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Question: calculate normal to a parabola
Thu, 2007-05-10 19:25 - Anonymous (not verified)
Hello:
I need advice on how to calculate the coordinates of a vector normal to a parabola section. The parabola is defined by 3 points on a plane: (x1,y1), (x2,y2), (x3,y3). I need to calculate the coordinates of the end point of the normal vector with the starting point (x2,y2) and a precribed magnitude A.
Any help would be greatly appreciated.
Thank you.
David.
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Normal to a parabola
The general equation for a parabola is
with the restriction that 
. That means there are four unknowns and you will need four points to specify a parabola uniquely. Otherwise you will get more that one parabola - see for example http://mathworld.wolfram.com/Parabola.html. It is often advantageous to convert the equation into parametric form.
Once you know the equation of the parabola, take the derivative of the equation to get
(you will need to check that for correctness). Evaluate the slope at the point (x2,y2) and create a line through that point with that slope that meets the x axis at (xp, 0). Then the line L1 = (x2,y2)-(xp,0) is a vector. Create another vector L2 = (x2,y2)-(xq,yq). Take the dot product of L1 and L2 and set the result to 0. Set xq = 0 and solve for yq. Once you have the vector L2 then scale it so that its magnitude is A.
This is the long and brute force route. You can simplify things by directly using the slope.
Normal to a parabola
If the principal axis of the parabola is parallel to the y-axis then the computation of the normal is quite simple: