SillyOldDuffer  11/10/2019 11:26:56 
5328 forum posts 1090 photos  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. 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". Dave Edited By SillyOldDuffer on 11/10/2019 11:27:23 
Kiwi Bloke  11/10/2019 11:37:45 
344 forum posts 1 photos  Dave (S.O.D), look up 'sine formula'. Not all trig needs rightangled triangles... 
Michael Gilligan  11/10/2019 11:43:13 
14963 forum posts 638 photos  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 coordinates' 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 Bloke  11/10/2019 11:50:41 
344 forum posts 1 photos  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, 'sineformula'like answer. Coordinate geometry is good, if only because it's easy to transfer that approach to machining coordinates. 
SillyOldDuffer  11/10/2019 12:12:44 
5328 forum posts 1090 photos  Posted by Kiwi Bloke on 11/10/2019 11:37:45:
Dave (S.O.D), look up 'sine formula'. Not all trig needs rightangled 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 Wyatt  11/10/2019 13:02:45 
Moderator 17326 forum posts 690 photos 77 articles  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 Wyatt  11/10/2019 13:03:24 
Moderator 17326 forum posts 690 photos 77 articles  Damn beat me to it!

Michael Gilligan  11/10/2019 13:11:17 
14963 forum posts 638 photos  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, 'sineformula'like answer. Coordinate geometry is good, if only because it's easy to transfer that approach to machining coordinates. . ... if you say so What I should perhaps have written is that the geometry only works accurately in polar coordinates ... but I thought that what I wrote, and the links that I provided, might help Dave's thinking. Oh well ... MichaelG. 
paul rayner  11/10/2019 13:23:32 
136 forum posts 40 photos  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 Farr  11/10/2019 13:43:53 
2114 forum posts 1025 photos  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. 
Andrew Johnston  11/10/2019 13:55:04 
5184 forum posts 599 photos  I'm mystified as to why polar coordinates should be inherently more accurate than Cartesian coordinates. 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 coordinate systems in one geometry have different accuracies how do different geometries, such as spherical, fit into the picture? Andrew 
SillyOldDuffer  11/10/2019 14:02:59 
5328 forum posts 1090 photos  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 Johnston  11/10/2019 14:17:32 
5184 forum posts 599 photos  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 NPcomplete, 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 Gilligan  11/10/2019 14:35:49 
14963 forum posts 638 photos  Posted by Andrew Johnston on 11/10/2019 13:55:04:
I'm mystified as to why polar coordinates should be inherently more accurate than Cartesian coordinates. 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 coordinate 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 coordinates can only be approximated by Cartesian; thanks to Pi MichaelG. Edited By Michael Gilligan on 11/10/2019 15:06:59 
Howard Lewis  11/10/2019 16:14:05 
2885 forum posts 2 photos  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 
blowlamp  11/10/2019 16:19:02 
1245 forum posts 82 photos  What about the large circle tangent to the three small circles, inside or outside the large circle? 
JasonB  11/10/2019 18:25:20 
Moderator 17280 forum posts 1859 photos 1 articles  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 Gilligan  11/10/2019 18:33:22 
14963 forum posts 638 photos  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. 
Andrew Johnston  11/10/2019 22:03:30 
5184 forum posts 599 photos  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 coordinates can only be approximated by Cartesian; thanks to Pi Indeed; in the majority of cases I'd expect nice round numbers in polar coordinates to translate to irrational numbers in Cartesian coordinates due to the circular trigonomic functions involved. I expect the reverse is also true, nice round numbers in Cartesian coordinates could translate to irrational numbers in polar coordinates. Andrew 
Michael Gilligan  11/10/2019 22:32:03 
14963 forum posts 638 photos  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 coordinates are ‘definitive’ For Rectangular jobs, rectangular coordinates are ‘definitive’ A classic example being the NEMA hole pattern for stepper motor flanges ... devised as holes on a pitchcircle, but often specified as holes at each corner of a square [which requires expedient approximation]. MichaelG.

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