Variable radius of curvature as a function of sagitta

[hupernikomen]

How can I express a radius of an arc R in terms of its sagitta H assuming that: (1) the arc follows a locus of a circle with radius R, (2) arc length S is constant. I’m interested in the region where the included angle of the arc is between but not equal to 0° and 4°.

 

[stapel]

How can I express a radius of an arc R in terms of its sagitta H assuming that: (1) the arc follows a locus of a circle with a variable radius, (2) arc length S is constant. I’m interested in the region where the included angle of the arc is between but not equal to 0° and 4°.

What do you mean by "a circle with a variable radius"? Thank you!

 

[hupernikomen]

What do you mean by "a circle with a variable radius"? Thank you!

Perhaps I should restate the assumption as follows: "the arc follows a locus of a circle with radius R".

 

[stapel]

Perhaps I should restate the assumption as follows: "the arc follows a locus of a circle with radius R".

Is the following correct?

You've got a circle with radius R. There are two points on the circle, A and B. Using these points as endpoints, there is a chord (let's call it C) which cuts an arc from the circle (let's call the arc AB_arc). There is a line from the center of this arc to the midpoint of L; if continued, this line would reach the center of the circle. Let's call this line H. The sagitta is the length of H. (definition of "sagitta") We will assume that none of A, B, C, L, or the circle changes in any way.
​

If the above is not correct, please reply with clarifications.

You then state:

Express a radius of an arc R in terms of its sagitta H assuming that: (1) the arc follows a locus of a circle with radius R, (2) arc length S is constant.

Since the arc is, by definition, part of the circle, what do you mean by "the arc follows a locus of a circle"? Also, for any given arc, its length (and all other aspects) are constant, so I'm not understanding this "restriction"...?

I’m interested in the region where the included angle of the arc is between but not equal to 0° and 4°.

How are you expecting this "restriction" to affect the computations? Also, if the arc length is fixes, then the angle cannot "range" between zero and four degrees. So is the arc fixed, or is it not? Also, the radius R of the circle (and thus also of the arc) is fixed, so how are you expecting to express the fixed value of R in terms of the (variable?) value of H?

Thank you!

 

[hupernikomen]

Is the following correct?

You've got a circle with radius R. There are two points on the circle, A and B. Using these points as endpoints, there is a chord (let's call it C) which cuts an arc from the circle (let's call the arc AB_arc). There is a line from the center of this arc to the midpoint of L; if continued, this line would reach the center of the circle. Let's call this line H. The sagitta is the length of H. (definition of "sagitta") We will assume that none of A, B, C, L, or the circle changes in any way.
​

If the above is not correct, please reply with clarifications.

You then state:


[/FONT][/COLOR]Since the arc is, by definition, part of the circle, what do you mean by "the arc follows a locus of a circle"? Also, for any given arc, its length (and all other aspects) are constant, so I'm not understanding this "restriction"...?


How are you expecting this "restriction" to affect the computations? Also, if the arc length is fixes, then the angle cannot "range" between zero and four degrees. So is the arc fixed, or is it not? Also, the radius R of the circle (and thus also of the arc) is fixed, so how are you expecting to express the fixed value of R in terms of the (variable?) value of H?

Thank you!


[/FONT][/COLOR]

Assume a line with constant length, S that is able to flex in such a way that it forms a circular arc with radius R. As the line flexes, both the sagitta of the arc, H and R change. For instance at H=0, the line is straight and R = ∞. Is it possible to express R as a function of H?

So far I have arrived at:
Cos (S/R) = 2(H/R)² - 4(H/R) + 1
But what I'm looking for is a way to calculate, if possible, R given a value of H.