Physically Based Simulation of Rainbows

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

¹University of California, San Diego ² Universidad de Zaragoza ³ Horley, UK ⁴Disney 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	25^o	25^o	25^o	25^o	25^o
Lens Type	Rectilinear	Rectilinear	Rectilinear	Rectilinear	Rectilinear
Sun's Inclination	N/A	N/A	N/A	N/A	0^o
Sun's Angular View	0^o	0.5^o	0.5^o	0.5^o	0.5^o
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	30^o	30^o	30^o	30^o	30^o
Lens Type	Rectilinear	Rectilinear	Rectilinear	Rectilinear	Rectilinear
Sun's Inclination	N/A	N/A	N/A	N/A	0^o
Sun's Angular View	0.5^o	0.5^o	0.5^o	0.5^o	0.5^o
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	20^o	100^o	30^o
Lens Type	Rectilinear	Fisheye	Rectilinear
Sun's Inclination	N/A	N/A	N/A
Sun's Angular View	0.5^o	0.5^o	0.5^o
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	10^o	100^o	30^o
Lens Type	Rectilinear	Fisheye	Rectilinear
Sun's Inclination	N/A	N/A	N/A
Sun's Angular View	0.5^o	0.5^o	0.5^o
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	120^o	120^o	120^o	120^o	120^o
Lens Type	Fisheye	Fisheye	Fisheye	Fisheye	Fisheye
Sun's Inclination	N/A	25^o	25^o	25^o	25^o
Sun's Angular View	0.5^o	0.5^o	0.5^o	0.5^o	0.5^o
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	120^o	120^o	120^o	120^o
Lens Type	Fisheye	Fisheye	Fisheye	Fisheye
Sun's Inclination	N/A	0^o	0^o	0^o
Sun's Angular View	0.5^o	0.5^o	0.5^o	0.5^o
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	120^o	120^o	120^o
Lens Type	Rectilinear	Rectilinear	Rectilinear
Sun's Inclination	0^o	20^o	40^o
Sun's Angular View	0.5^o	0.5^o	0.5^o
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	120^o	120^o	120^o
Lens Type	Rectilinear	Rectilinear	Rectilinear
Sun's Inclination	0^o	0^o	0^o
Sun's Angular View	0.5^o	0.5^o	0.5^o
Illumination	D65	D65	D65
Polarization	Circular	Linear 90^o	Linear 0^o

	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		30^o
Lens Type		Rectilinear
Sun's Inclination		0^o
Sun's Angular View		0.5^o
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