how can i get the length of a line that has turns? or waves?

Question

a line curved up then curve horizontal across a plateau, then curved back down and curved back horizontal. like the outline of a derby hat with a flat top.

what if the line in question is a bent piece of metal. requiring a specific base length, height, and top length.

also this is not a physical object yet. i need to know for manufacturing.

Answer 1

The actual distance your feet travel on a hike is usually greater than the distance measured from the map. Still, you can get a fairly accurate measurement - even along a curved line - using this technique.

Steps:
1. Locate and mark on the map the start and end points of the trail you'd like to measure.

2. Align the straight edge of a piece of paper on the map along the start of the trail and the first section of your planned hike.

3. Make a slash mark with a pencil at the start of the trail, leaving a tick mark on both the straightedge and the map.

4. Mark another pencil slash on the paper and map at the first point where the trail either bends away from the paper's edge or disappears underneath it.

5. Keep the second slash mark on the paper lined up with the second mark on the map, and pivot the paper until its edge is aligned with the next section of trail.

6. Make another slash across both the straightedge and map at the next place the trail bends away from the paper.

7. Pivot the paper again until it lines up with the successive section of trail, and make another slash.

8. Continue until you've reached the end of the distance you want to measure.

9. Measure that distance against one of the bar scales at the bottom of your map. (See "How to Measure a Straight Line Distance Using a Topo Map," under Related eHows.)

Tips:
The sum total of distances marked on the paper straightedge, from first slash to last, is an approximation of the trail's distance.

Be precise when measuring a curvy distance; the more twists of the straightedge and corresponding marks you make, the more accurate your measurement will be.

Answer 2

If it's perfectly round curves, then treat them as portions of a circle.

In the general case, you'd use integral calculus. The basic point of all calculus is dealing with things really close to zero. Here's how that happens:

Imagine you cut the line into a bunch of small pieces. What would be the length of each piece? Let us call delta-x or dx the width of a piece, and delta-y or dy the height of a piece. Then each piece, thanks to Pythagoras, would have length sqrt(dx^2 + dy^2).

So we're adding together a bunch of pieces of that length. Let's use the letter S for sum, and say that each dx is the same (and only the dy changes, depending on how curved the line is).

S sqrt(dx^2 + dy^2)

Now we can do some algebra and say that this is the same as:

S sqrt(1 + (dy/dx)^2) * dx

And we let all of these tiny pieces tend towards zero, and this sum become a continuous sum. That's called an integral (it's the slightly stretched S symbol you may have seen). The ratio dy/dx is nothing other than the slope of the line, rise over run. So it's a calculus function called a derivative, which represents the continuous slope of the function across every tiny piece.

This formula is the arc-length formula for calculus. Once you take calculus, you'll know how to find the derivative, and how to integrate functions. Then it's just algebra to find what exactly sqrt(1 + derivative^2) comes out to.

Answer 3

If your problem is a tough mathematical one, then you should read a little bit in differentiation and integration.

If you want to measure the length of a line in real-life, then simply use a string or thread to measure it.

Answer 4

if the turns are perfectly curved, you can find the radius of the partial circle each curve makes and get the length of each partial circle. If the curves are trigonumetric, use sine, cosine or tangent formulas

Answer 5

If you have mathematical formula(s) for the shape of the line, you can do a bit of integral calculus to figure it out. Otherwise, use a tape measure.

Answer 6

Use string and run it along the entire edge of the surface then use a yard stick(or any tool of measurement)and measure out the string.

Answer 7

Use a string to measure then flatten it out and measure.

Answer 8

That's called calculus. Can you calculate?

Answer 9

pull the line straight.

Answer 10

Use a string / thread