What Is the Pole of a Spherical Mirror?

The pole of a spherical mirror is the point where its principal axis meets the reflecting surface. In a centered mirror, this is the vertex of the spherical surface and lies at the center of the mirror’s aperture. The pole is a reference point for describing the mirror’s geometry and measuring object and image distances.

How to identify the pole

A spherical mirror is a portion of a spherical surface. The sphere of which it is a part has a center of curvature, usually labeled $C$. Draw a straight line from $C$ to the middle of the mirror’s reflecting aperture. That line is the principal axis. Where it meets the mirror is the pole, usually labeled $P$.

The axis is perpendicular to the mirror’s surface at the pole. This makes the pole the natural reference point for paraxial-ray diagrams and the standard mirror equation. A simplified layout is:

principal axis:  --------------------
                 C        F        P )

Here the mirror is concave and faces left; $F$ is its principal focus. The drawing is schematic, and the distances are not to scale. For a convex mirror facing left, the reflecting surface bulges toward the left and the center of curvature is behind the mirror.

Pole, center of curvature, and focus are different

These three points are easy to confuse:

Point	Meaning
Pole $P$	Where the principal axis intersects the mirror’s reflecting surface
Center of curvature $C$	Center of the sphere whose surface forms the mirror
Principal focus $F$	For paraxial rays parallel to the principal axis, the point on the axis where reflected rays converge, or appear to diverge from

For a spherical mirror of radius of curvature $R$, the distance from the pole to the center of curvature has magnitude $R$. Under the paraxial approximation, the focal length has magnitude $R/2$. Neither relationship makes the pole the focus or the center of curvature: they are separate locations.

A worked example

Suppose a concave mirror is part of a sphere with radius $30\,\text{cm}$. Place its pole at $x=0$ and let the principal axis point in the positive $x$ direction, toward the sphere’s center. Then the center of curvature is at $x=30\,\text{cm}$, so the pole-to-center distance is $30\,\text{cm}$, the radius of curvature.

For paraxial rays, the focus is halfway between the pole and center of curvature, at $x=15\,\text{cm}$. The pole remains at $x=0$; it does not shift when you locate the focus. If an object is $10\,\text{cm}$ in front of the mirror, its distance is measured from the pole, not from the center of curvature. Whether that object distance is written as positive or negative depends on the sign convention being used.

Why the pole matters in mirror calculations

In the usual spherical-mirror model, object distance $u$ and image distance $v$ are measured from the pole. The mirror formula is often written as

$$\frac{1}{f}=\frac{1}{u}+\frac{1}{v},$$

where $f$ is the focal length. This equation is used with a consistent sign convention; conventions differ in how they assign signs to distances. The pole supplies the distance reference, while the sign convention determines whether each measured distance is positive or negative. See the sign convention for mirror and lens formulas for those details.

Common mistakes

• Calling the center of curvature the pole. The center of curvature is in front of a concave mirror and behind a convex mirror; the pole is on the reflecting surface.
• Calling the focus the pole. The focus is generally displaced from the mirror’s surface. For a spherical mirror in the paraxial approximation, its distance from the pole is about half the radius of curvature.
• Treating the pole as the physical center of the sphere. The pole is the vertex of the mirror cap, not the center of the sphere from which the cap is taken.
• Measuring distances from the wrong point. In standard mirror equations, object and image distances are referenced to the pole. Apply the selected sign convention separately.

In elementary optics, “the center of the mirror” often means the pole: specifically, the center of the reflecting aperture where the principal axis meets the mirror. This terminology assumes a mirror centered on its principal axis.