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Figure 1.16 illustrates the reflection of a ray from a curved surface.

The incident ray is at an angle, θ, with respect to the optical axis and the reflected ray is at an angle, ϕ to the optical axis. If we designate the incident angle as θ₁ and the reflected angle as θ₂ (with respect to the local surface normal), then the following apply, assuming all relevant angles are small:

Figure 1.16 Reflection from a curved surface.

We now need to calculate the angle, ϕ, the refracted ray makes to the optical axis:

(1.17)

In form, Eq. (1.17) is similar to Eq. (1.14) with a linear dependence of the reflected ray angle on both incident ray angle and height. The two equations may be made to correspond exactly if we make the substitution, n₁ = 1, n₂ = −1. This runs in accord with the empirical observation made previously that a reflective surface acts like a medium with a refractive index of −1. Once more, the sign convention observed dictates that positive axial displacement, z, is in the direction from left to right and positive height is vertically upwards. A ray with a positive angle, θ, has a positive gradient in h with respect to z.

As with the curved refractive surface, a curved mirror is image forming. It is therefore possible to set out the Cardinal Points, as before: Cardinal points for a spherical mirror

Both Principal Points: At vertex
Both Nodal Points: At centre of sphere

The focal length of a curved mirror is half the base radius, with both focal points co-located. In fact, the two focal lengths are of opposite sign. Again, this fits in with the notion that reflective surfaces act as media with a refractive index of −1. Both nodal points are co-located at the centre of curvature and the principal points are also co-located at the surface vertex.