Geometry or Formuls to construct a circle to three entities.

[Neale]

This one doesn't need much more than Pythagoras to solve it.

Call the distance of the centre of a blue circle from the horizontal line h (note - this is not the centre-centre spacing of the two blue circles)
Call the distance of the centre of a blue circle from the vertical line x
Call the radius of the small circle r

Then the radius of the large circle is: (h^2+x^2-r^2)/(2*(x-r))

I would have shown my working but typing out equations like this is too tedious!

No guarantees, but I've checked the results for a few values against a sketch in Fusion 360 and it gives the same answers. F360 won't let you construct a construct a circle with those constraints either, but you can create a circle in roughly the right position and then apply appropriate constraints afterwards to get the same effect.

[theforester]

Thank you for your help, much appreciated. I'll try it out at work today.

I presume both the ^ and * symbols in the formula stand for squared?

[Neale]

Thank you for your help, much appreciated. I'll try it out at work today.

I presume both the ^ and * symbols in the formula stand for squared?

"^2" means squared; "*" means multiplication. That's one reason why I didn't want to write out the derivation - too much chance of getting it wrong!

[Kitwn]

Nice to have an excuse to get the old geometric grey matter working! I've just turned to paper and pencil and come up with the same answer as Neale. I don't fancy trying to work out the option where the two blue circles are not symmetrical about the horizontal line though. That turns it into a two dimensional problem instead of just one.

Kit

[cropwell]

This one doesn't need much more than Pythagoras to solve it.

Then I would have expected a square root in the answer. Interesting question, but I would have liked to know how the solution was determined (the method, not the equation derivation).

Cheers,

Rob-T

[Neale]

Then I would have expected a square root in the answer. Interesting question, but I would have liked to know how the solution was determined (the method, not the equation derivation).

Cheers,

Rob-T

Well, strictly, I suppose, Pythagoras was a means to an end, and it's one reason why there is a bunch of squared terms in the final answer.

I started by sketching the problem, putting in the dimensions as per my first response (above). You can draw a triangle with vertices at the intersection of horiz and vert lines, centre of small circle, and assumed centre of large circle (radius R). Then drop a perpendicular from centre of small circle to horiz line. You have a right-angled triangle with one side = h, and another = R-x.

Write out the equation for the square on the hypotenuse (length y in my sketch): y^2=h^2+(R-x)^2

But from the sketch you can see that y=R-r, so replace y in the Pythogoras equation with (R-r). Multiply it all out and the R^2 terms magically cancel each other out. Then it's just a matter of rearranging for R, which is where we came in. Technically, we should state some assumptions (like r<=x, r<=h) to make sure that we can actually draw the thing. I am also assuming that the large circle is tangent to the small circles on the side of the circle closer to the vertical line.

I would like to say that it was easy, but the real trick is spotting how to get some kind of relationship between the things we know and the one thing that we don't, and I happened to strike lucky with my "draw a triangle" approach.

[cropwell]

I googled the problem and found this https://planetcalc.com/1421/ you have to scroll down to the 'Segment defined by chord and height' and that will give you the radius of that circle that passes through the centers of the smaller circle. So you just add on the radius of the smaller circle.

Cheers,

Rob-T

[Neale]

I googled the problem and found this https://planetcalc.com/1421/ you have to scroll down to the 'Segment defined by chord and height' and that will give you the radius of that circle that passes through the centers of the smaller circle. So you just add on the radius of the smaller circle.

Cheers,

Rob-T

Rob - unfortunately, it's not quite that simple! I generally try to find some sort of sanity check when I'm doing sums like this - does it give the right answer for the extreme cases? Does the answer move the right way if I change one of the given values? In this case, if you increase the size of the small circle to be equal to the segment height, the answer should be "infinity" for the radius of the big circle as the circle becomes a straight line. With your approach (which will be fairly close if the size of the small circles is small). as the small circle size increases, R increases - but that means that it no longer touches the vertical line. What's needed is the chord length and height at the points of tangency with the small circles - which you don't know at this point.

If it's any consolation, I'm typing this rather than being in the workshop because I've just fouled up a job on the CNC router. I wanted to move the tool up and clear so that I could add an extra fixing screw towards the end of a 1.5hr machining operation. Due to a complete brain fade, I moved Z up using machine rather than work coordinates - which meant that it plunged down rather than going up before moving in X and dragging work and spoil board sideways, leaving a big gouge in the work. The only good thing was that at least it was a thick spoil board so that the cutter didn't reach the machine bed, and I didn't break the 2mm carbide cutter. We all have off days...

[cropwell]

I would have thought that, if the problem can be drawn on a sheet of paper then the calculator I found would be OK for all practical purposes. I don't think that extreme cases apply here. The chord length is the distance between the centres of the smaller circles. the height of the chord is the distance from the large circle minus the smaller circle radius. So we are calculating the radius of a circle concentric to the large one, where the periphery passes through the centres of the small circle. the large circle is this calculated radius plus the radius of the small circle.

I presume the OP knows all the dimensions and can substitute them in the calculation app.

The problem may be a fascinating exercise, but the practicalities of mathematics faded for me when it got to imaginary numbers. I was alright with algebra, geometry and calculus, but that was 60 years ago and I haven't had the need to do much more than arithmetic, sines, cosines and tangents and a bit of Pythagorean juggling in the intervening time.

This was my today project, from circuit diagram to pcb. It is a logic indicator circuit (Hi-Lo_Pulse) using a 4011.


Cheers,

Rob-T

[Doddy]

This was my today project, from circuit diagram to pcb. It is a logic indicator circuit (Hi-Lo_Pulse) using a 4011.


Cheers,

Rob-T

Thread-creep: I'm curious - I've had mixed results with isolation routing in the past (to the point that I refuse to entertain the idea any more and unlikely to change my mind) - but you go for complete copper removal... what cutting bits do you use/find useful for this?