#NURBS

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pafurijazA <a href="https://mastodon.social/tags/CAD" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#CAD</a> model made with <a href="https://mastodon.social/tags/Plasticity" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Plasticity</a> by Kuechmeister Swagger a German user with great skill in Plasticity and <a href="https://mastodon.social/tags/Moi3D" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Moi3D</a> <a href="https://mastodon.social/tags/nurbs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#nurbs</a> <a href="https://mastodon.social/tags/surfacing" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#surfacing</a> <a href="https://mastodon.social/tags/xnurbs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#xnurbs</a> <a href="https://mastodon.social/tags/car" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#car</a> <a href="https://youtu.be/P0l--9Oxokk?feature=shared" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://youtu.be/P0l--9Oxokk?feature=shared</a>

pafurijazSooner or later I will be able to do it, for now it seems that something is improving, I managed to create an <a href="https://mastodon.social/tags/IGES" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#IGES</a> file exported directly by <a href="https://mastodon.social/tags/Blender" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Blender</a>, and also a <a href="https://mastodon.social/tags/STEP" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#STEP</a> file always with the same <a href="https://mastodon.social/tags/Python" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Python</a> add-on. <a href="https://mastodon.social/tags/b3d" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#b3d</a> <a href="https://mastodon.social/tags/CAD" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#CAD</a> <a href="https://mastodon.social/tags/SubD" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#SubD</a> <a href="https://mastodon.social/tags/nurbs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#nurbs</a>

pafurijazI've asked to <a href="https://mastodon.social/tags/AI" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AI</a>, an add-on to export <a href="https://mastodon.social/tags/SubD" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#SubD</a> to <a href="https://mastodon.social/tags/NURBS" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#NURBS</a> from <a href="https://mastodon.social/tags/b3d" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#b3d</a>, the add-on managed to export something to both IGES and STEP, there are no errors, the exported files contain data structures but unfortunately any <a href="https://mastodon.social/tags/CAD" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#CAD</a> app managed to open those, although that seem like a failure, it's not, in fact it is the first time that I get an add-on that exports both in <a href="https://mastodon.social/tags/IGES" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#IGES</a> and in <a href="https://mastodon.social/tags/STEP" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#STEP</a> directly from <a href="https://mastodon.social/tags/Blender" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Blender</a> without using external library, maybe next time I could be able to have a working add-on.

SplinesThe classic <a href="https://pixelfed.social/discover/tags/IonicScroll?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#IonicScroll</a> is the most complex of all components in the <a href="https://pixelfed.social/discover/tags/IonicOrder?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#IonicOrder</a> mainly because it is poorly documented, if at all, and even poorly understood. It is as if the classical architects deliberately concealed its enigmatic design secrets within the confines of a smooth elegant shell that could only be revealed after intense study and analysis. I got this impression because I spent years searching for credible and actionable documentation on how to recreate this beautiful design in a <a href="https://pixelfed.social/discover/tags/CAD?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#CAD</a> tool. In the Age of Internet and Social Media, my web searches always disappointed me because the results lacked something vital in one respect or another. Over the years, I created hundreds of versions of the scroll that looked so perfect and pleasing that I thought I had cracked it, only to find some flaw or another in my work. So, it is with caution that I present my work on the scroll in the hopes that someone will build upon this knowledge and either validate the design, or correct it and share it with me and the rest of the world. Looking back at my progress, I'm now surprised at how remarkably simple and elegant the design is that defied familiar geometrical construction techniques I had been using until now. As I mentioned in my introductory post, this design can be recreated by drawing simple 2-dimensional lines and circular arcs, but instead of just <a href="https://pixelfed.social/discover/tags/primaryProfileCurves?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#primaryProfileCurves</a>, we will use up to three additional sets of curves — <a href="https://pixelfed.social/discover/tags/secondaryCurves?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#secondaryCurves</a>, <a href="https://pixelfed.social/discover/tags/tertiaryCurves?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#tertiaryCurves</a>, and <a href="https://pixelfed.social/discover/tags/quaternaryCurves?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#quaternaryCurves</a> — each derived from the previous set. I extracted the <a href="https://pixelfed.social/discover/tags/primaryCurves?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#primaryCurves</a> after a lengthy trial-and-error process that involved <a href="https://pixelfed.social/discover/tags/curveFitting?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#curveFitting</a> image scans from <a href="https://pixelfed.social/discover/tags/Vignola?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#Vignola</a>’s book, <a href="https://pixelfed.social/discover/tags/RegolaArchitettura?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#RegolaArchitettura</a>. I had to <a href="https://pixelfed.social/discover/tags/reverseEngineer?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#reverseEngineer</a> the details because the measurements have either been lost, or are locked away in some library. Even though we start with lines and arcs, the end results are always <a href="https://pixelfed.social/discover/tags/NURBS?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#NURBS</a> curves and surfaces, but everything is done by the CAD tool, and no additional math is needed.

