A Refractive Lens

P0 - P1 - P2 : {i0, j0, k0}, {i1, j1, k1}, {2a1 i1 t1 + i1, 2b1 j1 t1 + j1, k1 - t1}

n = Sin[φ]/Sin[δ]

M:{-i1,-j1,-k1}

{i0 - i1, j0 - j1, k0 - k1}, {0, 0, 0}, {2a1 i1 t1, 2b1 j1 t1, -t1}

R: x,y plane α1, y,z plane β1

α1 = ArcTan[(i0 - i1)/(j0 - j1)]

β1 = ArcTan[(1 + (i0 - i1)^2/(j0 - j1)^2)^(1/2) (j0 - j1)/(k0 - k1)]

    M = (1 + ((i0 - i1)^2 + (j0 - j1)^2)/(k0 - k1)^2)^(1/2) (k0 - k1)

({{Cos[γ1], -Sin[γ1], 0}, {Sin[γ1], Cos[γ1], 0}, {0, 0, 1}}) . ({{A}, {B}, {C}}) == ({{0}, {M1}, {M2}})

γ1 = ArcTan[A/B]

γ1 = ArcTan[(2 (a1 i1 Cos[α1] - b1 j1 Sin[α1]))/(2 Cos[β1] (b1 j1 Cos[α1] + a1 i1 Sin[α1]) + Sin[β1])]

Circle1 : ({{Cos[φ] Sin[γ1], Cos[φ] Cos[γ1], Sin[φ]}})

     ({{Cos[-γ1], -Sin[-γ1], 0}, {Sin[-γ1], Cos[-γ1], 0}, {0, 0, 1}}) . ({{0}, {Cos[φ]}, {Sin[φ]}})

R: -β1, -α1, M:{i1,j1,k1}

φ

u = {i0 - i1, j0 - j1, k0 - k1} ;

v = {2a1 i1, 2b1 j1 , -1} ;

Cos[δ] = u . v/((u^2 . {1, 1, 1})^(1/2) (v^2 . {1, 1, 1})^(1/2))

δ = π/2 - ArcCos[(2 a1 (i0 - i1) i1 + 2 b1 (j0 - j1) j1 - k0 + k1)/((1 + 4 a1^2 i1^2 + 4 b1^2 j1^2)^(1/2) ((i0 - i1)^2 + (j0 - j1)^2 + (k0 - k1)^2)^(1/2))]

n == Sin[φ]/Sin[δ]

Sin[φ] = n Sin[δ]

φ = (π/2 - δ) + γ = π/2 - δ + ArcSin[n Sin[δ]] ;

{i2,j2,k2}

a2 x^2 + b2 y^2 - (z - c2) == 0

Solve[{a2 (i1 + d1 T)^2 + b2 (j1 + e1 T)^2 - (k1 + f1 T - c2) == 0}, {T}]

T = (f1 - 2 (a2 d1 i1 + b2 e1 j1) - ((f1 - 2 (a2 d1 i1 + b2 e1 j1))^2 - 4 (a2 d1^2 + b2 e1^2) (c2 + a2 i1^2 + b2 j1^2 - k1))^(1/2))/(2 (a2 d1^2 + b2 e1^2))

Attach - Annex

Refraction Lens

N

Surface

S1 : a1 x^2 + b1 y^2 - (z - c1) == 0

S2 : a2 x^2 + b2 y^2 - (z - c2) == 0

Point

P3 : {i3, j3, k3}

P2 : {i2, j2, k2}

P1 : {i1, j1, k1}

P0 : {i0, j0, k0}

S1 : a1 x^2 + b1 y^2 - (z - c1) == 0

V1 : {2a1 i1, 2b1 j1, -1}

L1 : (x - i1)/(2a1 i1) == (y - j1)/(2b1 j1) == (z - k1)/-1 == t1

eP1 : {2a1 i1 t1 + i1, 2b1 j1 t1 + j1, k1 - t1}

    a1 (2a1 i1 t1 + i1)^2 + b1 (2b1 j1 t1 + j1)^2 - (k1 - t1 - c1) == 0

c1 + a1 i1^2 + b1 j1^2 == k1

P1 : {i1, j1, a1 i1^2 + b1 j1^2 + c1}

S2 : a2 x^2 + b2 y^2 - (z - c2) == 0

V2 : {2a2 i2, 2b2 j2, -1}

L2 : (x - i2)/(2a2 i2) == (y - j2)/(2b2 j2) == (z - k2)/-1 == t2

eP2 : {2a2 i2 t2 + i2, 2b2 j2 t2 + j2, k2 - t2}

    a2 (2a2 i2 t2 + i2)^2 + b2 (2b2 j2 t2 + j2)^2 - (k2 - t2 - c2) == 0

c2 + a2 i2^2 + b2 j2^2 == k2

P1 : {i2, j2, a2 (i2)^2 + b2 (j2)^2 + c2}