Asked by mikaela
the law of reflection
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Short statement
- The angle of incidence equals the angle of reflection: θi = θr.
- Angles are measured between the ray and the surface normal (the perpendicular to the surface at the point of contact), not relative to the surface itself.
Definitions
- Incident ray: incoming light ray that strikes the surface.
- Reflected ray: outgoing ray that leaves the surface.
- Normal n: unit vector perpendicular to the surface at the reflection point.
- θi: angle between the incident ray and n. θr: angle between the reflected ray and n.
Vector form (useful in calculations)
- If the incident direction is a unit vector d pointing toward the surface and n is the unit normal pointing out of the surface, the reflected direction r is
r = d − 2 (d · n) n.
This reverses the normal component of d while leaving the tangent component unchanged.
Why it holds (brief justification)
- Geometric: reflection from a plane mirror makes the incoming and outgoing rays symmetric about the normal, so the angles to the normal are equal.
- Variational (Fermat’s principle): the actual path taken makes the optical path stationary; that condition yields θi = θr.
- Vector derivation: decompose d into normal and tangential parts, reverse the normal component and keep the tangential component → r as above, which gives θr = θi.
Notes and consequences
- Applies locally to curved mirrors: use the normal at the point of contact.
- Distinguish specular reflection (law applies to each ray) versus diffuse reflection (microscopic facets scatter light in many directions; law still holds for each tiny facet).
- In plane mirrors an object appears at the same distance behind the mirror as it is in front (virtual image).
- The law is independent of wavelength; polarization and amplitude are handled by Fresnel equations but the geometric angles follow the law of reflection.
Examples of use
- Mirrors, periscopes, billiards, radar/sonar reflections, optical instrument design.
If you want, I can: show a quick derivation of the vector formula, draw an ASCII diagram, or give worked numerical examples.
- The angle of incidence equals the angle of reflection: θi = θr.
- Angles are measured between the ray and the surface normal (the perpendicular to the surface at the point of contact), not relative to the surface itself.
Definitions
- Incident ray: incoming light ray that strikes the surface.
- Reflected ray: outgoing ray that leaves the surface.
- Normal n: unit vector perpendicular to the surface at the reflection point.
- θi: angle between the incident ray and n. θr: angle between the reflected ray and n.
Vector form (useful in calculations)
- If the incident direction is a unit vector d pointing toward the surface and n is the unit normal pointing out of the surface, the reflected direction r is
r = d − 2 (d · n) n.
This reverses the normal component of d while leaving the tangent component unchanged.
Why it holds (brief justification)
- Geometric: reflection from a plane mirror makes the incoming and outgoing rays symmetric about the normal, so the angles to the normal are equal.
- Variational (Fermat’s principle): the actual path taken makes the optical path stationary; that condition yields θi = θr.
- Vector derivation: decompose d into normal and tangential parts, reverse the normal component and keep the tangential component → r as above, which gives θr = θi.
Notes and consequences
- Applies locally to curved mirrors: use the normal at the point of contact.
- Distinguish specular reflection (law applies to each ray) versus diffuse reflection (microscopic facets scatter light in many directions; law still holds for each tiny facet).
- In plane mirrors an object appears at the same distance behind the mirror as it is in front (virtual image).
- The law is independent of wavelength; polarization and amplitude are handled by Fresnel equations but the geometric angles follow the law of reflection.
Examples of use
- Mirrors, periscopes, billiards, radar/sonar reflections, optical instrument design.
If you want, I can: show a quick derivation of the vector formula, draw an ASCII diagram, or give worked numerical examples.
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