Ray Diagrams
A ray diagram is a geometric construction that lets us predict exactly where, how large, and what orientation an image will appear when light reflects off a spherical mirror. Rather than tracking every reflected ray, physicists identified three special rays — called principal rays — whose paths after reflection are perfectly predictable from the mirror's geometry alone. Where two or more of these reflected rays cross (or appear to diverge from), an image is formed. Ray diagrams are the visual language of geometric optics. Spherical mirrors come in two varieties: concave mirrors, which curve inward like the inside of a spoon and can converge light to a focus; and convex mirrors, which curve outward like the outside of a spoon and always diverge light. Both types share the same principal axis (the line through the centre of curvature C and the pole P of the mirror), the same focal point F (midway between C and P), and the same set of principal rays. The radius of curvature R and focal length f are related by f = R/2. Ray diagrams allow us to classify images as real or virtual, upright or inverted, magnified or diminished. A real image forms where reflected rays actually converge — it can be projected onto a screen. A virtual image forms where reflected rays appear to diverge from behind the mirror — it cannot be projected. Mastering the three principal rays gives you a reliable method to find any image without solving equations first.
- Set Up the Mirror Diagram
- Draw Principal Ray 1 — Parallel to the Axis
- Draw Principal Ray 2 — Through the Focal Point
- Draw Principal Ray 3 — Through the Centre of Curvature
- Locate the Image
Set Up the Mirror Diagram
Draw the principal axis as a horizontal line. Mark the pole P at the mirror's surface, the focal point F at distance f from P, and the centre of curvature C at distance 2f from P (so C = 2F). For a concave mirror, C and F lie in front of the mirror (the reflecting side); for a convex mirror, they lie behind the mirror surface. Place the object (usually an upright arrow) on the principal axis at the chosen object distance u, measured from P.
Draw Principal Ray 1 — Parallel to the Axis
Draw a ray from the tip of the object travelling parallel to the principal axis toward the mirror surface. After reflection, this ray passes through the focal point F for a concave mirror (the ray converges through F). For a convex mirror, the reflected ray diverges, but when extended behind the mirror it appears to come from F. This rule follows directly from the definition of the focal point: all rays parallel to the axis converge at (or appear to diverge from) F after reflection.
Draw Principal Ray 2 — Through the Focal Point
Draw a ray from the tip of the object directed through (or toward) the focal point F. After striking the mirror, this ray reflects parallel to the principal axis. This is simply the reverse of Ray 1 (optics is time-reversible). For a convex mirror, direct the incoming ray toward the virtual focal point F behind the mirror; it reflects parallel to the axis. This ray is particularly useful because it immediately gives you the direction of the reflected beam.
Draw Principal Ray 3 — Through the Centre of Curvature
Draw a ray from the tip of the object directed through the centre of curvature C (or toward it for a convex mirror). Because every radius of a sphere is perpendicular to the surface at the point of incidence, this ray strikes the mirror along the normal and reflects straight back along its own path through C. For a convex mirror, aim toward the virtual point C behind the mirror; it reflects back on itself. This ray confirms the image position independently of Rays 1 and 2.
Locate the Image
The image forms at the point where any two (or all three) reflected rays intersect in front of the mirror — this is a real image. If the reflected rays diverge, extend them behind the mirror as dashed lines; their intersection behind the mirror is a virtual image. Measure the image distance v from pole P. The image is inverted if it appears on the opposite side of the principal axis from the object, and upright if on the same side. The linear magnification m = image height / object height = −v/u (with sign convention).
The Key Equation
- f — Focal length (metres (m)): Distance from the pole P to the focal point F; positive for concave mirrors, negative for convex mirrors under the standard real-is-positive convention.
- v — Image distance (metres (m)): Distance from the pole P to the image; positive when the image forms in front of the mirror (real image), negative when behind (virtual image).
- u — Object distance (metres (m)): Distance from the pole P to the object; taken as positive when the object is placed in front of the mirror (the usual case).
Worked Example
Problem. An object 4 cm tall is placed 30 cm in front of a concave mirror of focal length 10 cm. Use the mirror formula to find the image distance and linear magnification, then describe the image.
Answer. Real, inverted image 15 cm in front of the mirror; height 2 cm (diminished by factor 0.5).
Origins · 1 minIbn al-Haytham (Alhazen) and how this was discovered
Ibn al-Haytham wrote Kitāb al-Manāẓir (Book of Optics), in which he gave the first systematic account of reflection from curved mirrors. He correctly identified that an image in a mirror appears to originate from a point behind the mirror surface and he analysed reflection from spherical, parabolic, and cylindrical mirrors using geometric methods that directly anticipate modern ray diagrams. To study optics rigorously, Alhazen invented the camera obscura — a darkened box with a pinhole — and used it to project real images of the Sun during a solar eclipse, nearly six centuries before the camera was reinvented in Europe.
Related Concepts
- Focal Length and Centre of Curvature — Ray diagrams rely on locating F and C precisely; understanding how f = R/2 arises from mirror geometry is essential before drawing any diagram.
- Mirror Formula and Magnification — The mirror formula 1/f = 1/v + 1/u lets you calculate image position numerically, complementing what ray diagrams show graphically.
- Refraction and Snell's Law — The same principal-ray technique used for mirrors is extended to lenses, where refraction rather than reflection governs ray bending.
- Law of Reflection — Every ray in a mirror diagram obeys the law of reflection (angle of incidence = angle of reflection) at the mirror surface — it underpins why principal rays behave as they do.