Bild:Rainbow formation.png
Wikipedia
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Diagram showing how primary and secondary rainbows are formed. Details follow soon!
Legend:
- Spherical droplet
- Places where internal reflection of the light occurs
- Primary rainbow
- Places where refraction of the light occurs
- Secondary rainbow
- Incoming beams of white light
- Path of light contributing to primary rainbow
- Path of light contributing to secondary rainbow
- Observer
- Region forming the primary rainbow
- Region forming the secondary raimbow
- Zone in the atmosphere holding countless tiny spherical droplets
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Three raytraced images, created using POV-Ray (see http://www.povray.org/), put together and numbers and black lines added in a graphics software package afterwards. Below is the "code" to create the diagrams aboce, and the landscape below, respectively:
POV-Ray "code" for the diagrams showing light propagation inside the droplets:
/* ================================================= Light propagation in a spherical droplet, forming primary and secondary rainbows ------------------------------------------------- Created by Søren Peo Pedersen - see my user page at http://da.wikipedia.org/wiki/Bruger:Peo ================================================= */ #declare PrimaryRays=yes; // Use "no" for light propagation in secondary rainbow, // or "yes" for light propagation in primary rainbow. #if (PrimaryRays) // Rendition of light propagation for primary rainbow's light propagation: #local pgmSpectrum1=pigment { // Pigment for first spectrum "fan-out" radial color_map { [0.0000 color rgbt <.5,0,1,1>] [0.1429 color rgbt <.5,0,1,.5>] [0.2857 color rgbt < 0,0,1,.5>] [0.4286 color rgbt < 0,1,1,.5>] [0.5714 color rgbt < 0,1,0,.5>] [0.7143 color rgbt < 1,1,0,.5>] [0.8571 color rgbt < 1,0,0,.5>] [1.0000 color rgbt < 1,0,0,1>] } frequency 200 rotate <90,0,-13.1> translate <-3.806586,1.70909106,0> scale 1/1.67 } #local pgmSpectrum2=pigment { // Pigment for second spectrum "fan-out" radial color_map { [0.0000 color rgbt <1,1,1,1>] [0.5000 color rgbt <1,1,1,0>] [1.0000 color rgbt <1,1,1,1>] } frequency 200 rotate <90,0,-13.1> translate <-3.806586,1.70909106,0> scale 1/1.67 } union { difference { // Incoming white light beam box {<-100,.67,-.0001>,<0,.83,.0001>} sphere {0,1} pigment { gradient y color_map { [0.0 color rgbt<1,1,1,1>] [0.2 color rgbt<1,1,1,.9>] [0.3 color rgbt<1,1,1,.5>] [0.5 color rgbt<1,1,1,0>] [0.7 color rgbt<1,1,1,.5>] [0.8 color rgbt<1,1,1,1,.9>] [1.0 color rgbt<1,1,1,1>] } scale .16 translate <0,.67,0> } finish { ambient 1 diffuse 0 } no_shadow no_reflection rotate <0,0,-20.825> } difference { // First "rainbow-colored" beam inside droplet cylinder {<0,0,-.0001>,<0,0,.0001>,1} // Disc w. same diameter as droplet plane {<0,1,0>,0 rotate <0,0,-14.9> translate <-0.7141428,.7,0>} // Cut-off plane {<0,-1,0>,0 rotate <0,0,-13.1> translate <-0.6,.8,0>} // boundaries pigment { gradient x pigment_map { [0 pgmSpectrum2] [1 pgmSpectrum1] } scale 1.67 translate <-.7,0,0> } finish { ambient 1 diffuse 0 } rotate <0,0,-20.825> no_shadow no_reflection } merge { // Second "rainbow-colord" beam inside droplet - "crossing over" itself difference { // The portion to the left of the "cross-over" point cylinder {<0,0,.0003>,<0,0,.0002>,1} // Droplet-sized disc plane {<0,-1,0>,0 rotate <0,0,45> translate <1,.12,0>} // Cut-off plane {<0,1,0>,0 rotate <0,0,25> translate <1,-.09,0>} // boundaries } difference { // The portion to the right of the "cross-over" point cylinder {<0,0,.0003>,<0,0,.0002>,1} // Droplet-sized disc plane {<0,1,0>,0 rotate <0,0,45> translate <1,.12,0>} // Cut-off plane {<0,-1,0>,0 rotate <0,0,25> translate <1,-.09,0>} // boundaries } pigment { radial color_map { [0.0000 color rgbt <.5,0,1,1>] [0.1429 color rgbt <.5,0,1,.5>] [0.2857 color rgbt < 0,0,1,.5>] [0.4286 color rgbt < 0,1,1,.5>] [0.5714 color rgbt < 0,1,0,.5>] [0.7143 color rgbt < 1,1,0,.5>] [0.8571 color rgbt < 1,0,0,.5>] [1.0000 color rgbt < 1,0,0,1>] } frequency 18 rotate <90,0,-35> translate <.606514871,-.273485129,0> } finish { ambient 1 diffuse 0 } no_shadow no_reflection } difference { // The "rainbow-colored light" leaving the droplet at lower left corner: box {<-100,0,-.0001>,<0,10,.0001> rotate <0,0,32.825> translate <-.113,-.994,0>} plane {<0,-1,0>,0 rotate <0,0,8.825> translate <-.565,-.805,0>} cylinder {<0,0,-1>,<0,0,1>,1} // Cut away part that would fall inside droplet pigment { radial color_map { [0.0000 color rgbt < 1,0,0,1>] [0.1429 color rgbt < 1,0,0,0>] [0.2857 color rgbt < 1,1,0,0>] [0.4286 color rgbt < 0,1,0,0>] [0.5714 color rgbt < 0,1,1,0>] [0.7143 color rgbt < 0,0,1,0>] [0.8571 color rgbt <.5,0,1,0>] [1.0000 color rgbt <.5,0,1,1>] } frequency 15 rotate <-90,0,20.825> translate <0.416125,-.65268,0> } finish {ambient 1 diffuse 0} no_shadow no_reflection } } #else // Rendition of light propagation for secondary rainbow's light propagation: #local pgmSpectrum1=pigment { // Pigment for "rainbow-colored" part of first beam inside droplet radial color_map { [0.0000 color rgbt < 1,0,0,1>] [0.1429 color rgbt < 1,0,0,.5>] [0.2857 color rgbt < 1,1,0,.5>] [0.4286 color rgbt < 0,1,0,.5>] [0.5714 color rgbt < 0,1,1,.5>] [0.7143 color rgbt < 0,0,1,.5>] [0.8571 color rgbt <.5,0,1,.5>] [1.0000 color rgbt <.5,0,1,1>] } frequency 200 rotate <90,0,-.9> translate <-3.7364,-0.69606,0> scale 1/1.7 } #local pgmSpectrum2=pigment { // Pigment for white part of first "rainbow-colored" beam inside droplet radial color_map { [0.0000 color rgbt <1,1,1,1>] [0.5000 color rgbt <1,1,1,0>] [1.0000 color rgbt <1,1,1,1>] } frequency 200 rotate <90,0,-.9> translate <-3.8364,-0.69606,0> scale 1/1.7 } union { difference { // White-to "rainbow-colored" beam inside droplet cylinder {<0,0,-.0001>,<0,0,.0001>,1} plane {<0, 1,0>,0 rotate <0,0,-.9> translate <-.657,-.757,0>} plane {<0,-1,0>,0 rotate <0,0, .9> translate <-.777,-.637,0>} pigment { gradient x pigment_map { [0 pgmSpectrum2] [1 pgmSpectrum1] } scale 1.7 translate <-.8,0,0> } finish {ambient 1 diffuse 0} no_shadow no_reflection } merge { difference { // Lower part of self-crossing beam at right-ahdn side inside droplet cylinder {<0,0,.0002>,<0,0,.0003>,1} plane {< 1,0,0>,0 rotate <0,0,-6 > translate <.622,-.692,0>} plane {<-1,0,0>,0 rotate <0,0, 6 > translate <.787,-.727,0>} } difference { // Upper part of self-crossing beam at right-ahdn side inside droplet cylinder {<0,0,.0002>,<0,0,.0003>,1} plane {<-1,0,0>,0 rotate <0,0,-6 > translate <.622,-.692,0>} plane {< 1,0,0>,0 rotate <0,0, 6 > translate <.787,-.727,0>} } pigment { radial color_map { [0.0000 color rgbt < 1,0,0,1>] [0.1429 color rgbt < 1,0,0,.5>] [0.2857 color rgbt < 1,1,0,.5>] [0.4286 color rgbt < 0,1,0,.5>] [0.5714 color rgbt < 