Diseño de Comics con Curvas de Bézier

Carlos César Mejía Toledo (1)        Robert Ipanaqué Chero (2)

Universidad Nacional de Piura, Perú
(1) car_mt123@hotmail.com
(2) robertchero@hotmail.com
http://www.unp.edu.pe/pers/ripanaque

Curvas de Bézier

Polinomios de Bernstein

Consideremos el espacio vectorial de los polinomios de grado menor o igual que n, en una variable t, que denotamos por (Subscript[P, n]). Una base de (Subscript[P, n]) está constituida por el conjunto de los polinomios {1, t,  ... , (t^n)} lo cual significa que todo polinomio P (t) ∈ (Subscript[P, n]) se puede expresar de manera única como combinación lienal de los polinomios de esta base.

Definición

Los polinomios de Bernstein de grado n, que denotamos por (Subscript[B, 0, n]) (t), (Subscript[B, 1, n]) (t),  ... , (Subscript[B, n, n]) (t), son

(Subscript[B, i, n]) (t)=(({{n}, {i}}))(((1 - t))^(n - i))(t^i), para i = 0,  ... , n

donde (({{n}, {i}}))=((n!)/(i! (n - i)!)).

Polinomios de Bernstein con Mathematica

Podemos obtener los polinomios de Bersntein con Mathematica mediante

B[t_, i_, n_] := Binomial[n, i]*(1 - t)^(n - i)*t^i

Por ejemplo obtengamos (Subscript[B, 0, 1]) (t)

B[t, 0, 1]

ó (Subscript[B, 2, 3]) (t)

B[t, 2, 3]

Polinomios de Bernstein de grado 1

B[t, #, 1] & /@ {0, 1}

Plot[% // Evaluate, {t, 0, 1}]

Polinomios de Bernstein de grado 2

B[t, #, 2] & /@ {0, 1, 2}

Plot[% // Evaluate, {t, 0, 1}]

Polinomios de Bernstein de grado 3

B[t, #, 3] & /@ {0, 1, 2, 3}

Plot[% // Evaluate, {t, 0, 1}]

Curvas de Bézier

Definición

Se denomina curva de Bézier asociada a n + 1 puntos (Subscript[P, 0]), (Subscript[P, 1]),  ... , (Subscript[P, n]) de (ℝ^2) ó (ℝ^3), a la curva parametrizada, definida para t ∈ [0, 1], cuyos puntos vienen dados mediante la siguiente expresión

α (t) = (∑_(i = 1)^n) (Subscript[B, i, n]) (t) 
            (Subscript[P, i]),

en la que los (Subscript[B, i, n]) (t) son los polinomios de Bernstein de grado n.

Los puntos (Subscript[P, 0]), (Subscript[P, 1]),  ... , (Subscript[P, n])  que determian una curva de Bézier se denominan puntode control, y la poligonal que los une es el polígono Bézier o B-polígono.

Curvas de Bézier con Mathematica

Podemos obtener curvas de Bézier con Mathematica mediante

BezierCurve[t_, P_] := (n = Length[P] - 1; 
        Sum[B[t, i, n]*P[[i + 1]], {i, 0, n}])

Por ejemplo, dados los puntos (Subscript[P, 0]) (1, 1), (Subscript[P, 1]) (2, 4), (Subscript[P, 2]) (5, 3), hallaremos la curva de Bézier asociada

BezierCurve[t, {{1, 1}, {2, 4}, {5, 3}}]

ParametricPlot[%, {t, 0, 1}]

Como otro ejemplo, hallemos la curva de Bézier asociada a los puntos (Subscript[P, 0]) (2, 4), (Subscript[P, 1]) (1, 1), (Subscript[P, 2]) (5, 3)

BezierCurve[t, {{2, 4}, {1, 1}, {5, 3}}]

ParametricPlot[%, {t, 0, 1}]

Diseños

Mathematica incluye un paquete para obtener una curva de Bézier asociada a un conjunto cualquiera de puntos dados por el usuario. Para nuestro propósito usaremos dicho paquete.

