The Mathematics of Arches

Arches are part of the design feature set of our house — so is a mansard roof, but I am not as keen to replicate it — and as I add or replace various features on the house, I would like to add the same kind of flattened arch that features on facade of the house and in some of the cabinetry. For that, I need math. In particular, given the width of a given opening and how high I would like the arch to be, I need to be able to calculate the length of material of the resulting arch.

For those who missed this particular part of geometry, here are the parts involved:

Arch

For the math, we need the following:

(x - x[0])^2 + (y - y[0])^2 = r^2

Or:

x[0] = c/2

y[0] = (s - x[0]^2/s) / 2

r^2 = x[0]^2 + y[0]^2

Y = y[0] + sqrt(r^2 - (x - x[0])^2)

Workbench

I’ve been meaning to build a workbench for a while now: [the one][] this Redditor build looks pretty amazing, though I think the one I have in mmd for myself will be about half the size.

[the one]: http://imgur.com/a/ou0rs

Retractable Screen

At some point, the back door is going to need a screen on it. When we do put one in/on, I think we are going to try out one of these [retractable screens from Home Depot](http://www.homedepot.com/h_d1/N-5yc1vZ1xg5/R-100656625/h_d2/ProductDisplay?langId=-1&storeId=10051&catalogId=10053):

The New Kitchen Window Begins

At some point we realized that the best compromise solution for our current living space was to put a window above the washer and dryer, which sit on the south wall of our kitchen. Our kitchen is quite large — one can imagine a real estate agent using something like “farm-style” to describe its size — but it is also fairly dark. In its current state, there is a window over the sink and a window cut into the door, but both of those give onto the carport, which means they will always bring in a fairly subdued light.

In other words, our kitchen is a large cave.

So the first thing the new window does is bring the hope of lightening the cave. The second thing it does is give us a view of the backyard from the kitchen, which is important when you have a growing girl in the house who, as summer approaches, wants to spend more and more time outside. And with our improvements to the backyard, we do too. The new window will, we hope, give us more communication between the backyard and the kitchen.

We also hope to put a counter in over the washer and dryer, simultaneously closing them in and removing them from view and giving us additional counter space — not, mind you, because we really, really need more counter space.

But before you get to the dream, you have to do a little destruction. Here is an image from the current work:

Laudun-2009-0296

I’ll post a slideshow of the project once it’s all done.