Part 1: Studying Parts and Proportions of a Four-Part Vault
The image below is a reflected ceiling plan of the Notre-Dame de Paris cathedral at ground level. Notice that it contains grid lines and dimensions. Even though this drawing is not fully accurate, it is a good reference for this article, as it is useful for identifying types of vaults and their location in the cathedral.
Highlighted above, between grid lines 7 & 8, and F & G, is an example of the first type of vault that we are going to study. This one is the simplest of all types. This one is known as a "four-part vault", and it is commonly known as "crossed vault", or "groin vault". To start getting familiar with these terms, see the following image and learn the names of some parts of a vault:
A = Arch (a bearing arch between two supports, on the edges of the vault). Also known as "double" arch, probably because it is most of the time next to the arch of the adjacent vault.
R = Rib (a bearing arch between two supports (usually semicircular) crossing the vault diagonally, forming an "x" in the reflected ceiling plan).
P = "part", commonly known as a vault between two arches. (The term "vault" can be used for a part or for the whole).
Geometric Proportions of a Four-part Vault
According to our illustrations above, our sample four-part vault is approximately 18 feet x 18 feet (5.48 m. x 5.48 m.). However, to study the geometric proportions, we will use a "prototype" that is a 1 x 1 square (either 1 foot or 1 meter).
This is a six-part series. Click the links below to continue working.
In Revit, to study these proportions, we can work in 2D first, using a generic model family and some geometry. In the plan view of a generic model family, I create reference planes separated by one unit (one meter, in my example), and then I draw some reference lines to define the location in plan view of edges and diagonals (arches and ribs) as shown in this illustration, where I have also identified all the corners and the center point with letters A, B, C, D, and E, and the center points of ridges with letters F, G, H, and I.
To find the apex or total height of the vault, in the plan view, in 2D, I create a semicircle with center in point e, going from point A to point C.
Notice that the radius of that semicircle is 0.7071, for a vault that is 1 x 1 square. For any vault of this type, square or rectangular, that distance is half of the hypotenuse of the triangle A D C. This distance, if we were to draw the semicircle following the diagonal in the same way, but in the vertical plane, will mark the highest point of the crossed vault, or apex. The following illustration shows the semicircles intersecting at an elevation of 0.7071, for an imaginary vault of 1 x 1 in plan.
In the construction of the actual vaults, these semicircular arches are the ribs, the first elements that need to be built.
Now, let's go back to our 2D study, the plan view, and let's find the geometry of the arches. There are several methods. Here I am following one method that Eugène Viollet Le-Duc describes in his dictionary of French Architecture as a method that became popular around the 13th century, because it was practical and easy. This was, most likely, the method that the builders of Notre Dame used to find the dimensions of a vault based on the available space between supports.
From the plan view, draw an auxiliary arc with center in point A, and with length equal to segment A-E, going to line A-B, and mark that intersection (point 2). Do the same from the opposite side, drawing an arc with center in point B, with length equal to segment B-E, going to line A-B, and mark that intersection (point 1) as shown in this image:
Now, use point 1 as the center of an arc that goes from point B towards the left, and use point 2 as the center of an arc that goes from point A towards the right. The intersection of these two arcs form the geometry of a pointed Gothic arc, one of several kinds. In our sample, this arc is the geometry of the "double" arch that goes on the edges of the crossed vault.
The proportions of this arc, for a vault of 1 x 1 square, are these:
Now, let's go back to a 3D view, and, following the same system, let's draw these arcs on all four edges, now that we know their proportions:
Now we need to draw the ridges. In the actual built vaults, the ridge is just the line of intersection between two sides of a vault (part) that is supported between two arches. At the intersection, the masons interlock blocks (in the same way as courses of bricks are interlocked). In Revit, we will see the ridges as hard lines, if we intersect solids and voids in generic families, or if use reference lines to create surfaces in adaptive families.
Notice, in the previous image, the difference between the height of the ribs 0.707, and the height of the pointed arches 0.676. The difference is 0.031.
Going back to the plan view of our generic family, just for understanding the geometry of the ridges, we can draw in 2D an arc that goes from point I to point G with a sagitta of 0.031, and another arc from point H to point F with the same sagitta (distance from the center of the arc to the center of its base).
If we pass this to the 3D view, we need to draw arcs in between the top of the pointed arcs, like this:
The elevations should look like this:
These descriptive geometry exercises, and these numbers that we have found, such as 0.707, 0.676, 0.031, for heights, and 0.293, 0.414, 0.293 for drawing the pointed arcs, will be useful to drive the geometry of vaults of this type for any size. This is a simple way to find parametric relationships.
This geometry is still very simplified, because it does not take into consideration some things, such as the thickness of ribs and arches, the vaults (surfaces) between ribs and arches, and the space they need to have on top of the capitals. Notice that ribs and arches start at points A, B, C, and D. In reality, there needs to be an offset around those points to accommodate the thickness of multiple arches coming to the same point.
Even though this exercise simplifies the geometry, it provides an understanding of the proportions, a first look at the relationships that we need to see before moving forward, in the same way as the builders at that time drew some triangles and arcs on the ground to figure out things with geometry, before building.
This is a six-part series. Click the links below to continue working.
Alfredo Medina is a very experienced and knowledgeable BIM / Revit professional with a background in architecture, high skills in training, troubleshooting, technical support, parametric modeling, extraction of quantities, definition of standards and best practices, clash detection, and coordination of large BIM projects. Alfredo has several years of experience and a reputation as an expert due to his participation in forums and international conferences.