Finding the length of a curved line between two points

[biker]

Hi,

In the drawing below ..

The distance of point "a" to point "c" is 8 meters

Point "b" is halfway between points "a" and "c" (4 meters for this example)

The distance between points "b" and "d" is 1 meter

What is the best method for determining the length of the curved line that contains points "a" "d" and "c" ?

There is a second part of this question, but wanted to begin with this.

Thank you!

 

[stapel]

Hi,

In the drawing below ..

The distance of point "a" to point "c" is 8 meters

Point "b" is halfway between points "a" and "c" (4 meters for this example)

The distance between points "b" and "d" is 1 meter

What is the best method for determining the length of the curved line that contains points "a" "d" and "c" ?

There is a second part of this question, but wanted to begin with this.

Thank you!


View attachment 36928

The best method would probably be to have an equation for the curve adc. Otherwise, any answer is going to have to involve guess-work.

What other information have you been given for this exercise?

 

[Dr.Peterson]

Hi,

In the drawing below ..

The distance of point "a" to point "c" is 8 meters

Point "b" is halfway between points "a" and "c" (4 meters for this example)

The distance between points "b" and "d" is 1 meter

What is the best method for determining the length of the curved line that contains points "a" "d" and "c" ?

There is a second part of this question, but wanted to begin with this.

Thank you!


View attachment 36928

Assuming the curve is an arc of a circle, you will find relevant formulas here:

Circular segment - Wikipedia

en.wikipedia.org

You want to find an arc length, given a chord and its sagitta (height of the segment).

If you have trouble putting it all together, show us your work as far as you get. Of course, if you don't know that you have a circular arc (could it be a parabola instead?), then whatever you calculate will just be an approximation.

 

[Steven G]

I would find the equation of the ellipse that fits the given information and find the arc length of half of the ellipse (that formula is known). Since your picture may not be an ellipse, then the length you'll get will approximate the actual arc length.

 

[Dr.Peterson]

I would find the equation of the ellipse that fits the given information and find the arc length of half of the ellipse (that formula is known). Since your picture may not be an ellipse, then the length you'll get will approximate the actual arc length.

Unfortunately, not only does the picture not look at all like half an ellipse, but there is no exact formula for the arc length (circumference) of an ellipse, making this a harder "solution" (approximation of an approximation).

 

[khansaheb]

Unfortunately, not only does the picture not look at all like half an ellipse, but there is no exact formula for the arc length (circumference) of an ellipse, making this a harder "solution" (approximation of an approximation).

I seem to recollect a whole section of calculus was invented (and bunch of tables generated ) while trying to calculate the arc length of a part of ellipse - elliptic integrals.

 

[blamocur]

The drawing looks rather close to circular arc :

 

[Steven G]

The drawing looks rather close to circular arc :
View attachment 36948

Which is a special ellipse!

 

[Dr.Peterson]

[circular arc] is a special ellipse!

But not ...

I would find the equation of the ellipse that fits the given information and find the arc length of half of the ellipse (that formula is known). Since your picture may not be an ellipse, then the length you'll get will approximate the actual arc length.

(And it's even harder to calculate the length of a partial arc of an ellipse.

 

[blamocur]

But not ...

(And it's even harder to calculate the length of a partial arc of an ellipse.

Exactly why I prefer to believe it is a circle