Transformations

Wikipedia · Geometric transformation · CC BY-SA 4.0

A transformation moves every point in the plane to a new position. The interesting ones preserve something: distances (isometries), angles (conformal maps), or straight lines (affine maps). The symmetries of a shape form a group under composition.

Translation

Slide every point by the same vector (dx, dy). Distances and angles are preserved. No fixed points.

Scheme

; Translation: shift every point by (dx, dy)
(define (translate point dx dy)
  (list (+ (car point) dx)
        (+ (cadr point) dy)))

(display "Translate (3,4) by (2,-1): ")
(display (translate (list 3 4) 2 -1))
(newline)

; Compose two translations: just add the vectors
(define (compose-translations dx1 dy1 dx2 dy2)
  (list (+ dx1 dx2) (+ dy1 dy2)))

(display "Translate by (2,3) then (1,-1): ")
(display (compose-translations 2 3 1 -1))

Rotation

Rotate every point around the origin by angle theta. The formulas: x' = x cos(theta) - y sin(theta), y' = x sin(theta) + y cos(theta). Preserves distances and angles. One fixed point: the center.

Scheme

; Rotation around the origin by angle theta (radians)
(define pi 3.141592653589793)

(define (rotate x y theta)
  (let ((c (cos theta))
        (s (sin theta)))
    (list (- (* x c) (* y s))
          (+ (* x s) (* y c)))))

; Rotate (1,0) by 90 degrees (pi/2)
(display "Rotate (1,0) by 90 deg: ")
(display (rotate 1 0 (/ pi 2)))
(newline)

; Rotate (1,0) by 45 degrees
(display "Rotate (1,0) by 45 deg: ")
(display (rotate 1 0 (/ pi 4)))
(newline)

; Four 90-degree rotations return to start
(let ((p (rotate 1 0 (/ pi 2))))
  (let ((p2 (rotate (car p) (cadr p) (/ pi 2))))
    (let ((p3 (rotate (car p2) (cadr p2) (/ pi 2))))
      (let ((p4 (rotate (car p3) (cadr p3) (/ pi 2))))
        (display "Four 90-deg rotations: ")
        (display p4)))))

Reflection

Flip every point across a line. Reflection across the x-axis: (x, y) maps to (x, -y). Across the y-axis: (-x, y). Across y=x: (y, x). Reflections preserve distances but reverse orientation.

Scheme

; Reflections
(define (reflect-x point)
  (list (car point) (- (cadr point))))

(define (reflect-y point)
  (list (- (car point)) (cadr point)))

(define (reflect-diagonal point)
  (list (cadr point) (car point)))

(display "Reflect (3,4) across x-axis: ")
(display (reflect-x (list 3 4)))
(newline)

(display "Reflect (3,4) across y-axis: ")
(display (reflect-y (list 3 4)))
(newline)

(display "Reflect (3,4) across y=x: ")
(display (reflect-diagonal (list 3 4)))
(newline)

; Two reflections = a rotation
; Reflect across x-axis, then y-axis = 180-degree rotation
(define (reflect-both p)
  (reflect-y (reflect-x p)))

(display "Two reflections of (3,4): ")
(display (reflect-both (list 3 4)))

Scaling and composition

Scaling multiplies coordinates by a factor. Uniform scaling (same factor for x and y) preserves angles but not distances. Transformations compose: applying rotation then translation is a new transformation. The set of all isometries of a shape forms its symmetry group.

Scheme

; Scaling
(define (scale point factor)
  (list (* (car point) factor)
        (* (cadr point) factor)))

(display "Scale (3,4) by 2: ")
(display (scale (list 3 4) 2))
(newline)

; Composition: apply transformations in sequence
(define (compose-transforms t1 t2)
  (lambda (point) (t2 (t1 point))))

; Rotate 90 degrees then translate by (5,0)
(define pi 3.141592653589793)

(define (rot90 p)
  (list (- (cadr p)) (car p)))

(define (shift-right p)
  (list (+ (car p) 5) (cadr p)))

(define rot-then-shift
  (compose-transforms rot90 shift-right))

(display "Rotate (1,0) 90 deg then shift right 5: ")
(display (rot-then-shift (list 1 0)))
(newline)

; Symmetry group of a square: 4 rotations + 4 reflections = 8 elements
(display "D4 symmetry group has 8 elements")

Neighbors

Cross-references

🔗 Abstract Algebra Ch.1 — groups: the symmetries of a shape form a group under composition

Foundations (Wikipedia)

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