |ax₀ + by₀ + c| / √(a² + b²)

Useful in geometry, optimisation, and machine learning (support vector machines find the line that maximises this distance to data points).

Conic Sections

All conic sections are intersections of a plane with a cone. In coordinate form:

Parabola: y = ax² + bx + c (or x = ay² + by + c for horizontal)

Vertex form: y = a(x − h)² + k, vertex at (h, k)

Ellipse: (x − h)²/a² + (y − k)²/b² = 1

a = semi-major axis, b = semi-minor axis
When a = b: circle

Hyperbola: (x − h)²/a² − (y − k)²/b² = 1

Two branches, opens left-right
Asymptotes: y − k = ±(b/a)(x − h)

All four (circle, parabola, ellipse, hyperbola) come from the same general second-degree equation:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

The discriminant B² − 4AC determines which:

B² − 4AC < 0: ellipse (or circle if B = 0 and A = C)
B² − 4AC = 0: parabola
B² − 4AC > 0: hyperbola

Transformations in the Coordinate Plane

Transformation	Rule
Translate by (a, b)	(x, y) → (x + a, y + b)
Reflect over x-axis	(x, y) → (x, −y)
Reflect over y-axis	(x, y) → (−x, y)
Reflect over y = x	(x, y) → (y, x)
Rotate 90° CCW	(x, y) → (−y, x)
Rotate 180°	(x, y) → (−x, −y)
Scale by factor k	(x, y) → (kx, ky)

Rotation by arbitrary angle θ:

x' = x cos θ − y sin θ
y' = x sin θ + y cos θ

This is the rotation matrix from trigonometry — coordinate geometry and trig unify here.

The Gradient of a Curve

At a point on a curve, the gradient (slope of the tangent) is what calculus computes — the derivative. Coordinate geometry sets up the language: a tangent line at (x₀, y₀) has the form y − y₀ = m(x − x₀), where m is the derivative at that point.

This is the bridge: coordinate geometry gives the framework; calculus gives the tool to find slopes of curves, not just lines.

Why Coordinate Geometry Matters

Descartes’ invention turned geometry into algebra. Every geometric problem — distance, area, intersection, curvature — became a computational problem. This is why computers can do graphics: every shape is an equation, and equations are just arithmetic.

All of machine learning, computer vision, and data science lives in high-dimensional coordinate space. The intuitions from 2D coordinate geometry — distance, projection, orthogonality — generalise directly to 1000-dimensional feature spaces.