Derive R = 2f for a spherical mirror, where the symbols have their usual meaning

Explanation

To derive the relationship \( R = 2f \) for a spherical mirror, we will consider a concave mirror and analyze the geometry involved in the reflection of light rays. Here are the steps to derive this relationship:

### Step-by-Step Solution

1. **Understanding the Setup**:
- Consider a concave mirror with its principal axis. Let \( P \) be the pole of the mirror, \( F \) be the focus, and \( C \) be the center of curvature. The distance from the pole to the focus is the focal length \( f \), and the distance from the pole to the center of curvature is the radius of curvature \( R \).

2. **Incident Ray Parallel to Principal Axis**:
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