# Raindrop Geometry

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**This is one of a set of almost 40 diagrams exploring Rainbows.**

**Each diagram appears on a separate page and is supported by a full explanation.**

- Follow the links embedded in the text for definitions of all the key terms.
- For quick reference don’t miss the summaries of key terms further down each page.

## Description

#### Raindrop Geometry

###### TRY SOME QUICK QUESTIONS AND ANSWERS TO GET STARTED

#### 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 20
^{0}C. - 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.

Label | Description |
---|---|

Air | Earth'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. |

Raindrop | 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. |

Impact parameter scale | An 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. |

A | The point at which an incident ray strikes a raindrop, |

B | The point at which the refracted ray reflects off the inside of a raindrop. |

C | The point at which the reflected ray strikes the inside of a raindrop undergoes refraction and exits towards an observer. |

D | The point on the rainbow axis at which the angular distance of the deflected ray is measured. |

E | The point of intersection between the original path of the incident rays prior to striking the raindrop and deflected ray. |

F | The centre of the raindrop. |

a | Equal angles |

a_{1} | Angle of incidence at point of impact of incident ray. |

b_{1} | Angle of refraction at point of impact of incident ray. |

b | Equal angles |

d | Angle of deviation |

y_{1} + y_{2} + y_{3} | Equal to angle of deviation (d) |

z | Angle of deflection |

z + d | Equal 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 90
^{0}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

Rainbows are at their best early morning and late afternoon when a shower has just passed over and the Sun is illuminating the curtain of raindrops formed on the trailing edge of the falling rain.

The angle of reflection is the angle between the incident light ray and the reflected light ray, both measured from an imaginary line called the normal.

- According to the law of reflection, the angle of incidence (the angle between the incident ray and the normal) is always equal to the angle of reflection.
- The angle of reflection is measured between the reflected ray of light and an imaginary line perpendicular to the surface, known as the normal.
- In optics, the normal is a straight line drawn on a ray-tracing diagram at a 90º angle (perpendicular) to the boundary where two different media meet.
- Expressed more formally, in optics, the normal is a geometric construct, a line drawn perpendicular to the interface between two media at the point of contact. This conceptually defined reference line is crucial for characterizing various light-matter interactions, such as reflection, refraction, and absorption.
- If the boundary between two media is curved, the normal is drawn perpendicular to the tangent to that point on the boundary.
- Reflection can be diffuse (when light reflects off rough surfaces) or specular (in the case of smooth, shiny surfaces), affecting the direction of reflected rays.

- The angle of reflection measures the angle at which reflected light bounces off a surface.
- The angle of reflection is measured between a ray of light which has been reflected off a surface and an imaginary line called the
*normal*. - See this diagram for an explanation: Reflection of a ray of light
- In optics, the
*normal*is a line drawn on a ray diagram perpendicular to, so at a right angle to (90^{0}), the boundary between two media. - If the boundary between the media is curved then the normal is drawn perpendicular to the boundary.

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

- The angle of refraction is measured between the bent ray and an imaginary line called the
*normal*. - In optics, the
*normal*is a line drawn on a ray diagram perpendicular to, so at a right angle to (90^{0}), the boundary between two media. - Snell’s law is a formula used to describe the relationship between the angle of incidence and the angle of refraction when light crosses the boundary between transparent media, such as water and air or water and glass.

- When discussing rainbows, angular distance is the angle between the line from the observer to the centre of the rainbow (rainbow axis) and the line from the observer to a specific colour within the arc of a rainbow.
- See this diagram for an explanation: Angular distance & Raindrop colour
- Angular distance is one of the angles measured on a ray-tracing diagram that illustrates the sun, an observer, and a rainbow from a side view.
- Think of angular distance as the angle between the line to the centre of a rainbow down which an observer looks and the line to a specific colour in its arc. The red light is deviated by about 42.4° and violet light by about 40.7°.

Rainbow colours are the bands of colour seen in rainbows and in other situations where visible light separates into its component wavelengths and the spectral colours corresponding with each wavelength become visible to the human eye.

