How to extract length along an angle that intercepts an ellipse.

I am working on a problem where I obtain a strain matrix and want to extract the magnitude of that strain along a specified angle. The way I envision it is that the strain matrix represents the vectors corresponding to the major and minor axis of an ellipse and I then want to find the magnitude of a line along my specified angle that intersects that ellipse.

For example above I have a strain matrix that would be defined as

E=[ 1 0;
    0 0.5];

This could be represented as the blue ellipse I plotted in the figure. In this scenario my then specified angle is 45˚ (pi/4) represented by the red line. In this scenario the magnitude I am looking for would be:

sqrt(0.45^2+0.45^2)=0.6364

Now I am trying to figure out how to do this via math. I imagine I should be able to simply rotate the ellipse by the negative of my given angle so that the point along the angle I am interested in is now along the x axis and I should be able to simply extract out the x component, as shown graphically:

So in doing this mathematically I figure I rotate the strain matrix as below:

E=[ 1 0;
  0 0.5];
rotation=[cos(-pi/4) -sin(-pi/4);sin(-pi/4) cos(-pi/4)];
rotated=rotation*E;

I then end up with:

rotated=[ 0.7071     0.3536;
         -0.7071    0.3536];

Which by my earlier definition of an ellipse gives the red rotated ellipse above (one axis point at 0.7071,-0.7071 and one axis at 0.3536,0.3536) which should intersect the X-axis at my desired point of x=0.6364.

So how do I extract out this value from the matrix:

rotated=[ 0.7071     0.3536;
         -0.7071    0.3536];

Is there a simpler or easier way to accomplish this same task? Thank you!

[James Tursa]

So you are starting with a 2x2 matrix, where the columns represent the endpoints of the principal axes of an ellipse. And you are given an arbitrary angle (e.g., from the x-axis). And you want the coordinates of the point where a line at that angle intersects the ellipse?

[JL]

Yes that is the goal.

[JL]

I guess another way of asking the question is: Given only the 2 endpoints of the vectors corresponding to the principal axis of an ellipse, can I generate an exact equation for that ellipse in order to calculate specific values on that ellipse?

Answer 1

[Roger Stafford]

You apparently have the center of your ellipse located at the coordinate origin. If [px,py] is the endpoint of one principal axis and [qx,qy] the other principal axis, then an equation of the ellipse is:

   (px*x+py*y)^2/(px^2+py^2)^2+(qx*x+qy*y)^2/(qx^2+qy^2)^2 = 1

for points (x,y) on the ellipse.

This is rather different from your original question of finding the length of a line from the origin to a point on the ellipse if the angle of the line with respect to the x-axis is known. The way to solve this second question is first to determine the angle between the one principal axis and the given line. For that see the thread:

   http://www.mathworks.com/matlabcentral/newsreader/view_thread/162503

in which I give a formula for this using the 'atan2' function.

Once this angle is determined, the distance from the center to a point on the ellipse is:

d = 1/sqrt(cos(t)^2/a^2+sin(t)^2/b^2);

where t is the angle of the line to the principal axis of length a and where b is the other orthogonal principal axis length.

[JL]

Thank you! I was able to solve the given equation for x at y=0 and obtain exactly what I was looking for. Much appreciated.

Answer 2

[James Tursa]

To get started, e.g.,

>> % Original data
>> E = [1 0; 0 0.5]
E =
    1.0000         0
         0    0.5000
>> rot_angle = -pi/4
rot_angle =
   -0.7854
>> % Create rotation
>> rotation = [cos(rot_angle) -sin(rot_angle); sin(rot_angle) cos(rot_angle)]
rotation =
    0.7071    0.7071
   -0.7071    0.7071
>> rotated = rotation*E
rotated =
    0.7071    0.3536
   -0.7071    0.3536

Now the recovery part:

>> % Recover principal axes lengths
>> v = sqrt(sum(rotated.*rotated))
v =
    1.0000    0.5000

>> % Recover principal axes rotation angle
>> rot_calculated = atan2(rotated(2,1),rotated(1,1))
rot_calculated =
   -0.7854

Now assume you have the following:

ang = angle from x-axis for the "line"

Then you have two equations to solve for x and y:

(x/a)^2 + (y/b)^2 = 1   % a and b are the elements of v above

and

x * tan(ang-rot_calculated) = y

So solve these two equations for x and y, then re-create the rotation matrix (from ang_calculated) and rotate x and y to give you the point coordinates.

If abs(tan(ang-rot_calculated)) > 1, then maybe work with the following equation instead:

x = cot(ang-rot_calculated) * y

I.e., plug either the x expression or the y expression into the ellipse equation and solve for the other.

[JL]

Thank you for the response, maybe there was a confusion but the angle of the "line" that I am looking for is the amount I rotated it to begin with so

 ang==rot_calculated

and therefore I end up with tan(0) and am unable to get any data.

The whole reason I rotated it in the first place was to make sure the intersection point I am interested in that was originally along the line defined by ang now lies on the x-axis.

[James Tursa]

That's why I posted a problem summary and asked for confirmation first. If that problem summary is not accurate, then please re-state your problem so we can address the real issue. Once you recover the principal axes lengths and the rotation angle, I think you have everything you need to get your answer. We simply need confirmation of what exactly this "line angle" is, I guess.

Are you saying that you actually know E up front, and all of this rotation business is something inserted by you as an attempt to solve the problem? I.e., do you just have E and a line angle and simply want to know where this line angle intersects the ellipse? If so, then just use those last two equations above.

[JL]

The problem summary you gave was correct and that is what I am looking for. I am confused by your solution in that you are stating there is a difference in the initial rotation angle and then the recovered rotation angle. In my problem, these will always be the same i.e.

 rot_angle=ang

thus

 rot_calculated=ang

and then

 tan(ang-rot_calculated)=0

I only used pi/4 as an example of the graphics that I displayed.

[James Tursa]

So you have "rotated", and you are simply trying to find the intersection of that "rotated" ellipse and the x-axis?

[JL]

Yes my known values are the angle I am interested in (ang) and the original matrix. I then rotated the original matrix by ang to get rotated. Now I want to find where the rotated ellipse crosses the x axis.