Paraboloids
Parabolas perfectly focus light sources that are infinately far away. The closer the light source, the more the distortion. This is not a problem when you are star gazing. A paraboloid is ideal for astronomy. The easiest way is to find the focal point of a parabaloid is to start with a light source that is infinately far away. This means that all incoming light rays are parallel to the why axis. So, for any given point P on a parabola, do the following.
A) Construct a line L1 parallel to the Y axis through P
B) Construct a line L2 perpendicular to the parabola at point P
C) Construct a line L3 that mirrors exactly the angle betweenL1 and L2 using the L2 (the perpendicular) as the mirror axis.
If everything is done right, L3 will intercept the why axis at the focal point.
There is an easier way. The slope of a parabola a*x^2 at any point is simply 2*a*x. When this slope equals exactly 1, the reflection will be perpendicular to the incident ray, or parallel to the x axis. This reflection will pass through the focal point, so if we can find this point, its why value will equal the focal length. Start with 2*a*x=1. Solving for x we get x=1/(2*a). Substituting in the original equation, we get y=a*(1/(2*a))^2. Squaring this reduces to y=a*(1/(4*a^2)) which reduces to y=1/(4*a) at the focal point. This makes the focal length (fl) fl=1/(4*a).
I hope this helps. :)
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