Working with Arcs

[ch10]

I have an arc of known distance (I don't know what other information is needed for an arc but I can supply it) with two lines of known distance that form a right angle.

From this arc I need two things. First I need a function that describes x for any h. Second I need to draw an arc which for any h has a measure of x + 2/3x. Any suggestions are appreciated.

 

[wjm11]

I have an arc of known distance (I don't know what other information is needed for an arc but I can supply it) with two lines of known distance that form a right angle. Arc.jpg

From this arc I need two things. First I need a function that describes x for any h. Second I need to draw an arc which for any h has a measure of x + 2/3x. Any suggestions are appreciated.

I made some assumptions. I assumed this was a circular arc. I assumed the segment of length 4 was on a diameter, i.e., that the segment would pass through the center of the circle if extended.

Using the segments of lengths 4 and 8, I calculated the circle to have a radius of 10.

With r = 10, I calculated an arc length of 9.273, which approximates your stated value of 9.342.

If these assumptions and approximations are sufficient, we can assume your arc is modeled by a circle formula: (x + 6)^2 + h^2 = r^2; ((x + 6) squared + h squared = radius squared):

(x + 6)^2 + h^2 = 10^2

or

x = (100 – h^2)^(.5) – 6

(x = the square root of (100 – h squared), then minus 6.)

check: When h = 0, x = 4. When h = 8, x = 0. Looks okay.

 

[ch10]

Thanks for the help! In this context using a circular arc is fine so I will update my arc length to 9.273. In the future, though, I won't be able to do this. I've attached another graphic which shows the arc I posted before as well as the radius of the circle . The radius doesn't run through the arc (r=9.3724 which intersects the vertical segment at 0.316). I realize this problem operates on a rather small scale so if there is another example you can use (or just lettered variables) that would be great. I just need to know how to solve this problem for non-circular arcs in the future. Thanks again!

 

[wjm11]

I just need to know how to solve this problem for non-circular arcs

In order to have a formula for your arcs, it is necessary that the arcs have some sort of mathematical definition, such as being part of a circle, ellipse, parabola, etc. It is insufficient to just call something an arc. Secondly, it will be necessary to know more details about the figure than were initially provided in this problem. For example, it would be necessary to know where the center of the circle is, what the radius is, etc.

I would suspect that some sort of Autocad program might be useful to you for solving future problems.

 

[ch10]

Yes, this is from autocad and is a simplification of a 3-d problem. I have a cone with a curved surface which models a tomb in Greece (I am an archaeologist). I need to ensure that the tomb walls at 2/3x thick to appropriately capture the shape of the tomb, most of which is hidden. I can provide any measurements necessary based on the arc given since that is known based on measurements taken on site. What data in particular is necessary?

 

[wjm11]

I have a cone with a curved surface

By cone with a curved surface, do you mean an inverted ice cream cone that does not come to a point at the top, but is rounded instead? If so, you might be able to approximate this shape with a parabola.

I need to ensure that the tomb walls at 2/3x thick

Does this mean that at the base, the wall thickness, t, is

t = (2/3)(x) = (2/3)(4) = 8/3

and that at the top of the 8 (meters or feet?) that

t = (2/3)(x) = (2/3)(0) = 0 ?

You can take the formula I gave you for x, ( x = (100 – h^2)^(.5) – 6 ), and put it into a spreadsheet (like Excel) and generate as many (h,x) values as you like. Likewise, in a column next to those calculations, you can use the calculated x values and generate a list of thicknesses for those same heights. You could then transfer those values to your Autocad drawing if you like. Make sense?