I need a mathematical solution to a very practical problem (laying a patio).
The attached will hopefully explain.
The center of the circle for the arc we wish to have is inaccessible (ie in the house behind walls).
There are 2 fixed points the arc must intersect.
There is an existing arc drawn using these 2 fixed points, but this circle is too small – radius 539cm
The desired arc is more shallow so would have a greater radius.
I think I need 10 or so measurements from the center of this existing circle, to the new arc.
These points can then be plotted on the ground and the dots joined up (obviously more than 10 will me more accurate, but 10 seems a sensible number)
I am sure this is possible, but my mathematical knowledge is not good enough, sorry. Answers or a simple formula I can apply would be very much appreciated.
Anyone out there to help?
Best Answer
In the picture above, I've drawn in the circle that I think you want, drawing it as a large pale-orange disk. The blue zig-zag line corresponds to the line in your figure, and the blue dots are the two points where you said the circle has to go. I'm making the assumption that the circle is supposed to have the top blue point as its uppermost point.
So here's what you do: extend the line of the upper walk 539 units to the right (the red segment) and mark a point there. That's the red dot in the picture. Notice that the green line coming down from it will NOT pass through the other blue point, so don't go trying to move things to make that happen ... it'll just make things wrong.
We're going to measure everything from this red point, either left-right (i.e., along the red segment) or up-down (along the green segment). It'll help if you actually draw a line parallel to the red one, but further down, and a line parallel to the green one, but further left, and fill in a kind of grid of chalk-lines or something to help guide you.
You want to find points on the circle arc like the aqua one. To help you do that, I'm going to give you "x" and "y" measurements, where "x" is "how far to the left of the green line" and $y$ is "how far down from the red line". They're computed like this:
$$ x = 539 (1 - \cos(t) ) \\ y = 539 (1 - \sin(t) ) $$
where $t$ is any number, and indicates the angle counterclockwise from horizontal (using a ray from the orange dot through the yellow one as a reference). Fortunately, you don't need to measure $t$. You just need a table, so I've appended it below. If you look about a third of the way down the table, you'll see a point $0, 539$. That means that if you go left from the red dot by 0 units, and down by 539 units, you've got a point on your circle....that's the yellow point in the figure!
To actually draw your curve, I'd take a few $x$ values that are all bigger than about 200 -- say 200, 250, 300, 350, 400, 450, 500 -- and use the table to find the corresponding $y$ values (or at least as near as you can come). That'd give me
For each of these, I'd measure $x$ units to the left of the red dot, along the red line, and then $y$ units down. So for the first one, I'd measure 199.79 units to the left of the red dot, and then move down 120.12 units, and I'd draw a point. To be honest, I'd measure 200 to the left and 120 down, rounding off to the nearest whole number, because I'm figuring that these numbers are probably cm, and little bits don't matter a lot.
Then I'd do the same thing again, but this time I'd pick $y$ values between 200 and 700, something like this:
For each of these, I'd measure $y$ units down from the red dot, and then $x$ units to the left. For the third row, for instance, I'd measure 605 units down and then 4 units to the left, which would give me a point a little below and to the left of the yellow dot.
This collection of points should give you enough to let you draw in the rest of the curve by bending a long thin piece of wood around some pegs set in the ground, etc. --- I'm assuming you can do this part on your own.
Here's the table of all the values: