Sector de una elipse

a = semieje mayor

b = semieje menor

y = 1 - (b/a)²

L/a = E(arcsin[x(c0)/a]; y) - E(arcsin[x(c1)/a]; y)

c = ángulo

c0 = angulo sustraido adyacene al semieje mayor a c1

c1 = angulo total adyacente al semieje mayor (a)

c2 = c1 - c0 = angulo del sector

c0 es adyacente al semieje mayor

c2 es adyacente a c0

c1 es la suma de c0 + c2, tambien es adyacente al semieje mayor

siempre c0 ≤ c1

x(c) = r(c) \* cos(c)

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

E(x,y) ≈ x - (y\*x³)/6 - (3y² - 4y)\*(x^5)/120 - (16y - 60y² + 45y³)\*(x^7)/5040 - y(- 64 + 1008y - 2520y² + 1575y³)\*(x^9)/362880

Sacamos el ultimo término para que sea mas fácil:

E(x,y) ≈ x - (y\*x³)/6 - (3y² - 4y)\*(x^5)/120 - (16y - 60y² + 45y³)\*(x^7)/5040

**Ejemplo**: segmento de elipse a = sqrt(2); b = 1; c0 = 15°; c1 = 45°

y = 1 - (b/a)² = 1 - (1/2) = 1/2

L/a = E(arcsin[x(c0)/a]; 1/2) - E(arcsin[x(c1)/a]; 1/2)

[r(c)]² = [sqrt2]²/([1\*cos(c)]² + [sqrt(2)\*sin(c)]²)

[r(c)]² = 2/([cos(c)]² + 2[sin(c)]²)

c0 = 15\*pi/180 = pi/12

c1 = 45\*pi/180 = pi/4

[r(c0)]² = 2/([cos(pi/12)]² + 2[sin(pi/12)]²) = 1.87443656

[r(c1)]² = 2/([cos(pi/4)]² + 2[sin(pi/4)]²) = 1.33333333 = 4/3

r(c0) = 1.36910064

r(c1) = 1.15470054

x(c0) = 1.36910064 \* cos(pi/12) = 1.32244905 = 1.32244967

x(c1) = 1.15470054 \* cos(pi/4) = 0.816496582

L/a = E(arcsin[x(c0)/a]; 1/2) - E(arcsin[x(c1)/a]; 1/2)

L/[sqrt(2)] = E(arcsin[1.32244967/sqrt2]; 1/2) - E(arcsin[0.816496582/sqrt2]; 1/2)

L/[sqrt(2)] = E(1.20857819 ; 1/2) - E(0.61547971 ; 1/2)

E(x; 1/2) ≈ x - (x³)/12 - (3/4 - 2)\*(x^5)/120 - (8 - 15 + 45/8)\*(x^7)/5040

E(x; 1/2) ≈ x - (x³)/12 - (3/4 - 2)\*(x^5)/120 - (8 - 15 + 45/8)\*(x^7)/5040 - (55/6)\*(x^9)/362880

E(1.20857819; 1/2) ≈ 1.20857819 - (1.20857819³)/12 + (5/4)\*(1.20857819^5)/120 + (11/8)\*(1.20857819^7)/5040 = 1.08935523 (exacto = 1.08916054267977)

E(0.61547971; 1/2) ≈ 0.61547971 - (0.61547971³)/12 + (5/4)\*(0.61547971^5)/120 + (11/8)\*(0.61547971^7)/5040 = 0.596979432 (exacto = 0.5969792386469983)

L/[sqrt(2)] ≈ 1.08935523 - 0.596979432 = 0.492375798 (exacto = 1.089316054 - 0.5969792386 = 0.492336815)

L ≈ 0.492375798\*sqrt2 = 0.696324531 (exacto = 0.492336815\*sqrt2 = 0.696269401)

**Ejemplo**: segmento de elipse a = 4; b = 1; c0 = 30°; c1 = 40°

y = 1 - (b/a)² = 1 - (1/16) = 15/16

L/a = E(arcsin[x(c0)/a]; 15/16) - E(arcsin[x(c1)/a]; 15/16)

[r(c)]² = [4]²/([1\*cos(c)]² + [4\*sin(c)]²)

[r(c)]² = 16/([cos(c)]² + 16[sin(c)]²)

c0 = 30\*pi/180 = pi/6

c1 = 40\*pi/180 = 2pi/9

[r(c0)]² = 16/([cos(pi/6)]² + 16[sin(pi/6)]²) = 3.36842105

[r(c1)]² = 16/([cos(2pi/9)]² + 16[sin(2pi/9)]²) = 2.22295127

r(c0) = 1.83532587

r(c1) = 1.4909565

x(c0) = 1.83532587 \* cos(pi/6) = 1.58943883

x(c1) = 1.4909565 \* cos(2pi/9) = 1.14213894

L/a = E(arcsin[x(c0)/a]; 15/16) - E(arcsin[x(c1)/a]; 15/16)

L/4 = E(arcsin[1.58943883/4]; 15/16) - E(arcsin[1.14213894/4]; 15/16)

L/4 = E(0.408637856 ; 1/2) - E(0.289564346 ; 1/2)

E(x; 15/16) ≈ x - 15\*(x³)/96 - (675/1256 - 15/4)\*(x^5)/120 - (15 - 3375/64 + 151875/4096)\*(x^7)/5040

E(0.408637856; 15/16) ≈ 0.408637856 - 15\*(0.408637856³)/96 - (675/1256 - 15/4)\*(0.408637856^5)/120 - (15 - 3375/64 + 151875/4096)\*(0.408637856^7)/5040 = 0.39828121992

E(0.289564346; 15/16) ≈ 0.289564346 - 15\*(0.289564346³)/96 - (675/1256 - 15/4)\*(0.289564346^5)/120 - (15 - 3375/64 + 151875/4096)\*(0.289564346^7)/5040 = 0.28582523566

L/4 ≈ 0.39828121992 - 0.28582523566 = 0.112455984

L ≈ 0.112455984\*4 = 0.449823936