SplinesThe bottom 1/3 of the <a href="https://pixelfed.social/discover/tags/columnShaft?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#columnShaft</a> for an <a href="https://pixelfed.social/discover/tags/IonicColumn?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#IonicColumn</a> is a perfect cylinder. So the line below point B is a straight line. In <a href="https://pixelfed.social/p/Splines/791723063470910081" rel="nofollow noopener noreferrer" target="_blank">https://pixelfed.social/p/Splines/791723063470910081</a>, we blended the bottom end of the 60° arc and the top end of the long interpolated curve between points J and K. Now blend the bottom end of the interpolated curve and the top end of the straight line between points B and C to obtain the 3rd and final <a href="https://pixelfed.social/discover/tags/NURBS?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#NURBS</a> segment for the <a href="https://pixelfed.social/discover/tags/primaryProfileCurve?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#primaryProfileCurve</a> of the shaft. Just like there's a <a href="https://pixelfed.social/discover/tags/cavetto?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#cavetto</a> and <a href="https://pixelfed.social/discover/tags/fillet?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#fillet</a> near the <a href="https://pixelfed.social/discover/tags/neck?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#neck</a> of the shaft, there is a fillet and cavetto near the foot of the shaft. However, there is a subtle difference between the two. The cavetto near the neck is tangential to the blended <a href="https://pixelfed.social/discover/tags/NURBS?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#NURBS</a> curve that is not a straight line. The profile curve for the cavetto near the foot is tangential to a straight line. There is a special name for a cavetto that is tangential to a straight line or flat surface, like the two cavetto moldings in the <a href="https://pixelfed.social/discover/tags/dado?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#dado</a> of the <a href="https://pixelfed.social/discover/tags/pedestal?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#pedestal</a>. It's called a <a href="https://pixelfed.social/discover/tags/conge?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#conge</a>. Another alternate name for the cavetto molding is <a href="https://pixelfed.social/discover/tags/cove?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#cove</a>, which is evocative of "cave" because of its concave profile curve. Above the neck is a fillet 8 units tall and an <a href="https://pixelfed.social/discover/tags/astragal?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#astragal</a> 16 units tall that <a href="https://pixelfed.social/discover/tags/Scarlata?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#Scarlata</a> puts in braces in the column shaft section within his tables of <a href="https://pixelfed.social/discover/tags/VignolaProportions?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#VignolaProportions</a>, with a note saying they are not counted as part of the shaft but are accounted for as part of the <a href="https://pixelfed.social/discover/tags/capital?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#capital</a>. I decided to include the top fillet as part of the shaft and keep the astragal with the capital. It does not change the design or alter the proportions in any way, but the inclusion of the fillet makes it more practical for <a href="https://pixelfed.social/discover/tags/3DPrinting?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#3DPrinting</a> and <a href="https://pixelfed.social/discover/tags/CNCMilling?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#CNCMilling</a> of the neck. This concludes the profile curve for the shaft with a height of 291 parts or 2328 units + 8 for fillet. The column shaft is tapered in the upper 2/3 due to <a href="https://pixelfed.social/discover/tags/entasis?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#entasis</a> whose purpose is to make optical corrections to the shape of the column which, without correction, appeared concave near the top.

SplinesThe section between points J and K is the <a href="https://pixelfed.social/discover/tags/neck?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#neck</a> of the <a href="https://pixelfed.social/discover/tags/shaft?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#shaft</a>. The blue <a href="https://pixelfed.social/discover/tags/primaryProfileCurve?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#primaryProfileCurve</a> below J is the <a href="https://pixelfed.social/discover/tags/interpolated?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#interpolated</a> <a href="https://pixelfed.social/discover/tags/NURBS?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#NURBS</a> curve we fit through 8 points in <a href="https://pixelfed.social/p/Splines/791526497210906825" rel="nofollow noopener noreferrer" target="_blank">https://pixelfed.social/p/Splines/791526497210906825</a>. The neck is conceptually divided into three bands, each 1 part (8 units) tall. In the top 2/3, we draw a circular 90° arc with radius of 16 units, divide it into thirds, and discard the lower 30° portion. Then, blend the lower end of the arc and upper end of the interpolated NURBS curve to create a new NURBS curve shown here in magenta. Zoom in, and you will see that it deviates slightly from the original 90° arc. This is because the blended curve is tangential to the 60° arc and the longer NURBS curve. When joined, the three sections form a smooth continuously differentiable NURBS curve. This level of precision is only needed for engineering work. If you just want a <a href="https://pixelfed.social/discover/tags/charcoal?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#charcoal</a> <a href="https://pixelfed.social/discover/tags/sketch?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#sketch</a>, <a href="https://pixelfed.social/discover/tags/draw?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#draw</a> in <a href="https://pixelfed.social/discover/tags/ink?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#ink</a>, <a href="https://pixelfed.social/discover/tags/paint?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#paint</a> in <a href="https://pixelfed.social/discover/tags/watercolor?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#watercolor</a>, or even make <a href="https://pixelfed.social/discover/tags/clay?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#clay</a> or <a href="https://pixelfed.social/discover/tags/ceramic?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#ceramic</a> <a href="https://pixelfed.social/discover/tags/basrelief?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#basrelief</a>, then you don't even need a <a href="https://pixelfed.social/discover/tags/CAD?src=hash" class="u-url hashtag" rel="nofollow noopener noreferrer" target="_blank">#CAD</a> program. A compass and protractor are sufficient. Just blend the shapes by hand as closely as you can. The imperfections, if any will be imperceptible. This brings us back to the previous post. If you're not using CAD, how do you obtain the 8 points C through J using manual tools? Look closely at the radiating lines, first of which passes through point B and the last one reaches point 8. An easy way to find the angle between these two lines is to use basic trigonometry. Focus on the center of the arc, follow up to point 8, and then drop down vertically where the horizontal line is split at 120 units, and close back to the origin. This is a right triangle whose hypotenuse is the radius of the arc. The cosine of the angle between the base and the hypotenuse is 120/144 = 0.83333333. So the angle itself is arc cosine of 0.83333333, or 33.55730976°. For hand drawing, round it off to 33.6°. Then divide that into 8 parts of 4.2° each to plot points 1 through 8.

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