First things first, please excuse my maths, its been over a decade since I did trig and the like at school and, real world, I haven’t much cause to use them every day…
Second the problem –
I need to find the centre points [or rather distance between them] we’ll call them A & B of an ellipse.
I have a plan which shows me a section of the ellipse but the only further info I can get from the architect is that “he made it using the standard ellipse tool on CAD” he doesn’t know what I’m talking about when I ask for the distance AB let alone why I need it to form an extrusion that follows the edge of his ellipse.
The plan shows me two dimensions, in Cartesian terms, x and (y/2).
Question one) I’m making the assumption that the section of the shape shown is half of it, cut along a the axis x (so values of x range from –x max to +x max, and values of y are >0). Do I need to make this assumption – I’m of the belief if I don’t my solution will work out circular as that’s the simplest solution to the equations (i.e. AB=0)
Question two) so… using the above assumptions and substituting some conveniently round numbers when needed I reckon the following should be correct:
X = 11, (y/2) = 7 :. y=14
For calculation purposes I need to have the points (Xmin,0) (Xmax,0) (0,Ymin) and (0,Ymax), for simplicity C, D, E and F respectively
A is located at (0,<0) B at (0,>0)
Given y > x then A & B lie on axis x=0 :. y = AB+ AE + BF
AE=BF so…
y=AB+2AE=14
——-
Given the shape is an ellipse: AB+AC+BC=2(AB)+2(AE) or AC+BC=AB+2(AE)
Using Pythagoras and that given C lies on the axis y=0 and that A and B are equidistant from the same (hence the half an ellipse assumption):
AC = ((AB/2)2+(x/2)2)0.5 = BC
So substituting in to the above:
2*((AB/2)2+(x/2)2)0.5 = AB+2(AE) or 2*((AB2/4)+(x2/4))0.5 = AB+2(AE)
given y = AB+2AE we have 2*((AB2/4)+(x2/4))0.5 = y
so square out to simplify:
4*((AB2/4)+(x2/4)) = y2
reduce
(4AB2)/4+(4×2/4) = y2
reduce
AB2+x2 = y2 or AB2 = y2-x2
Insert values for y and x to get AB2 = 142 – 112 Solve AB = (142-112)0.5 = 8.66 (ish)
Now that I have AB and I know that for values at y= +or- 7 x = 0 and I can plot my ellipse such that A is found at (0,-4.33) and B at (0,4.33)
Is this right?
Is there a simpler more elegant solution or way of getting there that doesn’t involve me having to inflict physical violence on the architect?
Thanks