Physically-Based Simulation of Rainbows

Iman Sadeghi 1 Adolfo Munoz 2 Philip Laven 3 Wojciech Jarosz 4
  Francisco Seron 2 Diego Gutierrez 2 Henrik Wann Jensen 1

1 University of California, San Diego  2 Universidad de Zaragoza  3 Horley, UK  4 Disney Research Zurich 





Abstract  In this paper we derive a physically based model for simulating rainbows. Previous techniques for simulating rainbows have used either geometric optics (ray tracing) or Lorenz-Mie theory. Lorenz-Mie theory is by far the most accurate technique as it takes into account optical effects such as dispersion, polarization, interference, and diffraction. These effects are critical for simulating rainbows accurately. Unfortunately, the shape of real raindrops is non-spherical, especially for larger raindrops, and there is no alternative theory that can handle this case. We present the first comprehensive technique for simulating the interaction of a wavefront of light with a physically-based water drop shape. Our technique is based on ray tracing extended to account for dispersion, polarization, interference, and diffraction. Our model matches Lorenz-Mie theory for spherical particles, but it also enables the accurate simulation of non-spherical particles. It can simulate many different rainbow phenomena including double rainbows and supernumerary bows. We show how the non-spherical raindrops influences the shape of the rainbows, and we provide the first comprehensive simulation of the rare twinned primary bow, which was believed to be caused by non-spherical water drops. 





  Fig. 1: Our rendering results for different types of rainbows: (a) Rainbow derived from Lorenz-Mie theory. (b) Single primary rainbow with considering the angular view of the sun. (c) Double rainbow with a flipped secondary rainbow. (d) Multiple supernumerary rainbows caused by small water drops with uniform sizes. (e) Twinned rainbow resulted from mixture of non-spherical water drops and spherical ones.
 
  (a) (b) (c) (d) (e)
Water Drop Size 0.4 mm 0.4 mm 0.4 mm 0.3 mm 0.40 mm
 0.45 mm
Phase Function Download Download I, II Download I, II Download I, II Download I, II
Download I, II
FOV 25o 25o 25o 25o 25o
Lens Type Rectilinear Rectilinear Rectilinear Rectilinear Rectilinear
Sun's Inclination N/A N/A N/A N/A 0o
Sun's Angular View 0o 0.5o 0.5o 0.5o 0.5o
Illumination D65 D65 D65 D65 D65


  Fig. 11: Comparison between our method (a) and Lorenz-Mie theory (b) for different water drop sizes. Top: Renders. Bottom: Plots of the phase function for the regions of the primary and secondary rainbows. From left to right: Water drops of radius 0.1mm, 0.2mm 0.3mm, 0.4mm and 0.5mm, respectively. Our method matches Lorenz-Mie theory for small water drops, which are spherical, but predicts different behavior as the radius increases by accounting for non-spherical drop shapes.
 
Water Drop Size 0.1 mm 0.2 mm 0.3 mm 0.4 mm 0.5 mm
Phase Function (a) Download I, II Download I, II Download I, II Download I, II Download I, II
Phase Function (b) Download Download Download Download Download
FOV 30o 30o 30o 30o 30o
Lens Type Rectilinear Rectilinear Rectilinear Rectilinear Rectilinear
Sun's Inclination N/A N/A N/A N/A 0o
Sun's Angular View 0.5o 0.5o 0.5o 0.5o 0.5o
Illumination D65 D65 D65 D65 D65


  Fig. 12 (top row): The inserts in these images show how our model can reproduce the rainbows in the underlying photographs. Only the background color of the insert has been matched to the specific photograph. From left to right: double rainbow (background reproduced with permission (c) Les Cowley - http://www.atoptics.co.uk/rainbows/adband.htm), full double rainbow (background reproduced with permission  (c) Karl Kaiser - http://home.eduhi.at/member/nature) and supernumerary bows.
 
Water Drop Size 0.4 mm 0.4 mm 0.3 mm
Phase Function Download I, II Download I, II Download I, II
FOV 20o 100o 30o
Lens Type Rectilinear Fisheye Rectilinear
Sun's Inclination N/A N/A N/A
Sun's Angular View 0.5o 0.5o 0.5o
Background Color (107,114,118) (183,202,212) (172,172,172)
Intensity 55% 100% 90%
Illumination D65 D65 D65


  Fig. 12 (bottom row): The inserts in these images show how our model can reproduce the rainbows in the underlying photographs. Only the background color of the insert has been matched to the specific photograph. From left to right: Multiple supernumerary bows, cloud bow (background reproduced with permission (c) Les Cowley - http://www.atoptics.co.uk/rainbows/cldbow.htm) and red bow.
 
