I'm trying to build a circular parabolic dish solar concentrator. So
far I've designed a simple hub to which I'll attach radial ribs. These
ribs will support reflective "petals" and an outer support ring.

I know that the general formula for a parabola is:

(x - h) ^ 2 = 4 * a * (y - k), where

(h,k) are the coordinates of the vertex and
a is the distance from the vertex to the focus and the distance from
the vertex to the directrix (and one-fourth the length of the latus
rectum).

What I need is an algebraic expression representing the distance along
the parabola from the vertex to any point (x,y) on the parabola so I
can lay out the shape of a petal for cutting from flat stock.

I'm not checking my work, but I'll give it a shot. First, let's simplify a
bit by locating your vertex at the origin. Your parabola is then y =
x^2/4a. The formula for arc length can be derived to be

L = int( x1, x2 ) sqrt( 1 + (dy/dx)^2) dx, where int( x1, x2 ) is the
definite integral from x1 to x2 on your parabola.

dy/dx = x/2a
(dy/dx)^2 = x^2/4a^2

so L = int( x1, x2 ) sqrt( 1 + x^2/4a^2 ) dx
= 1/2a int( x1, x2 ) sqrt( 4a^2 + x^2 ) dx

I looked up the indefinite integral in my CRC math tables to be

int() sqrt( x^2 + c^2) dx = 1/2[ x sqrt( x^2 + c^2 ) + c^2 ln( x + sqrt( x^2
+ c^2 ) ]

In our case c = 4a^2. Since you want your arc length to be from the vertex,
we can let x1 = 0. So our final formula becomes:

L = 1/4a[ x sqrt( x^2 + (2a)^4 ) + (2a)^4 ln( x + sqrt( x^2 + (2a)^4 ) ]

Check the algebra. I haven't had lunch yet so maybe I'm not thinking
clearly.

- Owen -