Raindrop Geometry



Raindrop Geometry

Rainbows are less common around midday because the higher the Sun is in the sky the lower the rainbow. If the Sun is too high, then by the time raindrops are in the right position to form a rainbow they are lost in the landscape.
Red always appears on the outside edge of a primary rainbow.
Yes! The centre of a rainbow is always below the horizon when an observer looks over water or level ground.
Seen from an observer's point of view, the angle between the centre of a rainbow and the coloured arcs is called the viewing angle. In diagrams, the same angle between the rainbow axis and a line extended from an observer's eyes to the arcs of a rainbow is called the angular distance.

About the diagram

About the diagram
  • This diagram shows some of the geometry that determines the path of light through a raindrop.
  • This is a measured diagram and the angles shown are indicative of the path of a single yellow ray with a wavelength of 589.29nm travelling through water at 200C.
  • The table below provides a key to the notation used on the diagram.
  • The aim of the diagram is to:
    • highlight the geometric points used to explain changes in the direction and speed of light when it strikes the surface of a raindrop.
    • Show where and how angles are measured.
    • Show connections between angles
  • Note that:
    • The notation used in the diagram is not associated with any other system of identification.
    • Standard mathematical symbols have not been used for this diagram.
AirEarth's atmosphere is generally considered to be composed of nitrogen, oxygen, argon with trace amounts of carbon dioxide, hydrogen, methane and neon.
Average humidity, pressure and a temperature of 20C produce a refractive index of 1.000293.
RaindropAn idealised raindrop forms a geometrically perfect sphere. Although such a form is one in a million in real-life, simplified geometrical raindrops help to make sense of rainbows and reveal general rules governing why they appear.
Impact parameter scaleAn impact parameter scale is used on a ray-tracing diagram to measure the point at which incident rays strike the surface of a raindrop. Rays are given a value between 0.0 and 1.0 depending upon their point of impact.
AThe point at which an incident ray strikes a raindrop,
BThe point at which the refracted ray reflects off the inside of a raindrop.
CThe point at which the reflected ray strikes the inside of a raindrop undergoes refraction and exits towards an observer.
DThe point on the rainbow axis at which the angular distance of the deflected ray is measured.
EThe point of intersection between the original path of the incident rays prior to striking the raindrop and deflected ray.
FThe centre of the raindrop.
aEqual angles
a1Angle of incidence at point of impact of incident ray.
b1Angle of refraction at point of impact of incident ray.
bEqual angles
dAngle of deviation
y1 + y2 + y3Equal to angle of deviation (d)
zAngle of deflection
z + dEqual to 180 degrees
>>The symbol used to mark parallel lines.

Key to Rainbow geometry diagram

About raindrop geometry

An idealised raindrop forms a geometrically perfect sphere. Although such a form is one in a million in real-life,  simplified geometrical raindrops help to make sense of rainbows and reveal general rules governing why they appear.

The insights that can be gained from exploring the geometry of raindrops apply to every rainbow, whilst the rainbows we come across in everyday life demonstrate that each individual case is unique.

Don’t forget that the idea of light rays is also a way to simplify the behaviour of light:

  • The idea that light is made up of rays is so commonplace when describing and explaining rainbows that it is easily taken for granted.
  • The idea of light rays is useful when trying to model how light and raindrops produce the rainbow effects seen by an observer.
  • Light rays don’t exist in the sense that the term accurately describes a physical property of light. More accurate descriptions use terms like photons or waves.
Basics of raindrop geometry
  • A line drawing of a spherical raindrop is the starting point for exploring how raindrops produce rainbows.
  • The easiest way to represent a raindrop is as a cross-section that cuts it in half through the middle.
  • A dot or small circle can be used to mark the centre whilst the larger circle marks the circumference.
  • Marking the centre makes it easy to add lines that show the radius and diameter.
  • Marking the centre also makes it easy to add lines that are normal to the circumference.
  • A normal (or the normal) refers to a line drawn perpendicular to and intersecting another line, plane or surface.
  • A normal is used in a diagram to connect the centre with a point where a ray strikes the circumference.
  • The diameter of a circle is a line that passes through its centre and is drawn from the circumference on one side to the other.
  • The radius of a circle is a line from the centre to any point on the circumference.
  • The horizontal axis of a raindrop is a line drawn through its centre and parallel to incident light. The vertical axis intersects the horizontal at 900 and also passes through the centre point.
  • The angle at which incident light strikes the surface of a raindrop can be calculated by drawing a line that shows where an incident ray strikes a droplet and then drawing the normal. The angle of incidence is measured between them.
  • The path of light as it strikes the surface and changes direction as it is refracted at the boundary between air and water can be calculated using the Law of Refraction (Snell’s law).
  • When light is refracted as it enters a droplet it bends towards the normal.
  • The law of reflection can be used to calculate the change of direction each time light reflects off the inside surface of the raindrop.
  • When light exits a raindrop the angle of refraction is the same as when it entered but this time bends away from the normal.

Some key terms

Angle of incidence

The angle of incidence measures the angle at which incoming light strikes a surface. The angle of incidence is measured ...

Angle of reflection

The angle of reflection measures the angle at which reflected light bounces off a surface. The angle of reflection is ...

When is the best time to see a rainbow?

Rainbows are at their best early morning and late afternoon when a shower has just passed over and the Sun ...

Angle of refraction

The angle of refraction measures the angle to which light bends as it passes across the boundary between different media ...

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