Coordinate Geometry
Key points
Distance between two points, and the midpoint
For two points A(x₁,y₁) and B(x₂,y₂), the distance is AB = √((x₂−x₁)²+(y₂−y₁)²). The midpoint is M = ((x₁+x₂)/2, (y₁+y₂)/2) — just the average of the x values and the average of the y values.
The equation of a line
A line with slope m and y-intercept b is written y = mx + b. If you only know two points (x₁,y₁) and (x₂,y₂), the slope of the line through them is m = (y₂−y₁)/(x₂−x₁): rise over run.
The equation of a circle
A circle with center (a,b) and radius r is described by (x−a)²+(y−b)² = r². When the center is the origin, this simplifies to x²+y² = r².
Distance from a point to a line
The shortest distance from a point P(x₀,y₀) to the line ax+by+c=0 is d = |ax₀+by₀+c| / √(a²+b²). Substitute the point into the line's equation, take the absolute value, and divide by √(a²+b²).
How GPS works (three circles crossing)
If you know your distances r₁, r₂, and r₃ from three GPS satellites, your position is the single point where those three circles meet. Solving the three circle equations together gives you that point.
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