Geometry help, please?

[CoreyK]

Haha - my geometry skills are not the best, to say the least.
I am building a canopy for my (36g corner) bowfront tank, however I don't have the tank with me to consult. I did take some measurements before leaving, however.
What I need to know is the angle/radius (whatever) of the arch of the bow.
These are the measurements I have:

The 7" refers to the middle, farthest point the arch bows out.
Is it possible to determine the angle/radius (whatever) of the arch (in red) from this information? If so, could anyone do me a HUGE favor and use their math skills to figure it out?
I should have paid attention in school, lol.
Thanks!

 

[ReefEnabler]

we know:
H=7
c=37.125

R= h + d

unfortunately we do not know d or R so its not that simple.

according to this site http://mathworld.wolfram.com/CircularSegment.html

we can also use

c = 2(sqrt[h(2R - h)])

so we can solve that one

37.125 = 2 (sqrt [7(2R - 7)])
18.5625 = sqrt [7(2R - 7)]
18.5625 = sqrt [14R - 49]

^2 to both sides

344.566 = 14R - 49
393.566 = 14R

R = 28.111
that should be your radius. I just followed the steps for the formula I linked to. on top of that its almost 2am and I haven't done this in years, so I'd double check the math if I were you.

some real simple guess and check by comparing to a 90 degree arc on the same tank:

if your bowfront was a perfect 90 degree arc, the radius would be 26.25". From the corner of the tank to the center of the chord is 18.56". 18.56+7= 25.56, or just a little bit smaller than 26.25 which means your arc is just a bit under 90 degrees, so an actual radius of 28.111 is consistent with that.

 

[ReefEnabler]

if you want to find the length of the arc the link above also has formulas to use for that. you should have the variables needed to solve anything else.

 

[saltyseaman]

We do know d, using the Pythagorean theorem. In this diagram, B=(C^2-A^2)^(1/2)--> B=d=18 9/16"

So d+h=18 9/16"+7"=25 9/16"

this means that the footprint of that tank isn't a perfect circular sector. If it was then B+7 would equal A, the radius of the circle. This leads me to believe that the curvature on that bowfront may be aspheric, making things a tad complicated.

 

[ReefEnabler]

no, B does NOT equal D!!!!

thats only the case if the arc were 90 degrees, which it is clearly not.

 

[ReefEnabler]

well now you got me questioning myself lol!!
its way too late

i resign my post for now

 

[ReefEnabler]

i think what youre misunderstanding from the diagram, is that the corner of his tank is NOT the center of the circle in the diagram I posted. its about an inch offset.

I showed that by comparing it to the 90 degree arc.

 

[saltyseaman]

Whoops, means to say B+7 would equal C not A.

This may be parabolic in nature, in which case you can use this

http://mathworld.wolfram.com/ParabolicSegment.html

 

[saltyseaman]

<a href=showthread.php?s=&postid=12022759#post12022759 target=_blank>Originally posted</a> by RyanBrucks
i think what youre misunderstanding from the diagram, is that the corner of his tank is NOT the center of the circle in the diagram I posted. its about an inch offset.

I showed that by comparing it to the 90 degree arc.

.... ok .. let me think about this for a second ... lol

 

[WuWu]

I used a formula that says

R=C(squared)/8m+m/2

C=chord (37 1/8)
m=middle ordinate (7)
R=radius of the curve (?)

1378.265625/56+7/2=R

24.61188616+3.5=28.11188616

I am by no means a math major, but thats a formula I learned for determining the radius of a curve. Anyways for what its worth you might want to check the work and make sure it works out correct, but I got the same as Ryan Brucks.

 

[ReefEnabler]

thats reassuring.

figuring out the angle of the arc should be easy then with some cosine stuff. i cant recall any of it

i need to reinstall photoshop. im horrible at explaining stuff without drawing my own pictures hehehe

 

[saltyseaman]

Isn't the R that you are using in your diagram the Radius of the circle? If so, R would have to be 26.25, right? The radius is measured from the center of the circle.

Assuming the curvature is circular, I think we are measuring something more like this in the brown

You're right, it's late :lol: I can't think straight.

I am using MS paint for my picture editing

 

[ReefEnabler]

youve got the right idea with that picture, but missing 1 thing I think... (reinstalling photoshop)...


it's more like this (exxagerated the difference)

R=28.111 (i hope :lol: )

 

[saltyseaman]

Ohhhh, I feel like an idiot. I think I misunderstood the "What I need to know is the angle/radius (whatever) of the arch of the bow" part.

you're right. Time for bed! :lol:

 

[ReefEnabler]

heres a question worth asking though.....

how do we know that bowfront tanks are manufactured with an actual circular curve and not just some arbitrary mould that just happens to have 7" of bow over the distance???????

 

[saltyseaman]

Yeah, that's what I was trying to say with the whole aspheric (meant not circular) and parabolic thing, albiet I was lead to that suspicion for the wrong reasons.

CoreyK, you could test this rather easily. Could you try it out and chime back in?

(It's hard to peel yourself away when you know the person on the other side is still awake )

 

[ReefEnabler]

you had a valid point but I was still focused on solving the question assuming the curve was an actual arc from a circle, which is an assumption we had to make in order to do any math, really.

now that weve got an idea about the math, making sure that the assumptions about the tank are correct is a must.


anyways, hope this helps; now I really am going to sleep since I do have to work a pretty long day tomorrow.

cheers,
Ryan

 

[Fiziksgeek]

I would recommend getting the tank, then building the canopy....problem solved.

 

[Chrsnwk]

I did a quick layout in CAD for you based on the dimensions you provided...

The length of the arc is ~40 - 9/16"

 

[logiktest]

<a href=showthread.php?s=&postid=12023293#post12023293 target=_blank>Originally posted</a> by Fiziksgeek
I would recommend getting the tank, then building the canopy....problem solved.

I think Fiziksgeek has this problem solved. You folks are obviously proficient with your math (pure genius if you ask me), but aren't you all over thinking this just a bit? Would it be more efficient to just to lay a board or a piece of paper on the top or bottom of the tank and trace it for a template? I understand that the tank cannot be consulted for more accurate measurements(right now), but it would be a shame to spend any amount of time building a canopy or stand just to realize that your math was wrong and it wont fit correctly.

If just for your own mental exercise, I can understand you trying to figure this out. I am enjoying watching your progress. Please create the "calculator" for bowfront, and corner tanks when your finished.