Precursor:

This tutorial may not necessairily the best or most appropriate way to accomplish the task to be detailed, but it is a method that works. Either way, the goal is to be able to create curved geomtery along other arched surfaces. To accomplish this task we'll be making use of the arch and vertex manipulation tools, so, it'll be helpful to be familiar with their functionality.

Let's start by making very basic framework, and make our threshold that will encapsulate our future architecture. In this particular case, I decided to make an opening that was 128x128 units with walls that were 32 units thick.

Now, it's somewhat heplful to be able to invision what we're trying to make as this will help you to understand what some of these imageless descriptions are all about. We're going to make the interior edge of a/any surface have a half-circle, convex, curve. The "curve" comes from continuing the half-circle, which will start on the ground, through an arch that will be at the top of our threshold. I know that still probably sounds somewhat cryptic, but hopefully it'll all come together by the end.

Step 1:

Well, the first part is easy enough, let's make an arch at the top of our threshold. (In actuality, we'll make half of an arch because we only need to make half of what will be the completed architecture.) After making the arch, you want to make all of the interior edges have a slope of 1, or put another way, make all of the interior edges have a 45 degree angle. The important thing to do, however, is to preserve the interior vertices that were first created when you made the arch.

So, why did we just do that? I'll point that out a little later down the road. The thing you need to be made aware of is that this arch that you just made is going to be our first "profile arch." What I mean by this is that this arch won't be apart of our final product; but rather, it will be used as a profile for creating future architecture in this tutorial.

Step 2:

Alright, well, now we need to make a pedastal, of sorts, that will be the base of our curved arch. In this case, I made a 90 degree arch, within the confounds of a 16x16 unit space, and copied it. The reason I used 3 sides for the arch is because when working with a smaller scale there's distortion, or sometimes useless faces, that may be the result of having too many faces on a small arch. At any rate, after creating the full 180 degree arch, I made it into one brush.

Step 3:

Now, this "base" arch is sitting on the inside frame of our door, and it extends from the ground to the bottom edge of our first profile arch we made earlier. The next step we want to do is to create another profile arch for each pair of vertices that was created from our base arch. Sound confusing? Well, let's look at a picture:

The top half of the picture is what you should have up until the last paragraph. The bottom half shows how we've created another profile arch for each edge of the base arch. Notice how all of the edge lines, of the profile archs, coincide with one another. Also, notice how the width between each point on the base arch is preserved around the edges of each profile arch. Still sound confusing? Well, take a closer look:

By now, you might start to see how our curved edge is going to take shape, but if not don't worry because we're getting close to finishing this all up.

Step 4:

In this step, what we need to do is to create brushes that aren't going to create a myriad of "Invalid Solid Structure" errors. The end product of this step will create a brush that we can easily manipulate to finish up the tutorial. This is a relatively simple process of taking two brushes, making them 4-point pyramids, and then manipuating them to form a brush with one full face. The idea is to replicate your base arch from step four, but divide each section of the arch into two triangular sections.

In the picture above, the highlighted brush is the one that was created from step four. Off to the side you'll see some grey brushes, which are the profile arches that have been referenced several times throughout this tutorial.

Now we can put all of our prep work into action and make that curved geometry.

Step 5:

Using vertex manipulation, select each group of vertices, and align that group of vertices to the corresponding vertex on the profile arches.

From there, all you need to do is is repeat the last part of step five for each section of the profle arch.

Finished:

Copy, paste, and flip horizontally.

There we have it, a curved piece of geometry that curves along the inside of another arch.

Caveats and other information

• There's several steps where you can diverge from this tutorial and make several changes. For example, you could add another profile arch for the last edge of the base arch, which would allow for a flat brush to go around the edge of the interior of the curved arch. As it is now, the interior part of the arch will curve all the way to the top. If this doesn't make sense, it's easy enough to add the other profile arch and see it for yourself.
• In step four, the "inside" of your triangulated arch sections will look like a mess, but none of that matters as it won't be visible in the end, (At least, not in this particular implementation.)
• Texturing may seem like it would be a nightmare for the end product, but it's really quite simple if you use "Align to View." Granted, that's another tutorial for another day.
• It almost goes without mentioning, but you won't want to leave the architecture as world brushes.
• This architecture is probably somewhat superfluous given that a prop would be a much more viable solution, but it's nice to have the option nonetheless.
• The labeling of the distance across the units in the last picture of Step 3 is labeled in such a way as to help illustrate a prior statement. The sum of their hypotenuses is greater than how it is labeled. The labeling is for the illustration of a concept as opposed to mathematical fact.
• The basic methodology behind this tutorial can be easily expanded to creating other geometry such as: hollow spheres, curved/slanted arches (concave & convex), etc.