0,1,1,.5>] [0.7143 color rgbt < 0,0,1,.5>] [0.8571 color rgbt <.5,0,1,.5>] [1.0000 color rgbt <.5,0,1,1>] } frequency 30 rotate <90,0,84> translate <.702661,0.075435,0> } finish {ambient 1 diffuse 0} no_shadow no_reflection } merge { // Right-hand part of self-crossing beam at top of droplet difference { cylinder {<0,0,.0004>,<0,0,.0005>,1} plane {<0,-1,0>,0 rotate <0,0, 6 > translate <.637,.777,0>} plane {<0, 1,0>,0 rotate <0,0,-6 > translate <.772,.642,0>} } difference { // Left-hand part of self-crossing beam at top of droplet cylinder {<0,0,.0004>,<0,0,.0005>,1} plane {<0, 1,0>,0 rotate <0,0, 6 > translate <.637,.777,0>} plane {<0,-1,0>,0 rotate <0,0,-6 > translate <.772,.642,0>} } pigment { radial color_map { [0.0000 color rgbt < 1,0,0,1>] [0.1429 color rgbt < 1,0,0,.5>] [0.2857 color rgbt < 1,1,0,.5>] [0.4286 color rgbt < 0,1,0,.5>] [0.5714 color rgbt < 0,1,1,.5>] [0.7143 color rgbt < 0,0,1,.5>] [0.8571 color rgbt <.5,0,1,.5>] [1.0000 color rgbt <.5,0,1,1>] } frequency 30 rotate <90,0,-6> translate <.06228,0.716595,0> } finish {ambient 1 diffuse 0} } no_shadow no_reflection rotate <0,0,5> } difference { // "Rainbow-colored" beam leaving the droplet box {<-10,0,.0002>,<.3,10,.0003> rotate <0,0,38.325> translate <-.827,.557,0>} plane {<0,-1,0>,0 rotate <0,0,14.325> translate <-.697,.717,0>} cylinder {<0,0,-1>,<0,0,1>,1} pigment { radial color_map { [0.0000 color rgbt < 1,0,0,1>] [0.1429 color rgbt < 1,0,0,0>] [0.2857 color rgbt < 1,1,0,0>] [0.4286 color rgbt < 0,1,0,0>] [0.5714 color rgbt < 0,1,1,0>] [0.7143 color rgbt < 0,0,1,0>] [0.8571 color rgbt <.5,0,1,0>] [1.0000 color rgbt <.5,0,1,1>] } frequency 15 rotate <90,0,-21.3> translate <-0.59003,0.744316,0> } finish {ambient 1 diffuse 0} no_shadow no_reflection } difference { // Incoming white light beam box {<-100,-1,-.0001>,<0,-.87,.0001>} sphere {0,1} pigment { gradient y color_map { [0.0 color rgbt<1,1,1,1>] [0.2 color rgbt<1,1,1,.9>] [0.3 color rgbt<1,1,1,.5>] [0.5 color rgbt<1,1,1,0>] [0.7 color rgbt<1,1,1,.5>] [0.8 color rgbt<1,1,1,1,.9>] [1.0 color rgbt<1,1,1,1>] } scale .13 translate <0,-1,0> } finish { ambient 1 diffuse 0 } no_shadow no_reflection rotate <0,0,-20.825> } #end difference { // Droplet - actually a hemisphere for appearance reasons sphere {0,1} plane {<0,0,1>,.001} pigment {color rgbt<.7,.8,1,.7>} finish { phong 1 phong_size 80 reflection .6 metallic } } camera { // Viewpoint up <0,1,0> right <1,0,0> // Assume square-shaped image format location <-.2,0,-2.5> look_at <-.2,0,0> } light_source { // Light <-10,0,-10> color rgb 1.5 rotate <0,0,-20.825> }
POV-Ray "code" for the landscape with rainbows, observer and "droplet cloud":
/* ================================================= "Landscape" showing the formation of two rainbows ------------------------------------------------- Created by Søren Peo Pedersen - see my user page at http://da.wikipedia.org/wiki/Bruger:Peo ================================================= */ plane { // Flat terrain in the foreground <0,1,0>,0 pigment {color rgb <.65,.7,.6>} finish {ambient .4} } plane {<0,0,-1>,0 // Invisible surface carrying the grey "cloud" pigment { cylindrical color_map { [0.0 color rgbt<1,1,1,1>] [0.5 color rgbt<.7,.7,.7,0>] [1.0 color rgbt<.5,.5,.5,0>] } rotate <90,0,0> scale 7 translate <8,8,0> turbulence .3 } finish {ambient 1 diffuse 0} } sky_sphere { // Provides a sky with light blue color gradient pigment { gradient