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There are many complex mathematical shapes like the cycloid, cissoid, umbilic torus (picture) or the shape of the Azadi tower. Sure one can design complex objects like this in CAD software today but how does one construct such objects accurately in the real world. What techniques do builders use to take an architect's zany design and accurately construct the prescribed shape?

I would like to know modern techniques to acccomplish this and also techniques used by pre 20th century engineers.

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  • $\begingroup$ What has your research shown you so far? $\endgroup$
    – Solar Mike
    Dec 28, 2017 at 6:17
  • $\begingroup$ i would suggest googling old blueprints. odd curves are broken down into a series of circles/arcs and coordinates. Once a person has those anything is possible. $\endgroup$ Dec 28, 2017 at 10:38
  • $\begingroup$ Buildings are hardly very accurate, not like machine parts. The frequently have tolerances in centimeters. $\endgroup$
    – joojaa
    Dec 28, 2017 at 16:04

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Most CAD software will allow you to produce arbitrary curves using control points eg NURBS and indeed this is analogous to the more traditional method of using a flexible strip of wood or steel held in place with pegs under tension.

The important point is to consider the manufacturing process and how you translate an arbitrary curve on a drawing or model into a real part.

The easiest type of curve to deal with is one made up of circular arcs and conic sections. These can be rolled fairy easily on roll benders, slip rollers or mandrel benders (for tight radius curves in hollow sections).

For more complex compound curves modern multi-axis CNC processes haev made a huge difference, indeed you can see this is the progression of mass produced car bodies since the 1960s. Having said that the traditional process of hand sculpting patterns in clay is still not entirely obsolete.

There are also a lot of artisan techiques for making complex forms in sheet metal. Hand built car bodies are generally constructed over a plywood 'buck'

enter image description here

In the background of that picture you can also see an 'English wheel' which is essentially a pair of adjustable rollers used to shape sheet metal as well as various specialist hammers and dies.

these sheet metal forming processes require a lot of skill from the operator both in terms of the physical process and having a good 'eye' for interpreting a drawing or model.

It is also interesting to look at Gothic cathedrals which mostly consist of intersecting circular arcs which can be generated using essentially quite simple geometry with dividers and a square (which is still the symbol of freemasonry).

Another important construction is the catenary curve , often seen in bridges, essentially the shape formed by a chain hanging under its own weight.


In many cases part of the design process is simplifying an arbitrary curve to elements which can be made by reasonably straightforward geometric constructions

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  • $\begingroup$ "In many cases part of the design process is simplifying an arbitrary curve to elements which can be made by reasonably straightforward geometric constructions" The "umbilic torus" the OP linked is a good example of that. If you zoom in you can see the individual squares it was made from. $\endgroup$
    – JMac
    Dec 28, 2017 at 17:43
  • $\begingroup$ "a CHAIN hanging under its own weight." $\endgroup$
    – Robin L.
    Jan 5, 2018 at 9:37
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This is a very broad question and here is my broad answer.

Regarding old methods(pre-CAD and CNC) you can look at how Gaudi designed and built many of his buildings. In particular he used Catenary curves. Also, there are many craftsman to create the features by hand. This is well documented.

New buildings leverage the power of the CAD/CNC systems aid in the complex fabrication. If everything is unique, so be it, and CAD/CNC/hard work is done to build the job. Depending on the materials, will drive the fabrication methods.

In the planning and detailing phase and/or quoting phase, typically some repetition is identified or created. The surfaces are rationalized into modules of repeating parts to simplify the entire system. If possible, splines are tried to be converted into sets of radial/tangential curves. Software and software automation is used to help with this rationalization.

This will drive down cost, decrease production time, decrease paperwork, simplify site work, decrease unique part count, decrease model size, etc, etc. In these projects Time is Money.

Regardless, this type of architecture requires highly skilled expert contractors to build this type of work.

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I was involved in the early stages of design of Azadi tower, my partners were building the study models for the project. The architect, Hossein Amanat had hired a famous structural engineering firm Ov Arup. They changed and modified the curves on the arcs and legs many times, working in close coordination and relying heavily on feed back from model making team.

The models where made of acrylic resin hardened into cylinders of roughly 12 inch diameter and where cut to rough dimensions by a dremel and then sanded down to precision by an assortment of custum build files or custum made tracks to slide the model by the band saw paper. But the finish shaping was done by hand by one of our friends, Amir Tabrizian, who was really knowledgeable of the geometry of ancient Iranian arcs and curves which were used in the Azadi design, and at the same time had perfect ambidexterity to hold and file or trim the model.

I heard his input and feed back as to using generating lines and curves was very useful for the study of the geometry of project.

Later they decided to use the white stone used to surface the building as the actual formwork for structural concrete, so they created detailed models and shop drawings for each individual piece of stone clading including instructions on how to cut them to form the intricate surfaces.

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