diameter calculation

[SillyOldDufferModerator @sillyoldduffer]

Posted by Kiwi Bloke on 11/10/2019 11:37:45:

Dave (S.O.D), look up 'sine formula'. Not all trig needs right-angled triangles…

Good clue: Wikipedia gives an explanation of the Law of Sines and this diagram (credit) shows where the diameter comes from:

Although I understand it now I shall have forgotten everything by bedtime. Sad isn't it.

Ta,

Dave

[Neil WyattModerator @neilwyatt]

Posted by SillyOldDuffer on 11/10/2019 11:26:56:

I can do trigonometry on right angled triangles, but ABC isn't right angled.

Draw a right angled triangle with AC as its base, and the other vertex, E on the circumference and A as the right angle.

The line CE is now a diameter, as the hypotenuse of a right angled triangle in a circle is always the diameter.

So from trig AC / CE is sin e

As CE = D and AC =B then, if the formula is correct, sin e = sin b.

Who can do the geometry that proves e= b ?

[Neil WyattModerator @neilwyatt]

Damn beat me to it!

[Michael GilliganParticipant @michaelgilligan61133]

Posted by Kiwi Bloke on 11/10/2019 11:50:41:

Michael: 'I think it's useful to remember that this is an example in polar geometry'. You mean 'R – theta' geometry? Not really, although I agree a bit of construction is required to get to this, 'sine-formula'-like answer. Co-ordinate geometry is good, if only because it's easy to transfer that approach to machining co-ordinates.

.

… if you say so

What I should perhaps have written is that the geometry only works accurately in polar co-ordinates

… but I thought that what I wrote, and the links that I provided, might help Dave's thinking.

Oh well …

MichaelG.

[paul raynerParticipant @paulrayner36054]

Hi all

worked a treat thanks guys.

I'm not very good at "sums" I've just read through the postings and I think my head is going to explode

thanks again.

regards

Paul

[Nicholas FarrParticipant @nicholasfarr14254]

Hi S.O.D., I guess I should have filled you in with the proceeding piece of text to example 26 above, which is;

Of course you need to understand the Sine rule, which I'm a bit rusty on now having not used it for a long time. Apologies if I've confused you and others of course. This rule though, would have solved Paul's question also, but the other diagram was already in my album.

Regards Nick.

[Anonymous]

I'm mystified as to why polar co-ordinates should be inherently more accurate than Cartesian co-ordinates. They are both Euclidian geometry and there are precise equations for converting from one to the other, albeit involving trigonometric functions which are inherently irrational. There are functions that are more elegantly expressed in polar terms, but that's not the same as accuracy.

A supplemental question is that if different co-ordinate systems in one geometry have different accuracies how do different geometries, such as spherical, fit into the picture?

Andrew

[SillyOldDufferModerator @sillyoldduffer]

Posted by Nicholas Farr on 11/10/2019 13:43:53:

Hi S.O.D., I guess I should have filled you in with the proceeding piece of text to example 26 above…

Of course you need to understand the Sine rule,

…

This rule though, would have solved Paul's question also…

Regards Nick.

No need to apologise Nick! It made me think a bit and I need the practice. When I've laid out a ring of bolt holes I've always started by knowing the PCD: give that it's straightforward to calculate the angles. Interesting to know how to get the PCD from the holes, and Paul shows it's sometimes necessary.

Another example of a calculation easier in one direction than the other: given a sequence of change wheels on a lathe's banjo, it's easy to calculate the pitch. Much harder to work out which change wheels are needed to produce a given pitch, or a reasonable approximation of it. Perhaps that's one for a new thread. Like Paul my brain has exploded!

Ta

Dave

[Anonymous]

Posted by SillyOldDuffer on 11/10/2019 14:02:59:

Much harder to work out which change wheels are needed to produce a given pitch, or a reasonable approximation of it.

For those whose brains have exploded don't bother reading this!

The problem of working out what change wheels are needed for a given pitch is an example of the travelling salesman problem. In mathematical terms it is NP-complete, in that a solution can be found by a brute force, but finite, search. However, an algorithm to reach the answer quickly doesn't exist.

Andrew

[Michael GilliganParticipant @michaelgilligan61133]

Posted by Andrew Johnston on 11/10/2019 13:55:04:

I'm mystified as to why polar co-ordinates should be inherently more accurate than Cartesian co-ordinates. They are both Euclidian geometry and there are precise equations for converting from one to the other, albeit involving trigonometric functions which are inherently irrational. There are functions that are more elegantly expressed in polar terms, but that's not the same as accuracy.

A supplemental question is that if different co-ordinate systems in one geometry have different accuracies how do different geometries, such as spherical, fit into the picture?

Andrew

.

Perhaps I am wrong there, as well, Andrew

… I’m having a bad day; so why not ?

My simple understanding though, is that anything which would be properly expressed in polar co-ordinates can only be approximated by Cartesian; thanks to Pi

MichaelG.

Edited By Michael Gilligan on 11/10/2019 15:06:59

[Howard LewisParticipant @howardlewis46836]

Being simple, I drew a circle with three chords 65 mm long. Then a 30/60/90 degree triangle, with one side being half the chord = 32.5 and used Sine or Cosine to calculate the Hypotenuse. Then doubled the answer to get the diameter, which my calculator said was 75.055535, if you want be frightfully accurate. Me? I'd settle for 75 as being about as good as i could get, although I might set the machine to 75.06 mm or 2.955 inches in old money.

