Characterize the geodesics of local surfaces in R^2 by the relative positions of their tangent n

Characterize the geodesics of local surfaces in R^2 by the...

Key Concepts

Surface Tangent and Normal Vectors

Every point on a smooth surface possesses a tangent plane and a corresponding normal vector. The tangent plane contains all the possible directions one can move tangentially on the surface, while the normal vector is perpendicular to this plane. When analyzing curves on surfaces, understanding how the curve’s tangent and normal directions relate to the surface’s tangent plane and normal vector is central to characterizing the curve’s geodesic properties.

Geodesics

Geodesics are the curves on a surface that locally minimize distance. They are characterized by having zero geodesic curvature, meaning that any deviation from a straight?line path on the surface is due entirely to the intrinsic curvature of the surface itself. As such, geodesics represent the natural, ‘straightest’ paths constrained to the surface, with their acceleration vectors always normal to the surface.

Frenet-Serret Frame

The Frenet-Serret frame is a moving coordinate system along a space curve that consists of the tangent vector, the principal normal, and the binormal vector. This frame encapsulates the curve’s geometric properties by describing how it twists and turns in space. In the context of curves on surfaces, comparing the Frenet-Serret frame to the surface’s tangent and normal directions allows one to relate the intrinsic curvature of the curve with the extrinsic curvature of the surface.

Geodesic and Normal Curvatures

The curvature of a curve on a surface can be decomposed into geodesic curvature (the component lying in the tangent plane) and normal curvature (the component in the direction of the surface normal). For geodesics, the geodesic curvature vanishes, meaning that the entire curvature is expressed as normal curvature. This relationship provides a way to characterize geodesics by studying how the acceleration vector of the curve projects onto the surface’s normal direction compared to its Frenet-Serret frame.

Transcript

00:01 In this question, we want to find a vector that is normal to the surface, x squared plus y squared minus z equals zero, at the point 3, 4, 25.

00:09 Then i'm going to find the equations of the tangent plane and the normal line to the surface at that point.

00:16 So here i have a surface that is of the form f of x, y, z equals some constant k.

00:25 And so in order to get my normal vector, i need the gradient of f.

00:30 So the gradient of f consists of the partial of f with respect to x, 2x, the partial with respect to y, 2y, and the partial with respect to z, negative 1.

00:43 Now i need to evaluate my gradient vector at the point i was given, 3, 4, 25.

00:50 When i do, i get 6, 8, negative 1, and that is my normal vector.

00:58 So that was the first part of this question.

01:00 Now, i want the tangent plane.

01:06 So to write the equation of a tangent plane, i need a normal vector, which i now have, and a point, which i have as well.

01:14 So i take the x component of the normal vector, 6, times the quantity of x minus the x coordinate, 6 times the quantity of x minus 3, plus the y component of my normal vector, 8, times the quantity of y minus 4.

01:32 And then it's minus, minus 1, times the quantity of z minus 25.

01:39 All of this equals 0.

01:41 That would be my tangent plane...

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