Mirror Equation

When you look into a curved mirror — the kind used in car wing mirrors, bathroom magnifying mirrors, or satellite dishes — the image you see is rarely where the object actually is. The mirror equation is the mathematical tool that tells you precisely where an image forms, given where an object is placed and how strongly curved the mirror is. It links three quantities: the object distance (u), the image distance (v), and the focal length (f) of the mirror, in one elegant relationship that applies to both concave and convex mirrors. The power of the mirror equation lies in its universality. Whether the mirror curves inward (concave) or outward (convex), whether the image is real or virtual, magnified or diminished, the same formula governs the geometry. This is possible because all spherical mirrors obey the same law of reflection at every point on their surface: the angle of incidence equals the angle of reflection. The mirror equation is derived directly from this law applied to paraxial rays — rays that travel close to and nearly parallel to the principal axis. Understanding the mirror equation requires a consistent sign convention. In the standard convention used in most physics courses (the New Cartesian convention), distances are measured from the pole of the mirror, with distances in the direction of the incident light taken as negative and distances opposite to the incident light taken as positive. Getting the signs right is as important as the arithmetic itself.

Key takeaways
  • The geometry of a curved mirror
  • Deriving the equation from similar triangles
  • Applying the sign convention correctly
  • Linear magnification

The geometry of a curved mirror

A spherical mirror is a portion of a hollow sphere. Its centre of curvature C is the centre of that sphere, and the radius of curvature R is the sphere's radius. The pole P is the midpoint of the mirror surface, and the principal axis is the straight line through P and C. Parallel rays striking a concave mirror converge at the principal focus F, which lies halfway between P and C, so the focal length f = R/2. For a convex mirror, F lies behind the mirror surface and f is negative under the New Cartesian convention.

Deriving the equation from similar triangles

Consider an object placed at distance u from the pole of a concave mirror. A ray parallel to the principal axis reflects through F. A ray through C reflects back on itself. These two rays meet to form the image at distance v. By examining the two pairs of similar triangles formed by these rays, the object height, image height, and the distances u, v, and f are related. After careful algebraic manipulation using the properties of similar triangles, the result is: 1/v + 1/u = 1/f. This derivation assumes paraxial rays — a small-angle approximation that keeps the geometry tractable and the formula accurate for most practical situations.

Applying the sign convention correctly

In the New Cartesian convention, all distances are measured from the pole P. The incident light travels from left to right. Distances measured in the direction opposite to incident light (i.e. to the left, in front of the mirror) are negative. So for a real object, u is always negative. For a concave mirror, f and R are negative (focus is in front). For a convex mirror, f and R are positive (focus is behind the mirror). A negative v means a real image (formed in front of the mirror); a positive v means a virtual image (behind the mirror). Always assign signs before substituting into 1/v + 1/u = 1/f.

Linear magnification

The mirror equation tells you where the image is, but not how big it is. Linear magnification m is defined as the ratio of image height (h') to object height (h). Using similar triangles from the same geometric derivation: m = h'/h = −v/u. A positive m indicates an upright (virtual) image; a negative m indicates an inverted (real) image. A magnitude greater than 1 means the image is magnified; less than 1 means it is diminished. For example, a concave shaving mirror produces a magnified, upright, virtual image when the object is placed inside the focal length.

The Key Equation

1v+1u=1f=2R\frac{1}{v} + \frac{1}{u} = \frac{1}{f} = \frac{2}{R}
  • v — Image distance (metres (m)): Distance from the pole of the mirror to the image. Negative for a real image (in front of mirror), positive for a virtual image (behind mirror).
  • u — Object distance (metres (m)): Distance from the pole of the mirror to the object. Always negative for a real object under the New Cartesian convention.
  • f — Focal length (metres (m)): Distance from the pole to the principal focus. Negative for a concave mirror, positive for a convex mirror.
  • R — Radius of curvature (metres (m)): Radius of the sphere of which the mirror forms a part. Related to focal length by f = R/2.

Worked Example

Problem. A concave mirror has a focal length of 15 cm. An object is placed 45 cm in front of the mirror. Find (a) the image distance and (b) the linear magnification. State the nature of the image.

v=−22.5 cm,m=−0.5v = -22.5\text{ cm},\quad m = -0.5

Answer. The image forms 22.5 cm in front of the mirror (real, inverted, diminished, with magnification −0.5).

Figure
Diagram
Principal axisMirrorPFCObject(h)Image (h')real, invertedfu (object distance)v(image distance)
Origins · 1 minIbn al-Haytham (Alhazen) and how this was discovered
IA
Ibn al-Haytham (Alhazen)
965–1040 · Arab (Iraqi)

Ibn al-Haytham, known in the West as Alhazen, made foundational contributions to optics in his monumental work Kitāb al-Manāẓir (Book of Optics, c. 1011–1021). He correctly described the law of reflection for curved mirrors, analysed concave and convex mirrors using geometric methods, and established that light travels from objects to the eye — overturning the ancient Greek 'emission' theory. His rigorous geometric treatment of image formation in spherical mirrors laid the groundwork from which the mirror equation was ultimately formalised. To avoid punishment by the Caliph al-Hakim, Ibn al-Haytham feigned madness and was placed under house arrest for a decade — during which time he produced most of his greatest scientific writings.

  • Laws of Reflection — The mirror equation is derived entirely from the law of reflection applied to paraxial rays at a curved surface.
  • Sign Convention for Mirrors — Correctly applying the New Cartesian sign convention is essential for using the mirror equation without error.
  • Magnification in Mirrors — Linear magnification m = −v/u is derived alongside the mirror equation and describes image size and orientation.
  • Lens Equation — The thin lens equation 1/v − 1/u = 1/f has an analogous form to the mirror equation and is derived by similar geometric reasoning.