Refractive Lens

Beam Point : {I0, J0, K0}

LED Angle : LED

Refraction Index : N

Curved Surface : a1, b1, c1

Curved Surface : a2, b2, c2

I0, J0, K0

LED

N

a1, b1, c1

a2, b2, c2

{I0,J0,K0},{az_,bz_,cz_},{dz_,ez_,fz_}

Beam Point : {I0, J0, K0}

Surface Point : {az_, bz_, cz_}

Normal Line : {dz_, ez_, fz_}

Solve[a1 (I0 + K0 Tan[LED] Cos AG_)^2 + b1 (J0 + K0 Tan[LED] Sin AG_)^2 - ((K0 - K0 AG_) - c1) == 0, AG_]

az_ = I0 + K0 Tan[LED] Cos[ZG_] AG_

bz_ = J0 + K0 Tan[LED] Sin[ZG_] AG_

cz_ = K0 - K0 AG_

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

dz_ = az_ + 2a1 az_ t1

ez_ = bz_ + 2 b1 bz_ t1

fz_ = cz_ - t1

Move:-{az_,bz_,cz_}, Rotation: gz_, hz_, iz_

{I0 - az_, J0 - bz_, K0 - cz_}, {0, 0, 0}, {dz_ - az_, ez_ - bz_, fz_ - cz_}

gz_ = ArcTan[(I0 - az_)/(J0 - bz_)]

hz_ = ArcTan[(1 + (I0 - az_)^2/(J0 - bz_)^2)^(1/2) (J0 - bz_)/(K0 - cz_)]

({{Cos[iz_], -Sin[iz_], 0}, {Sin[iz_], Cos[iz_], 0}, {0, 0, 1}}) . ({{α}, {β}, {γ}}) == ({{0}, {M1}, {M2}})

iz_ = ArcTan[α/β]

iz_ = ArcTan[((az_ - dz_) Cos[gz_] + (ez_ - bz_) Sin[gz_])/((bz_ - ez_) Cos[gz_] Cos[hz_] + (az_ - dz_) Cos[hz_] Sin[gz_] + (fz_ - cz_) Sin[hz_])]

Rotation: -iz_, -hz_, -gz_, Move:{az_,bz_,cz_}

jz_: Refraction Angle

Refraction Line : {kz_, lz_, mz_}

u = {I0 - az_, J0 - bz_, K0 - cz_} ;

v = {dz_ - az_, ez_ - bz_, fz_ - cz_} ;

u . v/((u^2 . {1, 1, 1})^(1/2) (v^2 . {1, 1, 1})^(1/2))

jz_ = π/2 - BG_ + ArcSin[Sin[BG_]/N]

{nz_,oz_,pz_}

Second Curved Surface Point : {nz_, oz_, pz_}

Second Curved Surface Normal Line : {qz_, rz_, sz_}

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

Solve[{a2 (az_ + kz_ CG_)^2 + b2 (bz_ + lz_ CG_)^2 - ((cz_ + mz_ CG_) - c2) == 0}, {CG_}]

CG_ = (mz_ - 2 (a2 az_ kz_ + b2 bz_ lz_) - (-4 (c2 + a2 az_^2 + b2 bz_^2 - cz_) (a2 kz_^2 + b2 lz_^2) + (2 (a2 az_ kz_ + b2 bz_ lz_) - mz_)^2)^(1/2))/(2 (a2 kz_^2 + b2 lz_^2))

nz_ = az_ + kz_ CG_

oz_ = bz_ + lz_ CG_

pz_ = cz_ + mz_ CG_

qz_ = 2a2 nz_ t2 + nz_

rz_ = 2 b2 oz_ t2 + oz_

sz_ = pz_ - t2

tz_ = ArcTan[(az_ - nz_)/(bz_ - oz_)]

uz_ = ArcTan[(1 + (az_ - nz_)^2/(bz_ - oz_)^2)^(1/2) (bz_ - oz_)/(cz_ - pz_)]

vz_ = ArcTan[((nz_ - qz_) Cos[tz_] + (-oz_ + rz_) Sin[tz_])/((oz_ - rz_) Cos[tz_] Cos[uz_] + (nz_ - qz_) Cos[uz_] Sin[tz_] + (sz_ - pz_) Sin[uz_])]

Second Refraction Angle : wz_

wz_ = π/2 - DG_ + ArcSin[Sin[DG_] N]

End Point : {xz_, yz_, zz_}

xz_ = nz_ + (Cos[wz_] (Sin[tz_] Cos[uz_] Cos[vz_] + Cos[tz_] Sin[vz_]) + Sin[wz_] Sin[tz_] Sin[uz_]) EG_

yz_ = oz_ + (Cos[wz_] (Cos[tz_] Cos[uz_] Cos[vz_] - Sin[tz_] Sin[vz_]) + Sin[wz_] Cos[tz_] Sin[uz_]) EG_

zz_ = pz_ + (-Cos[wz_] Sin[uz_] Cos[vz_] + Sin[wz_] Cos[uz_]) EG_