Is there a formula to calculate the dimensions of the arcs on a flat piece of paper needed to form a parabolic dish?
Say I was making a parabolic dish out of cardboard segments, would there be a way to calculate the size of the segments needed to form a parabolic dish with specific dimensions?
No
In ship building if building with plywood for example we draw what we call "developable surfaces" . Those are shapes that at all points only have curvature in one direction and you can draw a straight line along them in the other direction. You cannot bend a piece of plywood or paper around a surface with curvature in two directions. You will get a crease. You need to use a material you can mold like molten plastic or metal or cement or fiberglass.
PS
Try it with a piece of paper or plywood.
Yes, there is a formula to calculate the dimensions of the arcs needed to form a parabolic dish.
To calculate the dimensions of the arcs, you can use the equation that defines a parabola in its standard form y = ax^2, where "a" is a constant that determines the shape of the parabola. In the case of a dish, the parabola is oriented vertically, so the equation can be written as x = ay^2.
To determine the dimensions of the arcs, you will need to know the desired focal length (f) and the height (h) of the parabolic dish.
First, calculate the value of "a" using the formula a = 1 / (4f).
Next, calculate the width of the parabolic dish at the required height (x) using the formula x = a * y^2.
Now, let's assume you want to make a parabolic dish with a focal length of 10 units and a height of 20 units.
Using the formula, calculate the value of "a": a = 1 / (4 * 10) = 0.025.
Next, substitute the known values into the equation x = a * y^2 to find the width at the required height. For example, when y = 20, x = 0.025 * (20^2) = 0.025 * 400 = 10 units.
Therefore, to form the parabolic dish with a focal length of 10 units and a height of 20 units, you would need cardboard segments with a width of 10 units at a height of 20 units.
It's important to note that this calculation assumes a simplified parabolic shape and doesn't account for potential imperfections or factors such as material thickness and curvature accuracy.