Physics Textbook/Curved Mirrors

Reflection at curved surfaces

For studying such reflection, all we do is draaw normals at each point on the curved surface and apply i=r on each of them. The curved surfaces we'll study would be sphere cutouts. No parabolic/hyperbolic mirror.

Shed misconceptions

Not every ray parallel to the principal axis is sent to focus after reflection. Geometrical optics formulae are valid only for paraxial rays (rays incident close to the pole; close is not a scientific term; exactly what range is close to the pole is dependent upon size of mirror, distance involved, etc).
Not all rays are paraxial. The beam of ray (i) will be sent to focus, but (ii) will be sent to a point little away from the focus. We'll keep this in mind and developthese ideas little by little as we learn more of optics.
Even objects which are of comparable size to the mirror can't be dealt with the formulae we derive.
Formulae in geometrical optics are valid only for paraxial rays. Only the basic laws are universal.

What we have described above is exactly the reason why we use parabolic mirrors to focus parallel rays. Well, parabolic curves have the property of focusing each and every parallel ray of light to the axis on one point. Actually, a curve having this property is defined a parabolic curve [Ref. coordinate]

Checkpoint

Focusing. A beam of paraxial rays parallel to the principal axis converge to a point on the axis after reflection from a concave mirror [real image] [draw diagram].
When such rays reflection from a convex mirror, they diverge away. However, the one who sees these rays feels they're diverging from a point behind mirror [visual image] [draw diagram].

These points are focuses, and their distance from pole is focal length.. Now, it so happens that the focal length for convex/concave mirror comes out to be R/2.

Sign convention

Direction of incident rays --> +ve x direction
Principal axis --> x axis (-ve & +ve)
Pole is the origin
Height above principal axis, positive below principal axis, negative.