 
Water Drop Size 0.3 mm 0.1 mm 0.4 mm
Phase Function Download I, II Download I, II Download I, II
FOV 10o 100o 30o
Lens Type Rectilinear Fisheye Rectilinear
Sun's Inclination N/A N/A N/A
Sun's Angular View 0.5o 0.5o 0.5o
Background Color (69,99,112) (141,180,223) (154,83,58)
Intensity 80% 60% 80%
Illumination D65 D65

D65

D65 + Rayleigh



  Fig. 13: Comparison of renderings of rainbows owed to different water drop radii between Lorenz-Mie (left region on each image) and our solution (right region of each image). As the 0.4mm radius water drop is spherical, both algorithms lead to equal phase functions. As the particle gets bigger, the geometry becomes non-spherical and therefore Lorenz-Mie is unable to simulate it, while our solution takes it into account. Notice, also, that the variation on the secondary rainbow is quite unnoticeable compared to the variation on the primary rainbow, in agreement with the formation of twinned rainbows.
 
           
Water Drop Size 0.4 mm 0.5 mm 0.6 mm 0.7 mm 0.8 mm
Phase Function (right) Download Download Download Download Download
Phase Function (left) Download I, II Download I, II Download I, II Download I, II Download I, II
FOV 120o 120o 120o 120o 120o
Lens Type Fisheye Fisheye Fisheye Fisheye Fisheye
Sun's Inclination N/A 25o 25o 25o 25o
Sun's Angular View 0.5o 0.5o 0.5o 0.5o 0.5o
Illumination D65 D65 D65 D65 D65


   Fig. 14: The effect of different water drop radii on the apparent geometry of the rainbow.
 
Water Drop Size 0.4 mm 0.5 mm 0.6 mm 0.7 mm
Phase Function Download I, II Download I, II Download I, II Download I, II
FOV 120o 120o 120o 120o
Lens Type Fisheye Fisheye Fisheye Fisheye
Sun's Inclination N/A 0o 0o 0o
Sun's Angular View 0.5o 0.5o 0.5o 0.5o
Illumination D65 D65 D65 D65



   Fig. 15: The effect of the inclination of the sun on a non-spherical water drop (radius 0.5mm) alters the apparent geometry of the rainbow. This would not be the case for spherical water drops. The gray line indicates the horizon line.
 
  (a) (b) (c)
Water Drop Size 0.5 mm 0.5 mm 0.5 mm
Phase Function Download I, II Download I, II Download I, II
FOV 120o 120o 120o
Lens Type Rectilinear Rectilinear Rectilinear
Sun's Inclination 0o 20o 40o
Sun's Angular View 0.5o 0.5o 0.5o
Illumination D65 D65

D65



   Fig. 16: The effect of the different polarization states on the perception of a rainbow from 0.5mm radius non-spherical water drop.
       
  (a)  (b)  (c) 
Water Drop Size 0.5 mm 0.5 mm 0.5 mm
Phase Function Download I, II Download I, II Download I, II
FOV 120o 120o 120o
Lens Type Rectilinear Rectilinear Rectilinear
Sun's Inclination 0o 0o 0o
Sun's Angular View 0.5o 0.5o 0.5o
Illumination D65 D65

D65

Polarization Circular Linear 90o Linear 0o



  Fig. 17: Left: Photograph of a twinned rainbow, reproduced with permission (c) Benjamin Khne (www.nachtwolke.de/temp/regenbogen2.htm). Right: Twinned rainbow simulated using our algorithm, generated from a two showers of 0.4mm and 0.45mm radius water drops, respectively. 
     
Water Drop Size ________  0.40 mm
0.45 mm
Phase Functions   Download I, II
Download I, II
FOV   30o
Lens Type   Rectilinear
Sun's Inclination    0o
Sun's Angular View   0.5o
Background Color   (146,151,153)
Intensity 50%
Illumination   D65



Phase Function Specifications:

1D Lorenz-Mie Theory Phase Functions: File Format

2D Raytraced Phase Functions: File Format