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Tilted ellipse

As the above link figure shows, let's assume that we have a tilted ellipse, say "theta" angle from its original position. How can we get the coordinate of x1, x2 which have a same y value? Either getting it analytically, or numerically would be okay. But maybe numerical approaches using, for example, Python would be more appropriate for this community and promising, I guess.

tripleee
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Jinho Ryu
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  • Can you show the code you attempted to solve the problem? – thewaywewere Apr 20 '17 at 15:32
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1 Answers1

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Origin-centered ellipse, rotated by angle Theta, has equations

x = a * Cos(t) * Cos(theta) - b * Sin(t) * Sin(theta) 
y = a * Cos(t) * Sin(theta) + b * Sin(t) * Cos(theta)

We can introduce pseudoangle Fi and magnitude M

 Fi = atan2(a * Sin(theta), b * Cos(Theta))
 M = Sqrt((a * Sin(theta))^2 + (b * Cos(Theta))^2)

so

 y = M * Sin(Fi) * Cos(t) + M * Cos(Fi) * Sin(t)
 y/M = Sin(Fi) * Cos(t) +  Cos(Fi) * Sin(t)
 y/M = Sin(Fi + t) 

 Fi + t = ArcSin( y / M)
 Fi + t = Pi - ArcSin( y / M)
 t1 = ArcSin( y / M) - Fi        //note two values
 t2 = Pi - ArcSin( y / M) - Fi

Now substitute both values of t in the first equation and get values of X for given Y

If you have general ellipse equation like

 A*x^2 + 2*B*x*y + C*y^2 + D*x + E*y + F = 0

just substitute y by known value ans solve quadratic equation for x

MBo
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