Tangent Plane

f (x, y, z) == c

r (t) == x (t) i + y (t) j + z (t) k

r (t_0) = x_0 = (x_0, y_0, z_0)

(x - x_0) ∂f/∂x (x_0, y_0, z_0) + (y - y_0) ∂f/∂y (x_0, y_0, z_0) + (z - z_0) ∂f/∂z (x_0, y_0, z_0) == 0

Variable

Refractive Lens

Beam Point : {I0, J0, K0}

LED Angle : LED

Refraction Index : N

First Curved Surface Variable : a1, b1

Second Curved Surface Variable : c2

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

Beam Point : {I0, J0, K0}

First Curved Surface Point : {az_, bz_, cz_}

First Curved Surface Normal Line : {dz_, ez_, fz_}

Surface Equation : x^2/a1^2 + z^2/b1^2 == 1

Solve[ (I0 + K0 AG_ Tan[LED] Cos)^2/a1^2 + (K0 - K0 AG_)^2/b1^2 == 1, AG_]

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

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

cz_ = K0 - K0 AG_

dz_ = az_ + (2 az_)/a1^2 t1

ez_ = bz_

fz_ = cz_ + (2 cz_)/b1^2t1

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

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

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

π/2 - ArcCos[-u . v/((u^2 . {1, 1, 1})^(1/2) (v^2 . {1, 1, 1})^(1/2))]

First Refraction Angle : jz_

jz_ = π/2 - BG_ + ArcSin[Sin[BG_]/N]

First Refraction Line : {kz_, lz_, mz_}

kz_ = Cos[jz_] (Sin[gz_] Cos[hz_] Cos[iz_] + Cos[gz_] Sin[iz_]) + Sin[jz_] Sin[gz_] Sin[hz_]

lz_ = Cos[jz_] (Cos[gz_] Cos[hz_] Cos[iz_] - Sin[gz_] Sin[iz_]) + Sin[jz_] Cos[gz_] Sin[hz_]

mz_ = -Cos[jz_] Sin[hz_] Cos[iz_] + Sin[jz_] Cos[hz_]

{nz_,oz_,pz_}

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

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

(z - c2) == 0

Solve[{(cz_ + mz_ CG_) - c2 == 0}, {CG_}]

CG_ = (c2 - cz_)/mz_

nz_ = az_ + kz_ CG_

oz_ = bz_ + lz_ CG_

pz_ = cz_ + mz_ CG_

qz_ = nz_

rz_ = oz_

sz_ = pz_ - 1

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_

Induction - A Refractive Lens

Surface : x^2/a1^2 + z^2/b1^2 == 1

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

n = Sin[φ]/Sin[δ]

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

{i0 - i1, j0 - j1, k0 - k1}, {0, 0, 0}, {2 i1/Power[a1, 2] t1, 0, 2 k1/Power[b1, 2] t1}

R: x,y plane α1, y,z plane β1

α1 = ArcTan[(i0 - i1)/(j0 - j1)]

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

({{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[(b^2 i1 Cos[α1])/(b^2 i1 Cos[β1] Sin[α1] - a^2 k1 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]] ;