Okay. I’m down to one course; three of the four are finished. And I’m thinking about dropping the last one. Digital Signal Processing is an order of magnitude harder than any other MOOC (Massive Online Open Course) I’ve taken. Still, I ought to at least see where it all goes, and then drop it at the end.

I’m not working very hard at getting my tax stuff to my tax man, and I really need to get my act together on that, too. OTOH, “Warhorse” was a good movie to watch last night.

Back to the courses…. The image processing was a spectacular overview, and I really ought to do some coding for it – I have what looks like the perfect book The controls course was a marvelous insight into real control theory: we got a quick intro to PID and state space, and then focused on switching between different control strategies (go-to-goal, obstacle-avoidance, and follow-wall). I was impressed by how much controlling they did with so little theoretical machinery. I haven’t decided what I’ll do to follow up on all that. Maybe I’ll just keep on with what I (was) doing.

We’ll see whether I actually get a technical post done for this Monday….

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February 15, 2019 at 11:18 am

in case of polar coorinates for constrained coordinates are observed as prof dr mircea orasanu and special prof drd horia orasanu followed aspects To generalize the adage–and along the way to explain why planes travel this way–we will introduce a special class of curves on surfaces, called geodesics. Geodesics have the useful property that the shortest curve segment connecting two points on a surface is a segment of a geodesic. As we shall see, great circles are geodesics on the sphere, and they therefore have the property that they are the “shortest” curves on the sphere. To examine geodesics, we will develop connections between differential geometry, differential equations, and vector calculus. In order to see geodesics, even when they cannot be found explicitly, the computer algebra system Mathematica will be used.

1 INTRODUCTION

A surface in three-dimensional Euclidean space (R³) is a set of points in R³ that locally look like a plane-that is, given any point on the surface, there is a small neighborhood of that point which appears to be planar. Again, the earth’s surface taken as a sphere is a good example. The earth’s surface curves, yet by looking around, one cannot see this curvature. This is because the area of the earth one can see is a small enough neighborhood of the point where he/she is standing that this neighborhood appears flat. So the sphere is a surface in R³. More technically,

Definition: M R³ is a surface if for any x M, there exists an open neighborhood U R³ containing x, an open neighborhood W R², and a map x: W → U ∩ M that is differentiable with differentiable inverse. Such a function is called a parameterization or a coordinate patch since it allows us to assign coordinates to the surface corresponding to the coordinates of R².

Surfaces do not need to be in R3; replacing R3 with a general space X yields a valid definition. In fact, some of the surfaces we will examine cannot be placed in R3.

2 FORMULATION

For a sphere of radius r centered at the origin, a coordinate patch is x(u, v) = (rcos(u)cos(v), rsin(u)cos(v), rsin(v)), for – < u < , and – < v < A coordinate patch is said to be orthogonal if its first partial derivatives are orthogonal-that is, if xu•xv = 0. Clearly, for an orthogonal patch x, xu x xv is never zero. This means it is possible to construct a unit normal at any point on the surface. Also, because xu and xv vary smoothly on M, so will U. If any two points on a surface can be connected by a curve contained in the surface, the surface is said to be connected.

Using the sphere as an example, the parametrization of the sphere given above is orthogonal. xu = (-r sin(u) cos(v), r cos(u) cos(v), 0), and xv = (-r cos(u) sin(v), -r sin(u) sin(v), r cos(v)). So xu • xv = r2 sin(u) sin(v) cos(u) cos(v) – r2 sin(u) sin(v) cos(u) cos(v) + 0 = 0. The unit normal at x(u, v) is , as the reader can easily verify either by hand or using Mathematica.

A useful and important construct on a surface M is the tangent plane to the surface at a point p. Parameterize M in a neighborhood of p by x(u, v), with x(u0, v0) = p. Then, the tangent plane to M at p-denoted by TpM-is the two dimensional vector space spanned by {xu(u0, v0), xv(u0, v0)}. It is fairly easy to show that this