Arc Length (Pt. II)
Point of Post: This is a continuation of this post.
What are the properties of this arc length function? Well, perhaps one of the most obvious one is that the arc length function is invariant under isometries of . Recall that an isometry of is a distance preserving map, i.e. a map with the property that . A nice way to think about isometries are that they are the composition of a rotation and a translation (i.e. a rigid motion). Or, if one likes to think more about matrices then isometries are just affine maps where is an orthogonal matrix. We denote the set of all isometries of by . Now, as stated before we have the following result:
Theorem: Let be a smooth curve, , and . Then, .
Proof: It clearly suffices to prove that for an arbitrary partition in one has that . To do this we merely note that
from where the conclusion follows.
The next obvious geometric fact should be that the curve between any two points in which minimizes arc length is a straight line, i.e. backing up the oft spouted ‘the shortest path between two points is a straight line’.
Theorem: Let , then for any curve with and , with , one has that where starts from .
Proof: We merely note that if is a unit vector then
Taking then gives the desired result.
1. Carmo, Manfredo Perdigão Do. Differential Geometry of Curves and Surfaces. Upper Saddle River, NJ: Prentice-Hall, 1976. Print.
2. Montiel, Sebastián, A. Ros, and Donald G. Babbitt. Curves and Surfaces. Providence, RI: American Mathematical Society, 2009. Print.
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