<< Graphics`Spline`

Jonny Bravo

Graphics

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Figura

Show[%, AspectRatio -> Automatic];

Pedro Picapiedra

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              3.75`}, {2.75`, 2.8`}, {2.75`, 2.8`}, {2.75`, 2.8`}, {2.75`, 
              2.8`}, {2.75`, 2.8`}, {2.75`, 2.8`}, {3.2`, 1.35`}, {4, 
              1.8`}, {4, 1.8`}, {4, 1.8`}, {4, 1.8`}, {4, 1.8`}, {4, 
              1.8`}, {5.4`, 1}, {5.4`, 1}, {8, 2}, {8.5`, 6}, {8.5`, 
              6}, {8.5`, 6}, {8.5`, 6}, {8.5`, 6}, {8.8`, 15.1`}}, {{6.3`, 
              6.9`}, {8.5`, 7}, {7.3`, 4}, {6, 3}, {4.5`, 2.75`}, {3.5`, 
              3.75`}}, {{11.2`, 14}, {10.15`, 12}, {9.95`, 8.25`}}, {{9.95`, 
              8.25`}, {11.75`, 8.1`}, {14, 8.05`}, {14, 8.05`}, {15.75`, 
              7.95`}, {15.1`, 6.85`}}, {{8.5`, 6.5`}, {11, 5.25`}, {11, 
              5.25`}, {13.75`, 4.15`}, {15.1`, 6.85`}}, {{8.4`, 21}, {12.8`, 
              18}, {15.38`, 20.9`}}, {{7.75`, 19.4`}, {11, 15.5`}, {15.1`, 
              18.5`}}, {{12.75`, 19}, {13.5`, 18.85`}, {14.05`, 
              19.3`}, {14.05`, 19.3`}, {14.05`, 19.3`}, {14.05`, 
              19.3`}, {14.05`, 19.3`}, {14.05`, 19.3`}, {14.05`, 
              18.5`}, {14.05`, 18.5`}, {14.05`, 18.5`}, {14.05`, 
              18.5`}, {14.05`, 18.5`}, {14.05`, 18.5`}, {13.5`, 
              18.15`}, {12.35`, 18}, {12.35`, 18}, {12.35`, 18}, {12.35`, 
              18}, {12.35`, 18}, {12.35`, 18}, {12.35`, 18}, {12.75`, 
              19}}, {{12.5`, 30.5`}, {12, 30.4`}, {11.25`, 30.1`}, {11.25`, 
              30.1`}, {11.25`, 30.1`}, {11.25`, 30.1`}, {11.25`, 
              30.1`}, {11.5`, 29.75`}, {11.5`, 29.75`}, {11.5`, 
              29.75`}, {11.5`, 29.75`}, {11.5`, 29.75`}, {12.15`, 
              30}, {12.85`, 30.05`}}, {{13.05`, 29.4`}, {12.7`, 
              29.35`}, {11.85`, 29.2`}, {11.85`, 29.2`}, {11.85`, 
              29.2`}, {11.95`, 28.8`}, {11.95`, 28.8`}, {11.95`, 
              28.8`}, {12.55`, 28.92`}, {13.25`, 29.05`}}, {{13.55`, 
              28.5`}, {13, 28.35`}, {12.15`, 28.1`}, {12.15`, 
              28.1`}, {12.15`, 28.1`}, {12.4`, 27.8`}, {12.4`, 
              27.8`}, {12.4`, 27.8`}, {13.15`, 28}, {13.73`, 
              28.05`}}, {{9.95`, 8.2`}, {9, 7.5`}, {9, 7.5`}, {9, 7.5`}, {9, 
              7.5`}, {9, 7.5`}, {9, 7.5`}, {12.5`, 4}, {14.75`, 
              7.5`}}, {{6.45`, 9.75`}, {5.75`, 9.75`}, {5, 9}, {5, 9}, {6.3`, 
              8.5`}, {8.5`, 8.5`}, {9.75`, 9}, {9.75`, 9}, {8.75`, 
              9.75`}, {8.65`, 9.75`}}, {{6.45`, 8.5`}, {5.75`, 8.5`}, {5, 
              8}, {5, 8}, {5.3`, 7.5`}, {8.5`, 7.5`}, {9.75`, 8}, {9.75`, 
              8}, {8.75`, 8.5`}, {8.65`, 8.5`}}, {{8.5`, 10.45`}, {8.45`, 
              9.85`}, {10.25`, 9.85`}, {12.5`, 10.6`}, {10.3`, 10.9`}}, {{9, 
              9}, {10.5`, 9}, {10.5`, 9}, {11.75`, 9.5`}, {10.75`, 
              9.8`}, {10.15`, 9.8`}}
};

Figura

Show[%, AspectRatio -> Automatic];