the maximal deflection on an ellipse
DESCRIPTION
Mathematics Seminar- first presentationTRANSCRIPT
![Page 1: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/1.jpg)
{
The Maximal Deflection on an Ellipse
Stephanie Moncada
![Page 2: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/2.jpg)
Objectives- Find the maximal deflection between the radial direction and the normal direction.
- Define the normal direction using properties of gradients.
- Find the objective function to be maximized by using lagrange multipliers.
- Formulate the angle between the radial and normal direction by applying the dot product.
![Page 3: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/3.jpg)
- Consider:
An Ellipse centered at the origin, with semi-major axis a, semi-minor axis b, and with these axes along the x- and y- axes respectively.
![Page 4: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/4.jpg)
Maximal Deflection:- Located in the first quadrant- Angle between the normal and radial
vectors- Will not occur at either the X or Y intercepts
![Page 5: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/5.jpg)
The maximal deflection occurs where the ellipse meets the line from the origin to (a, b).
![Page 6: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/6.jpg)
![Page 7: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/7.jpg)
Lagrange Multipliers
To use Lagrange multipliers we need: - An objective function to be maximized.- A constraint.
![Page 8: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/8.jpg)
Since we consider only points of the ellipse, its equation defines the constraint. Accordingly, we define the function.
Where the constraint is g(x, y)=1
- CONSTRAINT
Constraint: a condition to an optimization problem that is required by the problem itself to be satisfied.
![Page 9: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/9.jpg)
For the objective function , we want the angle between the normal and the radial vectors at a point (x,y) on the ellipse.
We take r = (x,y) as the radial vector. For the normal vector, we take n = (x/a, y/b), which is one-half of the gradient of g.
Then, δ is determined by the equation.
- OBJECTIVE FUNCTION
![Page 10: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/10.jpg)
Now we can observe that r · n = g(x, y) = 1 for any point on the ellipse. Accordingly, we simplify matters by inverting and squaring to obtain
We define this to be our objective function. That is,
Given that δ is determined by the equation
![Page 11: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/11.jpg)
For (x, y) in the first quadrant and on the ellipse, we know that δ is between 0 and π/2. On this interval, sec^2 δ is an increasing function. Therefore, δ is maximized where f is.Our problem now is to maximize f subject to the constraint g = 1. The solution must occur at a point where ∇ f and ∇g are parallel.Thus, this leads to the single equation:
This shows that in the first quadrant, the solution to our optimization problem must lie on the line joining the origin to (a, b).
From this equation, it is straightforward to derive:
![Page 12: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/12.jpg)
As a first step, we compute the partial derivatives
![Page 13: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/13.jpg)
Combining these leads to
![Page 14: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/14.jpg)
Theorem
(1)
![Page 15: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/15.jpg)
Several methods to obtain the maximal deflection on an ellipse:
- Direct Parameterization: Using the standard parameterization of the ellipse
- Using Slopes: this method expresses everything in slopes.
- Symmetry: there is a symmetry that makes the location of the point of maximal deflection natural.
![Page 16: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/16.jpg)
ApplicationThe maximal deflection problem has one application.It concerns the ellipsoidal model of the Earth, and two ways to define latitude.
On a spherical globe, the latitude at a point is the angle between the equatorial plane and the position vector from the center of the sphere.
![Page 17: The maximal deflection on an ellipse](https://reader036.vdocuments.site/reader036/viewer/2022062313/55810b3ed8b42a05558b4b68/html5/thumbnails/17.jpg)
Sources- The Mathematical Association of America.
- Dan Kalman, Virtual Empirical Investigation: Concept Formation and Theory Justification, Amer: Math. Monthly 112 (2005), 786-798.
- William C. Waterhouse, Do Symmetric Problems have Symmetric Solutions?, Amer: Math. Monthly 90 (1983). 378-387.