Parabola

If you kick a soccer ball (or shoot an arrow, fire a missile or throw a stone) it will arc up into the air and come down again ...
... following the path of a parabola!
(Except for how the air affects it.) 

Definition
A parabola is a curve where any point is at an equal distance from:
 a fixed point (the focus), and
 a fixed straight line (the directrix)


Get a piece of paper, draw a straight line on it, then make a big dot for the focus (not on the line!).
Now play around with some measurements until you have another dot that is exactly the same distance from the focus and the straight line.
Keep going until you have lots of little dots, then join the little dots and you will have a parabola

Names
Here are the important names:
 the directrix and focus (explained above)
 the axis of symmetry (goes through the focus, at right angles to the directrix)
 the vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix.

Reflector
And a parabola has this amazing property:
Any ray parallel to the axis of symmetry gets reflected off the surface straight to the focus.
So the parabola can be used for:
 satellite dishes,
 radar dishes,
 concentrating the sun's rays to make a hot spot,
 the reflector on spotlights and torches,
 etc


And that explains why that dot is called the focus ... because that's where all the rays get focused! 


You can also get a parabola when you slice through a cone (the slice must be parallel to the side of the cone).
Therefore, the parabola is a conic section (a section of a cone). 

EquationsIf you place the parabola on the cartesian coordinates (xy graph) with:
 its vertex at the origin "O" and
 its axis of symmetry lying on the xaxis,
then the curve is defined by:
y^{2} = 4ax 

Example:
Where is the focus in the equation y^{2}=5x ?
Converting y^{2} = 5x to y^{2} = 4ax form, we get y^{2} = 4 (5/4) x,
so a = 5/4, and the focus of y^{2}=5x is:
The equations of parabolas in different orientations are as follows:

y^{2} = 4ax 
y^{2} = 4ax 
x^{2} = 4ay 
x^{2} = 4ay 
Measurements for a Parabolic Dish
If you want to build a parabolic dish where the focus is 200 mm above the surface, what measurments do you need?
To make it easy to build, let's have it pointing upwards, and so we choose the x^{2} = 4ay equation.
And we want "a" to be 200, so the equation becomes:
x^{2} = 4ay = 4 × 200 × y = 800y
Rearranging so we can calculate heights:
y = x^{2}/800
And here are some height measurments as you run along:

Distance Along ("x") 
Height ("y") 
0 mm 
0.0 mm 
100 mm 
12.5 mm 
200 mm 
50.0 mm 
300 mm 
112.5 mm 
400 mm 
200.0 mm 
500 mm 
312.5 mm 
600 mm 
450.0 mm 


If you build one tell me, and I can include a picture of it!