- The rainbow colours (ROYGBV) in order of wavelength are red (longest wavelength), orange, yellow, green, blue and violet (shortest wavelength).
- The human eye, and so human perception, is tuned to the visible spectrum and so to spectral colours between red and violet. It is the sensitivity of the eye to this small part of the electromagnetic spectrum that results in the perception of colour.
- Defining rainbow colours is a question more closely related to the relationship between perception and language than to anything to do with physics or scientific accuracy.
- Even the commonplace colours associated with the rainbow defy easy definition. They are concepts we generally agree on, but are not strictly defined by anything in the nature of light itself.
- Whilst the visible spectrum and spectral colour are both determined by wavelength and frequency it is our eyes and brains that interpret these and create our perceptions after a lot of processing.

The angle of incidence refers to the angle at which incoming light strikes a surface and is measured between a ray of incoming light and an imaginary line called the normal.

- In optics, the
*normal*is a line drawn on a ray diagram perpendicular to, so at a right angle to (90^{0}), the boundary between two media. - The angle at which incident light from the Sun or a light bulb strikes a surface can affect the outcome. For instance, when incident light hits a mirror, the angle of incidence determines the angle of reflection.

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

- When light is travelling towards something it is said to be incident to that surface or object.
- The angle of incidence is measured between a ray of incoming light and an imaginary line called the
*normal*. - In optics, the
*normal*is a line drawn on a ray diagram perpendicular to, so at a right angle to (90^{0}), the boundary between two media. - Expressed more formally, in optics, the normal is a geometric construct, a line drawn perpendicular to the interface between two media at the point of contact. This conceptually defined reference line is crucial for characterizing various light-matter interactions, such as reflection, refraction, and absorption.
- Incident light may have travelled from the Sun or a man-made source or may have already been reflected off another surface such as a mirror.
- When incident light strikes a surface or object it may undergo absorption, reflection, refraction, transmission or any combination of these optical effects.

- The angle of incidence measures the angle at which incoming light strikes a surface.
- The angle of incidence is measured between a ray of incoming light and an imaginary line called the normal.
- See this diagram for an explanation: Reflection of a ray of light
- In optics, the
*normal*is a line drawn on a ray diagram perpendicular to, so at a right angle to (90^{0}), the boundary between two media. - If the boundary between the media is curved, then the normal is drawn at a tangent to the boundary.

The angle of reflection measures the angle at which light rebounds from a surface after being reflected.

- The angle of reflection is measured between a ray of light which has been reflected off a surface and an imaginary line called the
*normal*. - In optics, the
*normal*is a line drawn on a ray diagram perpendicular to, so at a right angle to (90^{0}), the boundary between two media. - The angle of reflection can be used to understand how light will behave when it interacts with different types of surfaces and objects.

- When discussing the formation of rainbows, the angle of deflection measures the angle between the initial path of a light ray before it hits a raindrop, and the angle of deviation, which measures how much the ray bends back on itself in the course of refraction and reflection towards an observer.
- See this diagram for an explanation: Rainbow anatomy
- The angle of deflection and the angle of deviation are always directly related to one another and together add up to 180 degrees.
- The angle of deflection equals 180 degrees minus the angle of deviation. So, it’s clear the angle of deviation is always equal to 180 degrees minus the angle of deflection.
- In any particular case, the angle of deflection is always the same as the viewing angle because the incident rays of light that form a rainbow all follow paths that run parallel with the rainbow axis.

A rainbow is an optical effect produced by illuminated droplets of water. Rainbows are caused by reflection, refraction and dispersion of light in individual droplets and results in the appearance of an arc of spectral colours.

- Rainbows only appear when weather conditions are ideal and an observer is in the right place at the right time.
- Waterfalls, lawn sprinklers and other things that produce water droplets can produce a rainbow.
- A rainbow is formed from millions of individual droplets each of which reflects and refracts a tiny coloured image of the sun towards the observer.
- It is the dispersion of light as refraction takes place that produces the rainbow colours seen by an observer.
- When the sun is behind an observer then the rainbow will appear in front of them.