That is about as complicated as my puny brain can tolerate.

Howard

[blowlampParticipant @blowlamp]

What about the large circle tangent to the three small circles, inside or outside the large circle?

[JasonBModerator @jasonb]

I just looked in the Zeus book and took the figure for the distance between two holes in a three hole pattern and with a bit of simple maths got

1/0.86603 x 65 = 75.055

Every workshop should have one.

[Michael GilliganParticipant @michaelgilligan61133]

Posted by JasonB on 11/10/2019 18:25:20:

I just looked in the Zeus book and took the figure for the distance between two holes in a three hole pattern and with a bit of simple maths got

1/0.86603 x 65 = 75.055

Every workshop should have one.

.

Indeed it should

‘though I don’t recall it covering the question posed by Dave/S.O.D.

MichaelG.

[Anonymous]

Posted by Michael Gilligan on 11/10/2019 14:35:49:

My simple understanding though, is that anything which would be properly expressed in polar co-ordinates can only be approximated by Cartesian; thanks to Pi

Indeed; in the majority of cases I'd expect nice round numbers in polar co-ordinates to translate to irrational numbers in Cartesian co-ordinates due to the circular trigonomic functions involved. I expect the reverse is also true, nice round numbers in Cartesian co-ordinates could translate to irrational numbers in polar co-ordinates.

Andrew

[Michael GilliganParticipant @michaelgilligan61133]

I’m relieved to see that we are in agreement, Andrew

My earlier point about accuracy comes down [in practical terms] to:

For Polar jobs, polar co-ordinates are ‘definitive’

For Rectangular jobs, rectangular co-ordinates are ‘definitive’

A classic example being the NEMA hole pattern for stepper motor flanges … devised as holes on a pitch-circle, but often specified as holes at each corner of a square [which requires expedient approximation].

MichaelG.

[duncan webster 1Participant @duncanwebster1]

But if you work out the diameter for 3 holes on a pitch circle I suspect you get an irrational number whichever way, it's most unlikely to be an exact number

[Kiwi BlokeParticipant @kiwibloke62605]

This has got complicated and no doubt I'm out of my depth…

It seems to me that, if the problem is specified in Cartesian co-ordinates, such as could be the case for absolute hole positions, then Cartesian co-ordinate geometry is the way to go, even for problems involving positions on a circle. It's all done without Pi or trancendental functions poking their awkward noses in. However, if relative positions are specified, in terms of lengths and angles, then polar co-ordinate geometry is appropriate. Perhaps that's what Michael and Andrew are saying… For practical workshop applications, especially with simple co-ordinate tables and DROs, Cartesian rules, doesn't it?

[Kiwi BlokeParticipant @kiwibloke62605]

Duncan: 'But if you work out the diameter for 3 holes on a pitch circle I suspect you get an irrational number whichever way, it's most unlikely to be an exact number'

Most unlikely, yes, but there's a couple of solutions that spring to mind that are rational. I'll leave it as a teaser, for others to find…

[blowlampParticipant @blowlamp]

One coordinate system will always be an approximation of the other but for use in our workshops Cartesian will always be the best option. Positional errors in rotary division equipment change depending on the target radius, getting larger as the radius increases which doesn't happen in the same way when using coordinate equipment, although perpendicularity between the axes must be maintained for the machining to be reliable.

I would far rather make a division plate using a DRO on a milling machine than use a rotary table or dividing head if I wanted the best accuracy.

[Kiwi BlokeParticipant @kiwibloke62605]

Blowlamp: 'I would far rather make a division plate using a DRO on a milling machine than use a rotary table or dividing head if I wanted the best accuracy.'

I wouldn't, because the problem is effectively specified in polar co-ordinates, so errors are introduced by converting to cartesian. Plus, it's far more likely to suffer from operator error (well, at least in my case…).

[Former MemberParticipant @formermember19781]

[This posting has been removed]

[Michael GilliganParticipant @michaelgilligan61133]

Posted by Kiwi Bloke on 12/10/2019 00:29:10:

[…] For practical workshop applications, especially with simple co-ordinate tables and DROs, Cartesian rules, doesn't it?

.

Yes … ‘though I am currently tempted by the prospect of a DRO, or CNC, which is based on polar co-ordinates.

MichaelG.

.

P.S. I think you are in at just the right depth

[Michael GilliganParticipant @michaelgilligan61133]

Posted by 34046 on 12/10/2019 08:20:32:

My brain is on the same wavelength as Hpward's

I did it by drawing the equilateral triangle with 65mm sides then mid point of sides to halve the 60 angle gave me mid point of the triangle. Measure from centre to one of the triangle corners and got 37.5 mm radius. ie 75mm diameter.

Bill

.

”My brain is on the same wavelength as Hpward's”

Apparently not, Bill

You used elegant geometrical construction [which is subject, of course, to draughting & measurement errors]

Howard used calculated trigonometric functions to reach a good approximation of the target value.

MichaelG.

[Former MemberParticipant @formermember19781]

[This posting has been removed]

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