Centroid of a polygon in Clojure

I am studying Clojure and functional programming. I wrote this function to compute the centroid of a polygon specified as a vector of points, e.g. [[10 20] [0 11] [50 30]]).

Here you can read how to compute the centroid of a polygon.

Can you give me hints and suggestions on how to improve this code from a functional coding style POV?

(defn centroid [p]
  "Calcola il centroide del poligono p"
  (let [six*area     (* 6 (polygon-area p))
        n            (count p)
        first-vertex (p 0)
        polygon      (conj p first-vertex)
        x-terms      (atom [])
        y-terms      (atom [])]
     (dotimes [i n]
            (let [point_i   (polygon i)
                  point_i+1 (polygon (inc i))
                  x_i       (point-x point_i)
                  y_i       (point-y point_i)
                  x_i+1     (point-x point_i+1)
                  y_i+1     (point-y point_i+1)]
               (swap! x-terms conj (* (+ x_i x_i+1)
                                      (- (* x_i y_i+1)
                                         (* x_i+1 y_i))))
               (swap! y-terms conj (* (+ y_i y_i+1)
                                      (- (* x_i y_i+1)
                                         (* x_i+1 y_i))))))
     (make-point (/ (reduce + 0 @x-terms) six*area)
                 (/ (reduce + 0 @y-terms) six*area))))

You could replace the dotimes/atoms with more functional loop/recur recursion.

(defn centroid [p]
  (let [six*area     (* 6 (polygon-area p))
        n            (count p)
        first-vertex (p 0)
        polygon      (conj p first-vertex)]
     (loop [i       0
            x-terms []
            y-terms []]
       (if (< i n)
         (let [point  (polygon i)
               point' (polygon (inc i))
               x      (point-x point)
               y      (point-y point)
               x'     (point-x point')
               y'     (point-y point')
               dxy    (- (* x y')
                         (* x' y))]
           (recur (inc i)
                  (conj x-terms (* (+ x x') dxy))
                  (conj y-terms (* (+ y y') dxy))))
         (make-point (/ (reduce + 0 x-terms) six*area)
                     (/ (reduce + 0 y-terms) six*area))))))

You could also do it through sequences entirely:

(defn centroid [p]
  (let [six*area         (* 6 (polygon-area p))
        polygon          (map (juxt point-x point-y) p) ;; turns the points in [x y] pairs
        ;; `juxt` applies each function in order and outputs the results in a list.

        polygon'         (drop 1 (cycle polygon))       ;; circular permutation of polygon 
        ;; `cycle` lazily repeats and concats the input sequence.

        term           (fn [[[x y] [x' y']]]            ;; destructuring a pair of
                                                        ;; [x y] pairs
                         (let [dxy (- (* x y')
                                      (* x' y))] 
                           [(* dxy (+ x x')) (* dxy (+ y y'))]))
        [x-terms yterms] (->> (map list polygon polygon') ;; makes the [point point'] pairs
        ;; when `polygon` is exhausted, the generated sequence stops, so the fact that
        ;; `polygon'` is infinite has no effect.
                              (map term)
                              (apply (partial map list)))] ;; turns the sequence of pairs
                                                           ;; into a pair of sequences
        ;; `apply` turns the list argument in an argument list that it feeds to the function
        ;; - here, a sequence of pairs. `partial` takes a function and some of its arguments
        ;; and returns it with its first arguments set - here, `map list ...`. `map list`
        ;; applied to a list of pairs will call `list` with the first elements of all pairs,
        ;; then with the seconds.

    (make-point (/ (reduce + 0 x-terms) six*area)
                (/ (reduce + 0 y-terms) six*area))))

EDIT: a(n unefficient) version with map galore:

(defn centroid [p]
  (let [six*area (* 6 (polygon-area p))
        polygon  (map (juxt point-x point-y) p)
        polygon' (drop 1 (cycle polygon))
        x        (map first polygon)
        x'       (map first polygon')
        y        (map second polygon)
        y'       (map second polygon')
        x**x     (map * x x')
        y**y     (map * y y')
        x**y     (map * x y')
        y**x     (map * y x')
        dxy      (map - x**y y**x)
        x-terms  (map * x**x dxy)
        y-terms  (map * y**y dxy)]
    (make-point (/ (reduce + 0 x-terms) six*area)
                (/ (reduce + 0 y-terms) six*area))))

A full loop/recur version:

(defn centroid [[h & t :as p]]
  (let [six*area (* 6 (polygon-area p))
        a        (point-x h)
        b        (point-y h)]
    (loop [x      a
           y      b
           t      t
           x-term 0
           y-term 0]
      (if-let [[h & t] t]
        (let [x'     (point-x h)
              y'     (point-y h)
              dxy    (- (* x y')
                        (* x' y))
              x-term (+ x-term (* dxy (+ x x')))
              y-term (+ y-term (* dxy (+ y y')))]
          (recur x' y' t x-term y-term))
        (let [dxy    (- (* x b)
                        (* a y))
              x-term (+ x-term (* dxy (+ x a)))
              y-term (+ y-term (* dxy (+ y b)))]
          (make-point (/ x-term six*area)
                      (/ y-term six*area)))))))

StackExchange Code Review Q#39045, answer score: 4