## Parabola Geometry

### Equations

Some convenient equations for parabolas are:

x² = 4fy

where f = the focal length of the parabola. Click here to see where we get this equation from.

y = x²/4f = x²/2R

### Proof that a Parabola focusses parallel rays of light to a Point

How do we know that a parabola is the right shape? Light from a distant star arrives in parallel rays. We want the mirror to focus the light to a point, just like a lens. So here is the geometric proof:

The general equation for a parabola is quadratic: y = a x²

Calculus tells us that the slope of such a curve is given by: y' = 2ax

But we know that the slope is just the rise over the run, which is the same as the tangent. So y'= tanq, thus we have tanq = 2ax

Now tan 2q = x/d
so the distance d = x/ tan 2q

and, by trigonometric relationship, tan 2q = 2 tan q/(1-tan²q)
Subsitituting from above tanq = 2ax, we have
tan 2q = 2 (2ax) / [1-(2ax)²] = 4ax/(1-4a²x²)

So d = x (1-4a²x²) / 4ax = (1/4a) - ax²

Thus f = d+ax² = 1/4a and is invariant with x. Q.E.D.

It also gives a convenient formula for a parabola: y = x² /4f

How do we know that the radius of curvature R is twice the focal length?

Calculus tells us that for any curve, the radius of curvature of the curve at a particular point is given by
R(x) = [(1+y'(x)²)3/2]/y"(x)

where y' = the first derivative at the point
and y" = the second derivative at the point