A bit more on setting out curves.

[9fingers]

About 4 years ago I posted this viewtopic.php?f=20&t=387
on some of the maths behind setting out curves that are sections of circles but this post will be a bit more about application to a task and ending up with a neat method involving no maths at all.

I'm sure many of you sat in geometry lessons in your teens and wonder what is all this stuff for and when would I ever want to prove so and so's theorem in real life and yes I agree. However every now and then the results (not the proof!) come in handy in woodwork, metalwork and construction jobs.

This is all about one property of circles and chords. A chord is simply a line starting on the edge of a circle and going to another point on the edge of a circle (circumference). A special case of a chord which passes through the centre of the circle is a diameter. That's the definitions out of the way.



Here we have a couple of chords of a circle that cross each other at point X. The magic theorem is written below and holds true for any pair of intersecting chords. This might be handy for all sorts of things but becomes good for a common application when we add some restrictions on the chords shown below.



Here we make one chord be a diameter and the other one cross it at right angles ie the angle at X is 90 degrees.

The formula is the same but as AX=XB we can re-write it slightly.

So if we want to mark out a curve on a job to give a more pleasing line from A to B following the circle we need to know where the centre of that circle is in order to draw it.

Lets pick some numbers. I'm making a curved rail for a footstool that will be 500mm long ie AB=500 and I'd like the deviation of the curve from the boring straight line AB to be 25mm ie XD=25

From the formula we have that 250 * 250 = 25 * XC so XC= 250*250/25.

Nice easy numbers as it happens and XC = 2500mm. The diameter of the circle CD is therefore 2525mm and the centre of the circle lies 2525/2 = 1262.5mm from each of A, B and D.

So to mark out that curve is just about feasible on a big bench or wait till the mrs is out and use the living room floor.

The more gentle the curve you want the bigger the dimensions get.

Lets say you want to curve the long edges of a dining table 2m long and have it 50mm wider in the middle.

So AX =1m and DX = 0.025m Plug those into the formula and you get CX =1*1/0.025 =40m !! the diameter becomes 40.025 and the radius = 20.0125!

For some of us, our gardens are even big enough to mark that one out!

Next time I'll describe a method to set that curve out that involves no calculations and can be done in a realistic space.

Hope this is of interest so far.

Bob

 

[Malc2098]

Love it!

Takes me back to school and also measuring curved tyre marks from a car spun off the road!

We'd measure the chord mid-ordinate to feed into the long formula to establish a speed at the time of starting the skid.

 

[9fingers]

Part 2


This is a simple method to mark out an arc of a circle with no maths and the absolute minimum of measurement.

I've used the same annotation as the first post. It is best to make a template for the curve. a) if you make a mistake then the the job is not damaged and more often than not you will need the same curve in more than one location.

You will need some template material eg sheet of ply or mdf that is a least as long as the curve you want to create and two laths 40mm x 6mm x just over twice as long as the length of the chord.



Draw the chord AB, find the centre X and draw up to D. Partially put in 3 pins/nails as shown in red at A, B and D.



Lay the two laths on as shown with on resting on pins A and D and D and B.

Use nails, screws, hot melt glue etc to fix the two laths together as firmly as possible. The angle between them needs to be maintained.



Remove the nail at D

Take the two laths and hold a pencil point at the angled joint. Keeping the laths touching points A and B at all times, slide the joint and pencil along and the arc will be drawn out.

Half way along, the pencil should pass point D where the nail was. Endeavour not to break the pencil tip in the nail hole as you pass over it!

Assuming all is well, cut the template out and use to mark out the job.

Bob

 

[Malc2098]

That's neat. I used a similar 'trick' to create an ellipse with a piece of string and some nails.

To copy the arc on my kitchen doors, I had to find the radius and set the router up on the trammel, so used the first formula.

I've tried the old trick to rotate the images but it doesn't seem to work. Mods, could you rotate for me, please.

The practice piece



The finished piece



Mod Note: Malc we don't have a magic rotate button at our disposal. It is a tedious manual save, rotate, upload, reinsert and delete exercise for us too so
it would be helpful if you sorted this at your end please.

 

[Rod]

A very simple method is to make a bow out of a strip of wood and some string.
Tighten the string to achieve the desired curve then use the wood as a template.

Can’t you edit the photos and rotate - easy to do on the iPhone?

Rod

 

[9fingers]

That will give you a curve Rod but depending on the material not necessarily part of a circle.
If the feature is purely decorative then I agree it is an easily achieved curve.
Bob

 

[Malc2098]

Gents,

I will try again, hard, promise! I used to have a fix this end, but it doesn't seem to work now.

About the curve - I agree the curved steel rule is pretty good for a template, but there are times when you need to find the centre of the circle and therefore its radius (or is it the other way round) to proscribe the arc with, for example, a router.