Geodesic & Asymptotic Curves: Proving Segment of Straight Line

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In summary, a regular curve on a smooth surface is a geodesic and an asymptotic curve if and only if it is a segment of a straight line. This can be shown by considering the definitions of "geodesic" and "asymptotic curve" and using the fact that the normal curvature and surface normal have a relation of k<n,N> = a<N,N>, where a is a real number. This leads to the conclusion that if a curve is asymptotic, then it must also be a segment of a straight line.
  • #1
Dragonfall
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Homework Statement



Show that a regular curve on a smooth surface is a geodesic and an asymptotic curve if and only if it is a segment of a straight line.


The Attempt at a Solution



I did the <= implication, which is quite easy. I can't get the other one.
 
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  • #2
A unit speed curve [itex]\gamma[/itex] is a segment of a straight line iff [itex]\ddot{\gamma} = 0[/itex], and all regular curves can be reparametrised to be unit speed. What are the definitions of "geodesic" and "asymptotic curve"?
 
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  • #3
Suppose [tex]\alpha[/tex] a unit speed curve, then it is a geodesic if the covariant derivative [tex]D\alpha/dt=0[/tex] and it is an asymptotic curve if [tex]II(\alpha '(s))=0[/tex] for all s, where II is the second fundamental form and the normal curvature k_n at a point on the curve. Since [tex]k_n=k<n,N>=0[/tex] where k is the curvature of [tex]\alpha[/tex] at a point, n the curve's normal and N the surface's normal, we have to conclude that k=0 or <n,N>=0. This is where I am stuck. What if <n,N>=0?
 
  • #4
My book says that a curve is a geodesic if it's second derivative is zero or if it's second derivative is perpendicular to the surface. Since the second derivative is kn, this is equivalent to saying that kn = aN for some real a.

k<n,N> = <kn,N> = <aN,N> = a<N,N>

if the curve is asymptotic, then k<n,N> = 0, which would imply a = 0, implying that the second derivative is 0, which I claimed to be equivalent to saying that the curve is part of a line segment. So somehow, you need to show that your book's definition of "geodesic" is equivalent to my book's.
 

1. What is a geodesic curve?

A geodesic curve is the shortest path between two points on a curved surface. It is equivalent to a straight line on a flat surface.

2. How do you prove a segment of a straight line is also a geodesic curve?

To prove that a segment of a straight line is a geodesic curve, we use the definition of a geodesic curve which states that it is the shortest path between two points. Therefore, we need to show that any other curve connecting those two points is longer than the straight line segment.

3. What is an asymptotic curve?

An asymptotic curve is a path on a curved surface that approaches a straight line as the two points on the curve get farther apart. It is a curve that is always getting closer and closer to a straight line, but never actually becomes a straight line.

4. How is an asymptotic curve different from a geodesic curve?

An asymptotic curve and a geodesic curve are different in that an asymptotic curve approaches a straight line, while a geodesic curve is the straightest possible path between two points. Additionally, an asymptotic curve only approaches a straight line as the two points on the curve get farther apart, while a geodesic curve is always the shortest path between two points.

5. How can we use geodesic and asymptotic curves in real-life applications?

Geodesic and asymptotic curves have applications in various fields such as physics, engineering, and navigation. They are used in designing efficient transportation routes, constructing bridges and tunnels, and predicting the motion of planets and satellites. They are also used in computer graphics to create realistic 3D models of curved surfaces.

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