Mathematics Seminar- first presentation

1. The Maximal Deflection
on an Ellipse
{ Stephanie Moncada

2. 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.

3. - 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.

4. Maximal Deflection:
- Located in the first quadrant
- Angle between the normal and radial vectors
- Will not occur at either the X or Y intercepts

5. The maximal deflection occurs where the ellipse meets
the line from the origin to (a, b).

7. Lagrange Multipliers
To use Lagrange multipliers we need:
- An objective function to be maximized.
- A constraint.

8. - CONSTRAINT
Constraint: a condition to an optimization problem that is
required by the problem itself to be satisfied.
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

9. - OBJECTIVE FUNCTION
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.

10. Given that δ is determined by the equation
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,

11. 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:
From this equation, it is straightforward to derive:
This shows that in the first
quadrant, the solution to our
optimization problem must lie on the
line joining the origin to (a, b).

12. As a first step, we compute the partial derivatives

13. Combining these leads to

14. Theorem
(1)

15. 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.

16. Application
The 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.

17. 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.