Get intersection points on rotated ellipse with unrotated axis

conic sectionsgeometrylinear algebra

I'm trying to find intersection points of the unrotated axis on rotated ellipse as shown in below visual(black circles)

enter image description here

I have below information of ellipse:

u=3         #x-position of the center
v=6         #y-position of the center
a=np.sqrt(2)#radius on the x-axis
b=1         #radius on the y-axis
t_rot=pi/6  #rotation angle

Also I have bounding box lines:

xa = np.sqrt((a**2*np.cos(t_rot)**2)+(b**2*np.sin(t_rot)**2))
ya = np.sqrt((a**2*np.sin(t_rot)**2)+(b**2*np.cos(t_rot)**2))

as a result for given ellipse info bounding box intersection points:

 ya+v = 7.12 
-ya+v = 4.88
 xa+u = 4.32
-xa+u = 1.68

Is it possible to calculate black dots on the visual with these information?

I know it is quite a general question but I am not sure where to start.

Any suggestion would be helpful.

Thanks!

Best Answer

With respect to the center, the equation of the ellipse is: $$\frac{(x\cos\theta+y\sin\theta)^2}{a^2}+\frac{(x\sin\theta-y\cos\theta)^2}{b^2}=1$$ Your axes are $x=0$ and $y=0$. Plugging in $x=0$, you get $$y^2\left(\frac{\sin^2\theta}{a^2}+\frac{\cos^2\theta}{b^2}\right)=1$$ Can you solve it from here? Similar to $x$, you plug in $y=0$. Then the last step is to add the coordinates of the center.

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