Arch Forms for Oven Door - The Calculation?

[brickie in oz]

Re: Arch Forms for Oven Door - The Calculation?

I use a plastic strip or aluminium ruler or a thin piece of plywood, bend to shape with the aid of a few nails and mark.

[Powindah]

Re: Arch Forms for Oven Door - The Calculation?

Neil2:
The paragraph you quote from goes on to say "I just need to know how long the string should be." Another paragraph contains "What is the radius of the circle I would need to draw on the plywood? "

This indicates the original poster was happy to use string and pencil to draw the arch, but wanted to know how large a radius was required for his requirements. The calculated method would give him exactly that.

[Amac]

Re: Arch Forms for Oven Door - The Calculation?

Saves all that bother wih the nightmares of highschool trigonometry!

Very nice and elegant solution. Not only does it save bother with trigonometry but it is also more accurate:

Since nobody took the trouble to correct kemos original solution here is my go!
I'll use powindahs notations for the lines:

kemos original equations 1 and 2 are OK albeit with rounding errors becuase of the irrational numbers involved - eg sqrt of 104 etc. But I have no idea where he came up with equation 3.

anyway here goes.
Angle X = 11.31
Lets call A the angle between Y and R
and B the angle between Y and Z
and C the angle between G and R
A = B = 180-(90+11.31) = 78.69
C = 78.69 - 11.31 = 67.38
C is an angle of the right angled triangle hypoteneuse R and adjacent G,
cosine C = G/R = 10/R = .384
R = 10/cosC = 10/.384 = 26.0416667
and of course R=Z
more accurate but still not as good as powindahs solution which is exactly right
This thread btw is only for people with too much time on their hands
Amac

[Amac]

Re: Arch Forms for Oven Door - The Calculation?

I wouldn't dispute that too much wotavidone - just I do like to see puzzle solved in an elegant manner, and powindahs was elegant compared with the trigonometric calculations.
One thing though. Boat builders are probably trying to achieve a different curve than arch builders and will be limited by the strain that the lath will sustain without breaking. I would say it would be good for a gentle curve but for anything more rounded it is surely easier to work out the radius and use a compass to draw the outline. Certainly in my case since I made a completely semicircular arch and all I needed to know was what entry width I could tolerate.
Aidan

[Amac]

Re: Arch Forms for Oven Door - The Calculation?

no slur upon your trigonometric genius was intended

if there is any ingenuity involved I can asuure you it isn't mine - maybe powindah if he's still around might take umbrage.
I was looking at the original problem and powindahs solution and it struck me how coincidental that it should be an integer. It seemed to me to be highly unlikely but it is correct.
A side effect of this is that the right triangle whose hypoteneuse is the radius G (26) and also the other two sides (10 and 24) are an example of a pythagorean triple - i.e a right triangle whose sides are all integers.

In the list from wiki the second one in bold 5:12:13 (10:24:26) is the relevant ratio for this problem.

Common Pythagorean triples
The following are all the Pythagorean triple ratios expressed in lowest form with both non-hypotenuse sides less than 100:

3: 4 :5
5: 12 :13
8: 15 :17
7: 24 :25
9: 40 :41
11: 60 :61
12: 35 :37
13: 84 :85

Another interesting thing (I really do have too much time on my hands) is the relationship of these "triples" to fibonacci series.
Wiki again:

Starting with 5, every other Fibonacci number {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...} is the length of the hypotenuse of a right triangle with integral sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.

So in our case 13 is the hypoteneuse, the long leg is 3+4+5 (12) and the short leg = 8-3 (5)
This is all highly coincidental to the original problem but how cool is it that your arch is based on pythagorean triples, connected to fibonacci series which in turn has connections to the golden ratio 1.61801 which the fibonacci series approaches
I really must have a look to see how kemos oven turned out with all those magical properties built in

[C5dad]

Re: Arch Forms for Oven Door - The Calculation?

Heck, after developing a spreadsheet to do the calculation, I used a string and pencil and multiple iterations to get my curve right! More fun the old way than using the spreadsheet.

[david s]

Re: Arch Forms for Oven Door - The Calculation?

Speaking of the Fibonacci series and phi, if you look at the ideal door height to dome height ratio of 62% it is 1/1.618 the Golden Ratio.

[Amac]

Re: Arch Forms for Oven Door - The Calculation?

Just to add another method to get the radius into the mix. - here's a nice method needing just a compass and straight edge - no calculations - whch will work for any three points.
Draw a perpindicular line from the centre of Y, and another from its mirror on the right and where they intersect is the centre.
How to construct a circle through 3 points with compass and straightedge or ruler - Math Open Reference