# Viewing angle

The viewing angle of a rainbow is the angle between a line extended from an observer’s eyes to a bow’s centre point and a second line extended out towards the coloured arcs.

• Viewing angle, angular distance and angle of deflection all produce the same value measured in degrees.
###### Viewing angle and rainbows
• Viewing angle refers to the number of degrees through which an observer must move their eyes or turn their head.
• On the vertical plane, the viewing angle is a measure of how far an observer must raise their eyes or head to look from the centre of a rainbow out to the coloured arcs.
• On the horizontal plane, the viewing angle is a measure of how far an observer must look from the centre out to the side to see the coloured arcs.
###### Viewing angle and raindrops
• The idea of a viewing angle for a specific raindrop within a rainbow is nonsense really because they are too small to see. However, the viewing angle for a specific raindrop can be derived from the angle of deflection.
• The angle of deflection measures the degree to which a ray striking a raindrop is bent back on itself in the process of refraction and reflection towards an observer.
• Of all the rays deflected towards an observer by a single raindrop, there is always one that produces the most intense impression of colour for an observer at any specific moment. It is often called the rainbow ray.
• The term rainbow ray refers to the path taken by the deflected ray that produces the most intense colour experience for any particular wavelength of light passing through a raindrop.
• A ray-tracing diagram can calculate which of the rays of a specific wavelength, exiting a raindrop is the rainbow ray.
• If an observer could watch a single raindrop as it falls, they would see its viewing angle decrease and its colour change from red, through intermediate colours, to violet. With each change of viewing angle, colour and wavelength the exact trajectory of the rainbow ray must be recalculated.
###### Find the viewing angle
• To find the viewing angle as you look at a rainbow, trace two lines away from your eyes, one to the centre of the rainbow, and the other to any point on the coloured arcs. The viewing angle is between those two lines, which intersect within the lenses of your eyes.
• If you are not sure where the centre of the rainbow is, imagine extending the ends of the bow until they meet and form a circle. The centre (the anti-solar point) is right in the middle.
• For atmospheric rainbows seen from the ground, the anti-solar point is always below the horizon.
• The coloured arcs of a rainbow form the circumference of circles (discs or cones) and share centres at their anti-solar point.
• The viewing angle is the same whatever point is selected on the circumference of the circular arcs of the rainbow visible above the horizon.
• The viewing angle for a primary bow is between approx. 40.70 and 42.40 from its centre. The exact angle depends on which rainbow colour is selected.
• The viewing angle for a secondary bow is at an angle of between approx. 50.40 and 53.40 when you are looking outwards from its centre.
• The viewing angle can be calculated for any specific colour within a rainbow.
• The centre of a rainbow is always on its axis. The rainbow axis is an imaginary straight line that connects the light source, observer and anti-solar point.
• Considered from an observer’s viewpoint, it is clear that all incident rays seen by an observer run parallel with each other as they approach a raindrop.
• Most of the observable incident rays that strike a raindrop follow paths that place them outside the range of possible viewing angles. The unobserved rays are all deflected towards the centre of a rainbow.
• The viewing angles for all rainbow colours are constants determined by the laws of refraction and reflection.
• The elevation of the Sun, the location of the observer and exactly where rain is falling are all variables that determine where a rainbow will appear.
###### Viewing angle, angular distance and angle of deflection
• The term viewing angle refers to the number of degrees through which an observer must move their eyes or turn their head to see a specific colour within the arcs of a rainbow.
• The term angular distance refers to the same measurement when shown in side elevation on a diagram.
• The angle of deflection measures the degree to which a ray striking a raindrop is bent back on itself in the process of refraction and reflection towards an observer.
• The term rainbow ray refers to the path taken by the deflected ray that produces the most intense colour experience for any particular wavelength of light passing through a raindrop.
• The term angle of deviation measures the degree to which the path of a light ray is bent back by a raindrop in the course of refraction and reflection towards an observer.
• In any particular example of a ray of light passing through a raindrop, the angle of deviation and the angle of deflection are directly related to one another and together add up to 1800.
• The angle of deviation is always equal to 1800 minus the angle of deflection. So clearly the angle of deflection is always equal to 1800 minus the angle of deviation.
• In any particular example, the angle of deflection is always the same as the viewing angle because the incident rays of light that form a rainbow are all approaching on a trajectory running parallel with the rainbow axis.