Math help please - parabola

[Morris Dovey]

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.

-- Morris Dovey DeSoto Solar DeSoto, Iowa USA

[kzinNOSPAM99]

Hi Morris,

Answer is kinda mathy so I posted a jpg snapshot of the mathcad screen in abpw

ml

[David Merrill]

Merrill

[Owen Lawrence]

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 -

[Morris Dovey]

| On 17-Jul-2005, "Morris Dovey" wrote: | || 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 | | | | Answer is kinda mathy so I posted a jpg snapshot of the | mathcad screen in abpw

Mark...

Thank you - that's *exactly* what I need!

It's been more than three decades since I last needed to work with definite integrals; and a bit of review is in order - but your help will allow me to complete a prototype in just a day or two.

"Mathy" is ok. Trial and error cutting with a 16-petal assembly would have been really nasty. Having the general formula will allow me to make reflectors with /any/ number of "petals" and even build accurate reflectors with concentric rings of petals.

You made my day!

-- Morris Dovey DeSoto Solar DeSoto, Iowa USA

[Morris Dovey]

It's interesting to note that a query to rec.woodworking produced a usable solution immediately, while the same query to sci.math (where it's probably more topical) hasn't produced any response at all.

-- Morris Dovey DeSoto Solar DeSoto, Iowa USA

[kzinNOSPAM99]

glad to be of help. MathCad does all that symbolically, and I've got some ancient version of it. Can't imagine what it does now.

ml

[Morris Dovey]

| 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...

Thank you. Even without lunch you did better than I managed. :-)

-- Morris Dovey DeSoto Solar DeSoto, Iowa USA

[David Merrill]

You have probably already noted that there are several forms for expressing the equation of a parabola, some perhaps more convenient than others for deriving the associated arc length expression in a compact form for programming your Shopbot.

You might find these sites helpful for optical effects visualization:

telescope makers (ATM) are often very interested in parabolic mirrors as in Newtonian telescopes.

And you probably already know about this site that I stumbled across (DAGS "solar concentrators"):

Merrill