--- title: "Curved Mirrors, the Mirror Equation and Real Images" description: "For paraxial rays, a spherical mirror has focal-length magnitude equal to half its radius of curvature. Ray construction and the mirror equation locate images. Signed image distance and magnification " canonical: https://lightmysky.com/learn/science/curved-mirrors-the-mirror-equation-and-real-images-mt_7XqB2umthJ source: https://lightmysky.com/learn/science/curved-mirrors-the-mirror-equation-and-real-images-mt_7XqB2umthJ.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Curved Mirrors, the Mirror Equation and Real Images For paraxial rays, a spherical mirror has focal-length magnitude equal to half its radius of curvature. Ray construction and the mirror equation locate images. Signed image distance and magnification describe whether an image is real or virtual, upright or inverted. Subject: Science · Area: Waves, Light & Sound · Ages 19 to 20 Page: https://lightmysky.com/learn/science/curved-mirrors-the-mirror-equation-and-real-images-mt_7XqB2umthJ ## Ready when they can - Locates an image by ray construction for concave and convex mirrors and checks it against the mirror equation - Relates paraxial focal length to radius of curvature and distinguishes spherical aberration in an axial beam from off-axis coma - Explains why a shaving mirror magnifies while a car wing mirror shrinks the view and widens it ## Lesson: Where curved mirrors put images A concave spherical mirror reflects from a sphere's inside; a convex mirror reflects from its outside. The optical axis joins the vertex V to the centre of curvature C. For rays close to this axis at small angles, |f| = R/2, where R is the radius's magnitude. This is the paraxial approximation. The focus F is in front of a concave mirror and behind a convex mirror. *(drawing: Axial positions for a concave mirror: F is halfway to C. This is not a ray diagram.)* Trace two rays from the same object tip. For a concave mirror, an axis-parallel ray reflects through the focus F; a ray along the focal line reflects parallel to the axis. A ray along the line through the centre of curvature C retraces its path. For a convex mirror, F and C lie behind the surface. Reflected rays or their backward extensions follow these rules. Real images use ray intersections; virtual images use backward-extension intersections. For a real object inside a concave mirror's focal length, the image is virtual, upright and enlarged, as in a shaving mirror. Beyond the focus, it is real and inverted. At the focus, reflected rays are parallel and no finite image forms. A convex mirror gives a real object a virtual, upright, reduced image. Compared with an equal-sized plane mirror, its curved surface shows a wider field. Use 1/f = 1/do + 1/di and m = -di/do, measuring distances from the vertex. For a real object, do is positive. Take f positive for concave mirrors and negative for convex mirrors. Positive di means a real image in front; negative di means a virtual image behind. Positive m means upright; negative m means inverted. The magnitude |m| is the image-to-object height ratio. *(drawing: The reciprocal calculation gives image distance. Magnification then gives orientation and size.)* **Example.** Let a concave mirror have f = 12 cm and do = 18 cm. Then 1/di = 1/12 - 1/18 = 1/36 cm^-1, so di = +36 cm. Next, m = -36/18 = -2. The image is real, inverted and twice the object's height. Compare this result with ray construction. A disagreement calls for checking signs, arithmetic and ray paths; agreement is a useful check, not proof. Spherical aberration occurs even for an axial parallel beam: rays at different heights on a spherical mirror focus at different distances. A wider opening can admit more edge rays and increase this blur. An ideal paraboloid brings an axial parallel beam to one focus, removing spherical aberration. For light tilted to the axis it can still produce coma, a comet-like image blur. Telescope primaries need not all be paraboloids. For a paraxial paper diagram, put the vertex at (0,0). Coordinates are in cm: x increases in front of the mirror and y measures height. Plot the object tip (do,h) and focus (f,0), using a small h. Its axis-parallel ray strikes near (0,h), then reflects along the line through (f,0). Its ray to the vertex reflects along the line through (0,0) and (do,-h), with equal angles to the axis. Extend reflected lines behind the mirror when locating virtual images. **Example.** Concave example, in cm: f = 15, do = 20, h = 0.2. Draw the reflected parallel-ray line through (0,0.2) and (15,0). Draw the reflected vertex-ray line through (0,0) and (20,-0.2). Their forward intersection is (60,-0.6): the image is 60 cm in front and inverted. Check 1/di = 1/15 - 1/20 = 1/60, giving di = +60 cm. Also m = -60/20 = -3 agrees with the height -0.6 cm. A paper drawing gives an approximate intersection to compare with the equation. **Example.** Convex example, in cm: f = -18, do = 36, h = 0.3. Draw the reflected parallel-ray line through (0,0.3) and (-18,0). Draw the reflected vertex-ray line through (0,0) and (36,-0.3). Their backward extensions intersect at (-12,0.1): a virtual image 12 cm behind the mirror. Check 1/di = -1/18 - 1/36 = -1/12, giving di = -12 cm. The magnification m = -(-12)/36 = 1/3 agrees with the upright image height 0.1 cm. **Recap.** Use paraxial ray rules and signed equations together. Real images can reach a screen. A paraboloid corrects axial spherical aberration, not every kind of blur. *Lesson, questions and flashcards written for this dataset. OpenStax texts were consulted for physics facts; textbook passages are not reproduced.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Thin lenses and the lens maker's equation](https://lightmysky.com/learn/science/thin-lenses-and-the-lens-makers-equation-mt_FXOcC5mpIM) - [Huygens's principle and the laws of reflection and refraction](https://lightmysky.com/learn/science/huygenss-principle-and-the-laws-of-reflection-and-refraction-mt_xxdniF2__w) ## Opens up - [Magnifiers, Microscopes and Telescopes](https://lightmysky.com/learn/science/magnifiers-microscopes-and-telescopes-mt_f30rUsqYkN)