Coordinate Geometry

Wikipedia · Analytic geometry · CC BY-SA 4.0

Descartes married algebra to geometry. Every point is a pair of numbers. Every curve is an equation. Every geometric proof becomes a calculation. The Cartesian plane is the bridge between shape and formula.

The Cartesian plane

Two perpendicular number lines crossing at the origin (0, 0). The horizontal axis is x, the vertical is y. Every point in the plane has coordinates (x, y). This is the idea that turned geometry into algebra.

Distance and midpoint

The distance formula is the Pythagorean theorem in coordinates: d = sqrt((x2-x1)² + (y2-y1)²). The midpoint is the average of the coordinates: ((x1+x2)/2, (y1+y2)/2).

Scheme

; Distance formula: derived from Pythagorean theorem
(define (distance x1 y1 x2 y2)
  (sqrt (+ (* (- x2 x1) (- x2 x1))
           (* (- y2 y1) (- y2 y1)))))

; Midpoint formula
(define (midpoint x1 y1 x2 y2)
  (list (/ (+ x1 x2) 2)
        (/ (+ y1 y2) 2)))

(display "Distance (1,2) to (4,6): ")
(display (distance 1 2 4 6))
(newline)

(display "Midpoint (1,2) and (4,6): ")
(display (midpoint 1 2 4 6))
(newline)

; Distance from origin
(display "Distance (0,0) to (3,4): ")
(display (distance 0 0 3 4))

Slope and lines

Slope measures steepness: m = (y2-y1)/(x2-x1). A line through (x1, y1) with slope m satisfies y - y1 = m(x - x1). Two lines are parallel when their slopes are equal. Two lines are perpendicular when m1 * m2 = -1.

Scheme

; Slope between two points
(define (slope x1 y1 x2 y2)
  (/ (- y2 y1) (- x2 x1)))

(display "Slope from (1,1) to (3,5): ")
(display (slope 1 1 3 5))
(newline)

; Line equation: y = mx + b
; Given slope m and point (x1,y1), find b
(define (y-intercept m x1 y1)
  (- y1 (* m x1)))

(let ((m (slope 1 1 3 5)))
  (display "Line: y = ")
  (display m)
  (display "x + ")
  (display (y-intercept m 1 1))
  (newline))

; Parallel check: equal slopes
(display "Parallel? slopes 2 and 2: ")
(display (= 2 2))
(newline)

; Perpendicular check: product of slopes = -1
(display "Perpendicular? slopes 2 and -1/2: ")
(display (= (* 2 -1/2) -1))

Conic sections

Slice a cone at different angles and you get circles, ellipses, parabolas, and hyperbolas. Each has a standard equation in the coordinate plane:

Circle: x² + y² = r²
Ellipse: x²/a² + y²/b² = 1
Parabola: y = ax²
Hyperbola: x²/a² - y²/b² = 1

Scheme

; Conic sections as predicates
; Does a point lie on the curve?

; Circle: x^2 + y^2 = r^2
(define (on-circle? x y r)
  (= (+ (* x x) (* y y)) (* r r)))

(display "Is (3,4) on circle r=5? ")
(display (on-circle? 3 4 5))
(newline)

; Parabola: y = a*x^2
(define (on-parabola? x y a)
  (= y (* a x x)))

(display "Is (2,12) on y=3x^2? ")
(display (on-parabola? 2 12 3))
(newline)

; Ellipse: x^2/a^2 + y^2/b^2 = 1
(define (on-ellipse? x y a b)
  (= (+ (/ (* x x) (* a a))
        (/ (* y y) (* b b)))
     1))

(display "Is (3,0) on ellipse a=3,b=2? ")
(display (on-ellipse? 3 0 3 2))

Neighbors

Cross-references

∫ Calculus Ch.1 — functions and graphs: coordinate geometry is the prerequisite

Foundations (Wikipedia)

← Euclidean Geometry by june.kim Transformations →