Coordinate Geometry

Coordinate geometry connects algebra and shapes. Learn Cartesian coordinates, distance and midpoint, slope and straight lines (intercepts, parallel and perpendicular), circles, parabolas, ellipses and hyperbolas (conic sections), parametric curves and polar coordinates — used in CAD, surveying, robotics and optics.

CAD models, CNC tool paths, GPS positions, satellite dish shapes and planetary orbits are all described by equations of lines and curves in coordinates.

An analogy

Coordinates are like a city's street grid: any point is an address (x, y), and shapes are sets of addresses satisfying a rule.

Lines

Slope measures steepness (rise over run). Parallel lines share slope; perpendicular slopes multiply to −1. Point-slope form y − y₁ = m(x − x₁) builds a line from one point and a slope.

Circles and conics

Complete the square to find a circle's centre and radius. Parabolas focus parallel rays (dish antennas, headlamps); ellipses describe orbits and elliptical gears; hyperbolas appear in navigation (TDOA) and cooling towers.

Parametric curves

x = f(t), y = g(t) describe motion over time — a CNC tool path, a robot arm tip or a projectile. Eliminating t gives the Cartesian equation.

Polar coordinates

Natural for rotating systems and radiation patterns: antenna gain vs angle, cam profiles, spirals. Use atan2 (not atan) to get the correct quadrant.

Formulas

• Distance & midpoint: d = √((x₂ − x₁)² + (y₂ − y₁)²), M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Line: y = mx + c; m = (y₂ − y₁)/(x₂ − x₁); perpendicular: m₁m₂ = −1
• Circle: (x − h)² + (y − k)² = r²
• Conics: Parabola y² = 4ax; ellipse x²/a² + y²/b² = 1; hyperbola x²/a² − y²/b² = 1
• Polar ↔ Cartesian: x = r cos θ, y = r sin θ; r = √(x² + y²), θ = atan2(y, x)

Worked example: Distance and slope

Given: A(1, 2), B(4, 6)

Find: AB, midpoint, slope, perpendicular slope.

• d = √(9 + 16) = 5
• M = (2.5, 4)
• m = 4/3; perpendicular m = −3/4

Answer: 5, (2.5, 4), 4/3, −3/4

Worked example: Circle from general form

Given: x² + y² − 6x + 4y − 12 = 0

Find: Centre and radius.

• (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0
• (x − 3)² + (y + 2)² = 25

Answer: Centre (3, −2), r = 5

Common mistakes

• Wrong: Using atan(y/x) for angles. — Right: Use atan2(y, x) to get the right quadrant.
• Wrong: Forgetting vertical lines have undefined slope. — Right: Write them as x = constant.

Applications

• CAD/CAM: Lines, arcs and splines in part geometry.
• Antennas & optics: Parabolic reflectors focus signals.
• Robotics: Polar/parametric descriptions of arm motion.

Key takeaways

• Distance, midpoint, slope.
• Conics from quadratic equations.
• Parametric curves describe motion.
• Polar coordinates for rotation; use atan2.

Going further

• Homogeneous coordinates: Transformations in graphics and robotics.
• Splines & Bézier curves: Smooth CAD curves.
• 3D analytic geometry: Planes, lines and quadric surfaces.