Centro de masas

PorJosé A. Alonso 22-octubre-201529-octubre-2015

El centro de masas de un sistema discreto es el punto geométrico que dinámicamente se comporta como si en él estuviera aplicada la resultante de las fuerzas externas al sistema.

Representamos un conjunto de n masas en el plano mediante una lista de n pares de la forma ((a(i),b(i)),m(i)) donde (a(i),b(i)) es la posición y m(i) la masa puntual. Las coordenadas del centro de masas (a,b) se calculan por

   a = (a(1)*m(1) + a(2)*m(2) + ... + a(n)*m(n)) / (m(1) + m(2) +...+ m(n))
   b = (b(1)*m(1) + b(2)*m(2) + ... + b(n)*m(n)) / (m(1) + m(2) +...+ m(n))

1 2	a = (a(1)m(1) + a(2)m(2) + ... + a(n)m(n)) / (m(1) + m(2) +...+ m(n)) b = (b(1)m(1) + b(2)m(2) + ... + b(n)m(n)) / (m(1) + m(2) +...+ m(n))

Definir la función

   centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

1	centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

tal que (centrodeMasas xs) es las coordenadas del centro
de masas del sistema discreto xs. Por ejemplo:

   centrodeMasas [((-1,3),2),((0,0),5),((1,3),3)] == (0.1,1.5)

1	centrodeMasas [((-1,3),2),((0,0),5),((1,3),3)] == (0.1,1.5)

Soluciones

-- 1ª definición (por comprensión):
centrodeMasas :: [((Float,Float),Float)] -> (Float,Float) 
centrodeMasas xs = 
    (sum [a*m | ((a,_),m) <- xs] / t,
     sum [b*m | ((_,b),m) <- xs] / t)
    where t = sum [m | (_,m) <- xs]

-- 2ª definición (por recursión):
centrodeMasas2 :: [((Float,Float),Float)] -> (Float,Float)
centrodeMasas2 xs = aux xs (0,0,0)
    where aux [] (as,bs,ms)             = (as/ms, bs/ms)
          aux (((a,b),m):ys) (as,bs,ms) = aux ys (as+a*m,bs+b*m,ms+m)

-- 1ª definición (por comprensión):

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas xs =

(sum [a*m | ((a,_),m) <- xs] / t,

sum [b*m | ((_,b),m) <- xs] / t)

where t = sum [m | (_,m) <- xs]

-- 2ª definición (por recursión):

centrodeMasas2 :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas2 xs = aux xs (0,0,0)

where aux [] (as,bs,ms) = (as/ms, bs/ms)

aux (((a,b),m):ys) (as,bs,ms) = aux ys (as+a*m,bs+b*m,ms+m)

10 Comentarios

Usando un monoide

-- Float es un monoide
instance Monoid Float where { mempty = 0; mappend = (+) }

-- Aplica f con el segundo a cada uno del primero
ap2 f ((a, b), c) = ((f a c, f b c), c)

centrodeMasas :: [((Float, Float), Float)] -> (Float, Float)
centrodeMasas = fst . ap2 (/) . mconcat . map (ap2 (*))

-- Float es un monoide

instance Monoid Float where { mempty = 0; mappend = (+) }

-- Aplica f con el segundo a cada uno del primero

ap2 f ((a, b), c) = ((f a c, f b c), c)

centrodeMasas :: [((Float, Float), Float)] -> (Float, Float)

centrodeMasas = fst . ap2 (/) . mconcat . map (ap2 (*))

Responder

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)
centrodeMasas xs = (sum as/sum ms, sum bs/sum ms)
    where (as,bs,ms) = unzip3 [(a*m, b*m, m) | ((a,b), m) <- xs]
    

-- Versión recursiva
centrodeMasas2 :: [((Float,Float),Float)] -> (Float,Float)
centrodeMasas2 xs = aux xs (0,0,0)
    where aux [] (as,bs,ms) = (as/ms, bs/ms)
          aux (((a,b),m):ys) (as,bs,ms) = aux ys (as+a*m,bs+b*m,ms+m)

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas xs = (sum as/sum ms, sum bs/sum ms)

where (as,bs,ms) = unzip3 [(a*m, b*m, m) | ((a,b), m) <- xs]

-- Versión recursiva

centrodeMasas2 :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas2 xs = aux xs (0,0,0)

where aux [] (as,bs,ms) = (as/ms, bs/ms)

aux (((a,b),m):ys) (as,bs,ms) = aux ys (as+a*m,bs+b*m,ms+m)

Responder

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)
centrodeMasas xs = (ca,cb)
    where ca = sum (map producto (listaAM xs)) / sum (listaM xs)
          cb = sum (map producto (listaBM xs)) / sum (listaM xs)

