General Optimization Question

[sdobbers]

How would you go about finding the highest point on the curve of intersection of an ellipsoid and a plane? Given: x^2 + y^2 + z^2 = k and ax + by + cz = j. I was thinking about using Lagrange Multipliers but I would always get stuck.

 

[EnumaElish]

How did you set up the problem and where did you get stuck?

 

[tacman]

One approach to finding the highest point on the curve of intersection between an ellipsoid and a plane could be to first solve for one of the variables in terms of the other two equations. For example, we could solve for z in terms of x and y in the equation of the ellipsoid, giving us z = √(k - x^2 - y^2). Then, we can substitute this expression for z into the equation of the plane, giving us ax + by + c√(k - x^2 - y^2) = j.

Next, we can use calculus to find the maximum value of this function by taking the partial derivatives with respect to x and y, setting them equal to 0, and solving for x and y. This will give us the coordinates of the highest point on the curve of intersection.

Alternatively, as mentioned, Lagrange Multipliers could also be used to find the maximum value of this function subject to the constraint of the ellipsoid equation. However, as you mentioned, this method may be more complicated and may require more advanced mathematical techniques.

Overall, the specific approach to finding the highest point on the curve of intersection will depend on the specific context and problem at hand. It may be helpful to consult with a mathematical expert or utilize online resources to explore different techniques and determine the most effective approach for the given situation.