Arch Calculator

First things first, what is an arch? Most of us have an idea of what it is, some sort of vertical opening in a wall with a curved top. But of course, If we’d like to be more formal and precise, we might come across definitions like this one:

“An arch consists of an opening in a structure used to span a doorway, window, or any other space. The curved top helps support and evenly distribute the structure’s weight through the compression of the elements that compose the arch around its curved profile, resulting in a combination of horizontal but mostly vertical loads.”

Why do we use arches? Arches are an elegant solution that architects from ancient civilizations came up with when faced with the problem of covering long spans between two pillars without the structure collapsing. Before the regular use of arches, and in order to achieve great constructions such as the Parthenon or Stonehenge, the main system used was post-and-lintel, which are straight horizontal elements supported by two vertical ones. Arch Calculator.

The span between the two vertical elements could be shorter or longer depending on the materials used. For ancient civilizations, where the primary construction materials were sticks and stones, the distances were rather short.

And how to cover longer spans with fewer vertical elements? This is where the arch makes its appearance. Some archaeologists attribute the creation of the arches to the Sumerians (around 3000 b. C.), but it was the Romans (between 509 b. C. and 476 a. C.) who perfected the technique and used them widely in different structures, such as aqueducts, bridges, doors, windows, and in 3D versions of arches, such as vaults and domes.

Now you might wonder why these curved structures work better than straight ones. It’s unclear whether either the Sumerians or the Romans really knew of the physics behind what was happening — it seems to have been the result of many trials and errors. Nowadays, we can understand why arches work better.

Many ancient constructions were made of stones, clay, or concrete, materials that are resistant to compression but not to tension. These materials are easier to break by pulling from their ends than by pressuring them or stepping on them.

Semi-circular or Roman arches (half-circle arches) were initially made of pieces of stone, shaped as wedges or trapezoids (the voussoirs), and a top central wedge (the keystone). From this top wedge, the weight of the above structure gets distributed by compression from one wedge stone to the next one, undergoing only compression stresses and no tensile stresses. Given the great resistance to compression of stones and concrete, this type of construction results in great resistance and stability, allowing larger spans between columns.

Despite its great advantages and all the advances in architecture and construction that this type of arch has signified over history, they’re tricky to use in your home. If you’d like to install a semi-circular arch above an already existing door frame, you’ll notice that they end up quite tall, which could result in a very small gap between the top of the arch and the ceiling — or not enough space at all. These semi-circular arches are called high-rise arches due to their height.

So instead of semi-circular arches, you could go for a low-rise arch, such as the segmental arch (a section of an arc of a circumference smaller than 180°) or the elliptical arch. With these, you can adjust the rise to any custom desired measure.

But should you go for a segmental or for an elliptical one? 🤔 The main advantage of an elliptical over a segmental one is that while both can achieve low rises and long spans, the elliptical ones exhibit a relatively lower amount of horizontal loads. This is because, at the joint of the ends of the arch and the pillars, the load becomes completely vertical, which means that less material is required to counteract any horizontal load.

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An elliptical arch, or semi-elliptical arch, is shaped as half of a horizontal ellipse. An ellipse has an oval shape, and in contrast to a circle, an ellipse’s radius is constantly changing. This particular characteristic is what makes it more challenging to draw. So, our goal here will be to understand how to draw an elliptical arch.

Before continuing, let’s take a moment to review some basic terms related to an ellipse. Note in the image below, we have a horizontal ellipse centered at point $C$. The longest diameter, which is horizontal in our case, is known as the major axis (the distance ∣𝑉2 𝑉1∣∣V2​V1​∣), and we call the shortest one the minor axis (distance ∣𝑉3 𝑉4∣∣V3​V4​∣).

Other elements we should consider are the focus points 𝐹1F1​ and 𝐹2F2​. The distance between the focal points and any point along the ellipse’s perimeter ($P$) is what we call the focal radii. In the image, this is depicted by the distances ∣𝑃 𝐹2∣∣PF2​∣ and ∣𝑃 𝐹1∣∣PF1​∣.

A very useful property of ellipses is that the sum of the two focal radii is constant and is always equal to the total length of the arch. In general terms, we can express this as:

If we were to select $P$ as the highest point of the ellipse (i.e., 𝑃2P2​), this same property remains. We then obtain:

This last equation will come in handy when drawing the ellipse.

Now, we can introduce the ellipse formula, which has the following general form:

where:

• $(x,y)$ – Coordinates of an arbitrary point on the ellipse;
• (𝑐1,𝑐2)(c1​,c2​) – Coordinates of the center point;
• $a$ – Distance from the center to the ellipse’s horizontal vertex; and
• $b$ – Distance from the center to the ellipse’s vertical vertex.