y color_map { [0 color rgb <.1,.3,.2>] [.5 color rgb <.65,.7,.6>] [.5 color rgb <.8,.9,1>] [1 color rgb <.2,.5,1>] } translate -.5 scale 2 turbulence .1 } } union { // The observer in the lower, left-hand corner sphere {0,1 scale <.2,.8,.2> pigment {color rgb <0,0,1>} finish {ambient .5}} sphere {<0,1,0>,.2 pigment {color rgb <1,.7,.4>} finish {ambient .5}} translate <-6.5,0,-4> } #macro ColorFunction(Plads) // Creates a color from the spectrum (from 0=red to 1=purple) #local U=4.9999*(Plads-int(Plads*4.9999)/4.9999); #switch (Plads) #range (0.0,0.2) color rgb < 1 , U , 0 > #break #range (0.2,0.4) color rgb <1-U, 1 , 0 > #break #range (0.4,0.6) color rgb < 0 , 1 , U > #break #range (0.6,0.8) color rgb < 0 ,1-U, 1 > #break #range (0.8,1.0) color rgb < U , 0 , 1 > #break #end #end #macro Beam(Num,Primay) // Renders a white incoming beam, and a colored "returned" beam #local R=seed(145*Num); #local Lgd=11+rand(R)*3; #if (Primay) #local Vinkel=42.3-20.825-1.7*Num; #else #local Vinkel=50.7-20.825+2.9*Num; #end merge { cylinder {0,<Lgd,0,0>,.01 pigment {ColorFunction(Num)} finish {ambient 1 diffuse 0} rotate <0,0,Vinkel> translate <-6.5,1,-4> no_shadow } cylinder { <-100,0,0>,0,.01 rotate <0,0,-20.825> translate <-6.5+Lgd*cos(radians(Vinkel)),1+Lgd*sin(radians(Vinkel)),-4>} pigment {color rgb 1} finish {ambient 1 diffuse 0} } #end // Light beams forming the primary rainbow: #object {Beam(0.00,yes)} #object {Beam(0.25,yes)} #object {Beam(0.50,yes)} #object {Beam(0.75,yes)} #object {Beam(1.00,yes)} // Light beams forming the secondary rainbow: #object {Beam(0.00,no)} #object {Beam(0.25,no)} #object {Beam(0.50,no)} #object {Beam(0.75,no)} #object {Beam(1.00,no)} // Totally transparent pigment set of primary and secondary arc #local ArcPgmt0=pigment { cylindrical color_map { [0.0 color rgbt<1,1,1,1>] [0.00001 color rgbt<1,0,1,1>] [0.022 color rgbt<0,0,1,1>] [0.044 color rgbt<0,1,1,1>] [0.066 color rgbt<0,1,0,1>] [0.088 color rgbt<1,1,0,1>] [0.11 color rgbt<1,0,0,1>] [0.11 color rgbt<1,1,1,1>] [0.39 color rgbt<1,1,1,1>] [0.39 color rgbt<1,0,0,1>] [0.40 color rgbt<1,1,0,1>] [0.41 color rgbt<0,1,0,1>] [0.42 color rgbt<0,1,1,1>] [0.43 color rgbt<0,0,1,1>] [0.44 color rgbt<1,0,1,1>] [0.44 color rgbt<1,1,1,1>] [1.0 color rgbt<1,1,1,1>] } } // Slightly non-transparent pigment set of primary and secondary arc #local ArcPgmt1=pigment { cylindrical color_map { [0.0 color rgbt<1,1,1,1>] [0.00001 color rgbt<1,0,1,.7>] [0.022 color rgbt<0,0,1,.7>] [0.044 color rgbt<0,1,1,.7>] [0.066 color rgbt<0,1,0,.7>] [0.088 color rgbt<1,1,0,.7>] [0.11 color rgbt<1,0,0,0>] [0.11 color rgbt<1,1,1,1>] [0.39 color rgbt<1,1,1,1>] [0.39 color rgbt<1,0,0,.2>] [0.40 color rgbt<1,1,0,.2>] [0.41 color rgbt<0,1,0,.2>] [0.42 color rgbt<0,1,1,.2>] [0.43 color rgbt<0,0,1,.2>] [0.44 color rgbt<1,0,1,.2>] [0.44 color rgbt<1,1,1,1>] [1.0 color rgbt<1,1,1,1>] } } // Surface carrying the two rainbows in front of the grey "cloud" plane {<-1,0,0>,0 pigment { radial pigment_map { [0.0 ArcPgmt1] [0.2 ArcPgmt0] [0.8 ArcPgmt0] [1.0 ArcPgmt1] } rotate <0,0,90> scale <4.18495,4.18495,7> translate <0,1,-4> } finish {ambient 1 diffuse 0} hollow no_shadow } // Viewpoint camera { location <-5,5,-15> look_at <1,4,0> } // "Infinitely" remote light source with parallel rays: light_source { <-1000,0,0> color rgb 1.5 rotate <0,0,-20.825> parallel }
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