--xs :: [(Float,Float),Float] -- [((a,b),m)]
puntosA ((a,_),_) = a
puntosB ((_,b),_) = b
masasM  ((_,_),m) = m

listaA xs = map puntosA xs
listaB xs = map puntosB xs
listaM xs = map masasM  xs

listaAM xs = zip (listaA xs) (listaM xs)
listaBM xs = zip (listaB xs) (listaM xs)

producto (a,b) = a*b

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas xs = (ca,cb)

where ca = sum (map producto (listaAM xs)) / sum (listaM xs)

cb = sum (map producto (listaBM xs)) / sum (listaM xs)

--xs :: [(Float,Float),Float] -- [((a,b),m)]

puntosA ((a,_),_) = a

puntosB ((_,b),_) = b

masasM ((_,_),m) = m

listaA xs = map puntosA xs

listaB xs = map puntosB xs

listaM xs = map masasM xs

listaAM xs = zip (listaA xs) (listaM xs)

listaBM xs = zip (listaB xs) (listaM xs)

producto (a,b) = a*b

Responder

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)
centrodeMasas xs = (a,b)
    where a = sum [x*m | ((x,y),m) <-xs] /sum [m |((x,y),m) <- xs]
          b = sum [y*m | ((x,y),m) <-xs] /sum [m |((x,y),m) <- xs]

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas xs = (a,b)

where a = sum [x*m | ((x,y),m) <-xs] /sum [m |((x,y),m) <- xs]

b = sum [y*m | ((x,y),m) <-xs] /sum [m |((x,y),m) <- xs]

Responder

Solución por comprensión:

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)
centrodeMasas xs = (a,b)
    where a = sum [a*m | ((a,_),m) <- xs] / sumaMasas xs
          b = sum [b*m | ((_,b),m) <- xs] / sumaMasas xs
          sumaMasas xs = sum [m | ((_,_),m) <- xs]

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas xs = (a,b)

where a = sum [a*m | ((a,_),m) <- xs] / sumaMasas xs

b = sum [b*m | ((_,b),m) <- xs] / sumaMasas xs

sumaMasas xs = sum [m | ((_,_),m) <- xs]

Responder

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)
centrodeMasas xs = (a,b)
    where a = sum [x*m | ((x,_),m) <- xs] / s
          b = sum [y*m | ((_,y),m) <- xs] / s
          s = sum [m | ((_,_),m) <- xs]

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas xs = (a,b)

where a = sum [x*m | ((x,_),m) <- xs] / s

b = sum [y*m | ((_,y),m) <- xs] / s

s = sum [m | ((_,_),m) <- xs]

Responder

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)
centrodeMasas xs = (ca,cb)
    where ca = sum o / sum t
          cb = sum e / sum t
          o  = [a*m | ((a,_),m) <- xs]
          e  = [b*m | ((_,b),m) <- xs]
          t  = [m   | ((_,_),m) <- xs]

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas xs = (ca,cb)

where ca = sum o / sum t

cb = sum e / sum t

o = [a*m | ((a,_),m) <- xs]

e = [b*m | ((_,b),m) <- xs]

t = [m | ((_,_),m) <- xs]

Responder

centroDeMasas :: [((Float,Float),Float)] -> (Float,Float)
centroDeMasas xs = (x,y)
    where x = sum [ a * m | ((a,_),m) <- xs] / s
          y = sum [ b * m | ((_,b),m) <- xs] / s
          s = sum [ m | ((_,_),m) <- xs]

centroDeMasas :: [((Float,Float),Float)] -> (Float,Float)

centroDeMasas xs = (x,y)

where x = sum [ a * m | ((a,_),m) <- xs] / s

y = sum [ b * m | ((_,b),m) <- xs] / s

s = sum [ m | ((_,_),m) <- xs]

Responder

centrodeMasas :: [((Float, Float), Float)] -> (Float, Float)
centrodeMasas xs = 
    ((sum [a*m | ((a, _), m) <- xs] / sum [m | ((_, _), m) <- xs]), 
     (sum [b*m | ((_, b), m) <- xs] / sum [m | ((_, _), m) <- xs]))

centrodeMasas :: [((Float, Float), Float)] -> (Float, Float)

centrodeMasas xs =

((sum [a*m | ((a, _), m) <- xs] / sum [m | ((_, _), m) <- xs]),

(sum [b*m | ((_, b), m) <- xs] / sum [m | ((_, _), m) <- xs]))

Responder

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)
centrodeMasas xs = (a,b)
    where a = sum [mi*ai/m | ((ai, _),mi) <- xs]
          b = sum [mi*bi/m | ((_, bi),mi) <- xs]
          m = sum [mi | (_,mi) <- xs]

centrodeMasas :: [((Float,Float),Float)] -> (Float,Float)

centrodeMasas xs = (a,b)

where a = sum [mi*ai/m | ((ai, _),mi) <- xs]

b = sum [mi*bi/m | ((_, bi),mi) <- xs]

m = sum [mi | (_,mi) <- xs]

Responder

Centro de masas

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