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diameter calculation

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SillyOldDuffer11/10/2019 11:26:56
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Posted by Nicholas Farr on 11/10/2019 10:05:06:

Hi S.O.D., I've found this in one of my books, but it does give a couple of dimensions and an angle, but you've asking about random placed points and I suspected it would involve Algebra as Andrew has said. However I could never get to grips with Algebra, which seems that MichealG's search is full of.

example26001.jpg

Regards Nick.

This simple example has left me in the dust! I've confirmed it works by drawing it with QCAD but why is the diameter of the circle given by dividing distance AC by sin( 41° ) ?

I don't understand the method. The calculation isn't solving the triangle ABC, or calculating the length of line AB.

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

Andrew says it's simples and I believe him. He said:

"The Equation of a circle has three unknowns. Three unique points give you three indpendent equations. Three equations and three unknowns allows a unique solution for the centre and radius, with a bit of algebraic manipulation." It's the 'bit of algebraic manipulation' that finishes me off. Michael's Wolfram and Weebly examples may help clear the fog.

Meanwhile I found this discussion on StackExchange. It goes with Quadratic equations and matrices.

It's like being back at school. I did a class called 'Maths for Scientists'. This was a euphemism. It should have been called "Remedial Maths for Silly Boys Who Should Have Chosen an Arts Subject".

sad

Dave

Edited By SillyOldDuffer on 11/10/2019 11:27:23

Kiwi Bloke11/10/2019 11:37:45
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Dave (S.O.D), look up 'sine formula'. Not all trig needs right-angled triangles...

Michael Gilligan11/10/2019 11:43:13
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Dave,

I think it's useful to remember that this is an example in polar geometry ... so the correct solution will only be obtained by polar 'construction'.

Any 'rectangular co-ordinates' solution will inevitably be an approximation: When I was using Autocad it worked to fourteen decimal places [it may have improved since then], but that's still an approximation.

MichaelG.

.

"Michael's Wolfram and Weebly examples may help clear the fog."

... I sincerely hope so.

Edited By Michael Gilligan on 11/10/2019 11:45:47

Kiwi Bloke11/10/2019 11:50:41
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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.

SillyOldDuffer11/10/2019 12:12:44
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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 Wyatt11/10/2019 13:02:45
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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 Wyatt11/10/2019 13:03:24
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Damn beat me to it!

Michael Gilligan11/10/2019 13:11:17
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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.

.

dont know ... 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 rayner11/10/2019 13:23:32
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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 explodesurprise

thanks again.

regards

Paul

Nicholas Farr11/10/2019 13:43:53
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Hi S.O.D., I guess I should have filled you in with the proceeding piece of text to example 26 above, which is;

example26a002.jpg

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.

Andrew Johnston11/10/2019 13:55:04
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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

SillyOldDuffer11/10/2019 14:02:59
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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

Andrew Johnston11/10/2019 14:17:32
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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 Gilligan11/10/2019 14:35:49
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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 Lewis11/10/2019 16:14:05
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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

blowlamp11/10/2019 16:19:02
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What about the large circle tangent to the three small circles, inside or outside the large circle? devil

JasonB11/10/2019 18:25:20
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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 Gilligan11/10/2019 18:33:22
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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 yes

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

MichaelG.

Andrew Johnston11/10/2019 22:03:30
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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 Gilligan11/10/2019 22:32:03
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I’m relieved to see that we are in agreement, Andrew yes

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.

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