Proving length of Polygon = length of smooth curve

In summary, the problem asks to show that the length of a polygonal curve and a smooth curve are equivalent if we consider an infinite amount of partitions of the interval [a,b]. I would approach the problem as follows: I would show that the total length of the curve was the sum of the integrals of each equidistant partition.
  • #1
MxwllsPersuasns
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Homework Statement


The problem statement is in the attached picture file and this thread will focus on question 7

Homework Equations


The length of a curve formula given in the problem statement
Take a polygon in R^n as an n-tuple of vectors (a0,...,ak) where we imagine the vectors, ai, as the vertices of the polygon and two successive vertices (ai and ai+1) be connected by the corresponding segment of straight line.
- Thus, the length of a polygon is the sum of the lengths of the line segments: Σk-1i=0 { absval(ai+1-ai)}

The Attempt at a Solution


Basically this question is asking to show that the length of a polygonal curve and a smooth curve are equivalent once we consider an infinite amount of partitions of the interval [a,b]. I'm not exactly sure how to write this mathematically but I would approach the problem as follows:

I would show using the length of the curve formula that the total length of the curve was the sum of the integrals of each equidistant partition i.e., take an integral from (a=)t0 to t1 of γ'(t) then another integral from t1 to t2 all the way to the last integral of tn-1 to tn(=b) and then try to show that the line segment joining the vertices γ(t0) and γ(t1), say, had the same length as the integral of the curve from t0 to t1 and then basically just say as you get closer to ∞ line segments the two lengths converge.

Is this on the right track in terms of concept? Anyone who can help me with the approach or helping to translate it into a mathematical argument would be very much appreciated!
 

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  • #2
This is almost impossible to read and I doubt that many members will download and resize it. Can you type question 7 in LaTex here instead?
 
  • #3
Sure I will give it my best try, here goes:

Let γ: [a,b] →ℝn be a smooth curve and a = t0 < t1 < ... < tn = b an equidistant partition of [a,b], i.e., tk+1-tk = 1/n for all k = 0,...,n-1. Consider the open polygon Pn with vertices γ(t0), γ(t1),...,γ(tn) and denote its length by L(P). Show that...

limn→∞ L(P) = L(γ) = ∫AbsVal{γ'(t)}dt where the limits of integration are from a to b
 
  • #4
Anyone have any ideas on how to proceed?
 
  • #5
Never mind I figured it out, thanks anyways all!
 

Related to Proving length of Polygon = length of smooth curve

1. What is the importance of proving the length of a polygon is equal to the length of a smooth curve?

The length of a polygon and the length of a smooth curve are fundamental concepts in geometry and calculus. Proving that they are equal provides a deeper understanding of the relationship between these two geometric shapes and can be used in various mathematical applications.

2. How is the length of a polygon and the length of a smooth curve defined?

The length of a polygon is defined as the sum of the lengths of all its sides. On the other hand, the length of a smooth curve is defined as the integral of its derivative over a given interval. This means that the length of a smooth curve is calculated by finding the area under its derivative curve.

3. Can a polygon have the same length as a smooth curve even if they have different shapes?

Yes, it is possible for a polygon and a smooth curve to have the same length even if they have different shapes. This is because the length of a smooth curve is not dependent on its shape, but rather on the area under its derivative curve.

4. What mathematical techniques are used to prove the equality of the length of a polygon and the length of a smooth curve?

The equality of the length of a polygon and the length of a smooth curve can be proven using calculus techniques such as integration and the Fundamental Theorem of Calculus. Additionally, geometric proofs involving congruent triangles and the Pythagorean Theorem can also be used.

5. Are there any real-world applications of proving the equality of the length of a polygon and the length of a smooth curve?

Yes, there are many real-world applications of this concept. For example, it can be used in engineering to calculate the length of a curved road or bridge, in physics to determine the arc length of a projectile's path, and in computer graphics to accurately render smooth curves on digital images. It also has applications in fields such as architecture, robotics, and animation.

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