Sector de una elipse III

**L/a = E[z(c0), k] - E[z(c1), k]**

k = sqrt[1 - (b/a)^2]

x(c) = sqrt[([cos(c)]²\*[ab]²)/([b\*cos(c)]² + [a\*sin(c)]²)]

x(c)/a = z(c) = sqrt[([cos(c)]²\*[ab]²)/([ab\*cos(c)]² + [a²\*sin(c)]²)]

z(c) in degrees = sqrt[([cos(c\*pi/180)]²\*[ab]²)/([ab\*cos(c\*pi/180)]² + [a²\*sin(c\*pi/180)]²)]

E(z,k) ≈ (z/60)\*(1 + 4\*sqrt[(1 - [k^2\*(z/20)^2])/(1 - (z/20)^2)] + 2\*sqrt[(1 - [k^2\*(2z/20)^2])/(1 - (2z/20)^2)] + 4\*sqrt[(1 - [k^2\*(3z/20)^2])/(1 - (3z/20)^2)] + 2\*sqrt[(1 - [k^2\*(4z/20)^2])/(1 - (4z/20)^2)] + 4\*sqrt[(1 - [k^2\*(5z/20)^2])/(1 - (5z/20)^2)] + 2\*sqrt[(1 - [k^2\*(6z/20)^2])/(1 - (6z/20)^2)] + 4\*sqrt[(1 - [k^2\*(7z/20)^2])/(1 - (7z/20)^2)] + 2\*sqrt[(1 - [k^2\*(8z/20)^2])/(1 - (8z/20)^2)] + 4\*sqrt[(1 - [k^2\*(9z/20)^2])/(1 - (9z/20)^2)] + 2\*sqrt[(1 - [k^2\*(10z/20)^2])/(1 - (10z/20)^2)] + 4\*sqrt[(1 - [k^2\*(11z/20)^2])/(1 - (11z/20)^2)] + 2\*sqrt[(1 - [k^2\*(12z/20)^2])/(1 - (12z/20)^2)] + 4\*sqrt[(1 - [k^2\*(13z/20)^2])/(1 - (13z/20)^2)] + 2\*\*sqrt[(1 - [k^2\*(14z/20)^2])/(1 - (14z/20)^2)] + 4\*sqrt[(1 - [k^2\*(15z/20)^2])/(1 - (15z/20)^2)] + 2\*sqrt[(1 - [k^2\*(16z/20)^2])/(1 - (16z/20)^2)] + 4\*sqrt[(1 - [k^2\*(17z/20)^2])/(1 - (17z/20)^2)] + 2\*sqrt[(1 - [k^2\*(18z/20)^2])/(1 - (18z/20)^2)] + 4\*sqrt[(1 - [k^2\*(19z/20)^2])/(1 - (19z/20)^2)] + sqrt[(1 - [k^2\*z^2])/(1 - (z^2)])

**Ejemplo: a=5, b=3, c1 = 75, c0 = 25**

k = sqrt[1 - (3/5)^2] = sqrt[1 - 9/25] = sqrt[16/25] = 4/5 = 0.8

c1 = 75pi/180 = 25pi/60 = 5pi/12

c0 = 25pi/180 = 5pi/36

z(c0) = sqrt[([cos(5pi/36)]²\*15²)/([15\*cos(5pi/36)]² + [25\*sin(5pi/36)]²)]

z(c0) = 0.7895811160709928

z(c1) = sqrt[([cos(5pi/12)]²\*15²)/([15\*cos(5pi/12)]² + [25\*sin(5pi/12)]²)]

z(c1) = 0.15873125170110722

L/a = E[z(c0), k] - E[z(c1), k]

L/5 = E[0.78958, 0.8] - E[0.15873, 0.8]

E(0.78958, 0.8) ≈ (0.78958/60)\*(1 + 4\*sqrt[(1 - [0.8^2\*(0.78958/20)^2])/(1 - (0.78958/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(2\*0.78958/20)^2])/(1 - (2\*0.78958/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(3\*0.78958/20)^2])/(1 - (3\*0.78958/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(4\*0.78958/20)^2])/(1 - (4\*0.78958/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(5\*0.78958/20)^2])/(1 - (5\*0.78958/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(6\*0.78958/20)^2])/(1 - (6\*0.78958/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(7\*0.78958/20)^2])/(1 - (7\*0.78958/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(8\*0.78958/20)^2])/(1 - (8\*0.78958/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(9\*0.78958/20)^2])/(1 - (9\*0.78958/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(10\*0.78958/20)^2])/(1 - (10\*0.78958/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(11\*0.78958/20)^2])/(1 - (11\*0.78958/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(12\*0.78958/20)^2])/(1 - (12\*0.78958/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(13\*0.78958/20)^2])/(1 - (13\*0.78958/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(14\*0.78958/20)^2])/(1 - (14\*0.78958/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(15\*0.78958/20)^2])/(1 - (15\*0.78958/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(16\*0.78958/20)^2])/(1 - (16\*0.78958/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(17\*0.78958/20)^2])/(1 - (17\*0.78958/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(18\*0.78958/20)^2])/(1 - (18\*0.78958/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(19\*0.78958/20)^2])/(1 - (19\*0.78958/20)^2)] + sqrt[(1 - [0.8^2\*0.78958^2])/(1 - (0.78958^2)]) = 0.8369263553352945

E(0.15873, 0.8) ≈ (0.15873/60)\*(1 + 4\*sqrt[(1 - [0.8^2\*(0.15873/20)^2])/(1 - (0.15873/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(2\*0.15873/20)^2])/(1 - (2\*0.15873/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(3\*0.15873/20)^2])/(1 - (3\*0.15873/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(4\*0.15873/20)^2])/(1 - (4\*0.15873/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(5\*0.15873/20)^2])/(1 - (5\*0.15873/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(6\*0.15873/20)^2])/(1 - (6\*0.15873/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(7\*0.15873/20)^2])/(1 - (7\*0.15873/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(8\*0.15873/20)^2])/(1 - (8\*0.15873/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(9\*0.15873/20)^2])/(1 - (9\*0.15873/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(10\*0.15873/20)^2])/(1 - (10\*0.15873/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(11\*0.15873/20)^2])/(1 - (11\*0.15873/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(12\*0.15873/20)^2])/(1 - (12\*0.15873/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(13\*0.15873/20)^2])/(1 - (13\*0.15873/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(14\*0.15873/20)^2])/(1 - (14\*0.15873/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(15\*0.15873/20)^2])/(1 - (15\*0.15873/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(16\*0.15873/20)^2])/(1 - (16\*0.15873/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(17\*0.15873/20)^2])/(1 - (17\*0.15873/20)^2)] + 2\*sqrt[(1 - [0.8^2\*(18\*0.15873/20)^2])/(1 - (18\*0.15873/20)^2)] + 4\*sqrt[(1 - [0.8^2\*(19\*0.15873/20)^2])/(1 - (19\*0.15873/20)^2)] + sqrt[(1 - [0.8^2\*0.15873^2])/(1 - (0.15873^2)]) = 0.15897331073319956

L/a = E[z(c0), k] - E[z(c1), k]

L/5 ≈ (0.8369263553352945 - 0.15897331073319956)

L ≈ 5\*(0.8369263553352945 - 0.15897331073319956)

L ≈ 3.38976522