Simple(?) parabolic dish design question

In summary, the conversation discusses the problem of specifying both the diameter and focal length of a parabolic mirror, with the focal length only being able to be specified as a perpendicular distance from the diameter. Various equations and solutions are proposed, including using a Maple CAS equation and a general solution involving the parameters r (radius of dish) and g (distance from front face of dish to focus). Ultimately, the solution involves finding the parameter a using the equation a = ( t - g ) / ( 2 · r² ) and then calculating the depth of the dish d as d = f - g where f = 1 / ( 4 · a ) and t = √ ( r² + g² ).
  • #1
TL;DR Summary
Not feeling too good, slowing me down a bit :).
I want to use this to design a parabolic (optical) mirror;

Capture.GIF

The problem is that in my application I need both D and f to be a parameter, but I need to specify f only as a perpendicular distance from D. In other words, I need to specify some f_2=f-d, and calculate d. I can't seem to come up with a way to do this without self-reference.

Thanks so much

Joe

ETA N/M it just took me an extra second. I'll post the solution shortly

ETA II Nope, sorry, still self reference problems. It's so annoying cause it seems so simple :/

ETA III Maple CAS says it's
Capture.GIF

(where f=f_2)but I'll be durned if I know how it got there.
 
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  • #2
Specify the distance from front edge of dish to focus as g.
Then; f = d + g; d = f - g;
You specify the paraboloid size by radius; r = diameter / 2.
Given r and g, solve for parameter a of the parabola equation.
y = a * x^2

The slope of surface is; y' = 2*a*x
When 45°, slope = 1, y = f
1 = 2 * a * x
x = 1 / ( 2 * a )
f = y = a / ( 4*a^2 )
f = 1 / ( 4 * a )
For the rim of the dish.
d = a * r^2
f - g = a * r^2
f = g + ( a * r^2 )
equate the two equations for f.
g + ( a * r^2 ) = 1 / ( 4 * a )
( r^2 * a^2 ) + (g * a) - 1/4 = 0
Then solve that quadratic for the parameter a.
 
  • #3
I think this is a general solution.
Specify radius of dish; r = diam / 2
Specify distance from front face of dish to focus; g
Vertex is at the origin so surface eqn; y = a · x²
Temporary; t = √ ( r² + g² ) ; which is distance from focus to lip
Parameter; a = ( t - g ) / ( 2 · r² )
Position of focus on y axis; f = 1 / ( 4 · a )
Depth of dish; d = f - g
 
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1. What is a simple parabolic dish?

A simple parabolic dish is a reflective dish that is shaped like a parabola. It is used to collect and focus incoming electromagnetic waves, such as radio waves or light.

2. How does a simple parabolic dish work?

A simple parabolic dish works by reflecting incoming waves off its curved surface and focusing them at a single point, known as the focal point. This allows for a more concentrated and powerful signal or beam.

3. What are the applications of a simple parabolic dish?

A simple parabolic dish has a wide range of applications, including in satellite dishes for TV and internet communication, solar cookers, telescopes, and radar systems.

4. What factors affect the design of a simple parabolic dish?

The design of a simple parabolic dish is affected by several factors, including the size and shape of the dish, the curvature of the parabola, the material used for the reflective surface, and the location and orientation of the focal point.

5. Can I build my own simple parabolic dish?

Yes, it is possible to build a simple parabolic dish at home using materials such as cardboard, aluminum foil, and glue. However, it may not be as efficient as commercially made dishes and requires precise measurements and calculations for optimal performance.

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