Suppose a satellite dish had the following surface patch parametrisation:
$\textbf{r} : \{ (x,y) \in \mathbb{R}^{2} : x^2+y^2 \leq 1 \} \rightarrow \mathbb{R}^3 ; \; (x.y) \mapsto (x, y, \frac{1}{4} (x^2 + y^2)) \in \mathbb{R}^3.$
https://w-dog.net/wallpaper/satellite-dish-sky-communication/id/202218/
In which point do you have to mount the actual antenna in order to guarantee an ideal signal? Explain and compute, proving first all that all incoming rays parallel to the central axis of symmetry are reflected into a common point.
I was told by my lecturer to first find the unit normal vector field to the surface patch:
$\textbf{N} (x,y) = \frac{1}{\sqrt{\frac{1}{4} x^2 + \frac{1}{4} y^2 + 1}}(-\frac{1}{2}x, - \frac{1}{4}y,1)$
I am also aware that rays parallel to the central axis of symmetry, will hit the dish at an angle $\alpha$ to $\textbf{N}$ and will bounce off at an angle $\alpha$ to $\textbf{N}$.
I am not really sure how to proceed from here. How do I answer the question in bold?
The property of the parabola is that all parallel rays reflect in one point that is its focus. If the equation is $y=\dfrac{1}{4}\,x$ focus is $1$ unit from the vertex
At point $(0,0,1)$
Hope This helps
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