Splines (@[email protected])

#Braids #3StrandBraids After creating the two #helix curves as described in https://pixelfed.social/p/Splines/797732962403957263, switch to the front view and #project the smaller blue helix on the vertical "wall" of the XZ plane. Hide the original helix. Then switch to the top view and project the larger magenta helix on the "ground" or XY plane and hide the original helix. Now compare the figure in this post with that in the previous post. Both curves have now been #flattened from 3D helix to 2D #sinusoid. When viewed from the front (top-left portion of the diagram), the blue curve is still visible as a #sinusoidal waveform but the magenta appears as a straight line flattened on the ground. When viewed from the top, the magenta curve is still visible as a sinusoid but the blue appears as a straight line clinging to the vertical wall. In the view from side (bottom-left portion of diagram), neither waveform is apparent, and both curves appear as perpendicular straight lines. Only in the perspective view you can see both waveforms, but even here it is clear that they are both flat 2D curves oriented perpendicular to each other in 3D space. Our goal is to convert these two flat sinusoids back into a single composite 3D curve that shows the smaller waveform in the front view and larger one in the top view. In acoustics, a sinusoid represents a pure tone with a single frequency. The tone varies with frequency and its perceptibility varies with amplitude. Musicians and people familiar with acoustic physics will immediately recognize that the blue curve has twice the frequency (or pitch) of the magenta curve, while the magenta curve has twice the amplitude (loudness) of the blue curve. We can divide the period or wavelength into phases. For the blue one, we divide the wavelength into 4 phases of 6 units each and shift the magenta curve left by that amount. Later, we will divide the magenta one into 3 phases — one for each strand, and shift each rightward by that.

Splines (@[email protected])

#Braids #3StrandBraids After creating the two #helix curves as described in https://pixelfed.social/p/Splines/797732962403957263, switch to the front view and #project the smaller blue helix on the vertical "wall" of the XZ plane. Hide the original helix. Then switch to the top view and project the larger magenta helix on the "ground" or XY plane and hide the original helix. Now compare the figure in this post with that in the previous post. Both curves have now been #flattened from 3D helix to 2D #sinusoid. When viewed from the front (top-left portion of the diagram), the blue curve is still visible as a #sinusoidal waveform but the magenta appears as a straight line flattened on the ground. When viewed from the top, the magenta curve is still visible as a sinusoid but the blue appears as a straight line clinging to the vertical wall. In the view from side (bottom-left portion of diagram), neither waveform is apparent, and both curves appear as perpendicular straight lines. Only in the perspective view you can see both waveforms, but even here it is clear that they are both flat 2D curves oriented perpendicular to each other in 3D space. Our goal is to convert these two flat sinusoids back into a single composite 3D curve that shows the smaller waveform in the front view and larger one in the top view. In acoustics, a sinusoid represents a pure tone with a single frequency. The tone varies with frequency and its perceptibility varies with amplitude. Musicians and people familiar with acoustic physics will immediately recognize that the blue curve has twice the frequency (or pitch) of the magenta curve, while the magenta curve has twice the amplitude (loudness) of the blue curve. We can divide the period or wavelength into phases. For the blue one, we divide the wavelength into 4 phases of 6 units each and shift the magenta curve left by that amount. Later, we will divide the magenta one into 3 phases — one for each strand, and shift each rightward by that.

Splines (@[email protected])

#3StrandBraids As I mentioned in https://pixelfed.social/p/Splines/797555432624206626, the geometry of braid strands is not at all obvious despite how familiar they look. The geometry is defined in a series of steps starting with two coil-shaped #helix curves that we project on two perpendicular flat surfaces to create two flat #sinusoidal curves. Then, we transform the two flat sinusoids back into a composite 3D curve that acts as the rail for each strand. Then, we arrange three strands to form a braid. If you have never seen the geometry of a braid strand, you might be surprised at how it looks. A braid strand is #periodic just like the #sinusoid curves it is based on. That means the pattern repeats at a fixed period and the motif can go on forever. For our purposes, a braid that is µ = 144 units long is sufficient, but to get that, we have to start with a helix that is longer than µ. In the front view, start at the origin and draw a helix of length µ*4/3 = 192 units. This is shown in blue in the top-left portion of the figure. Directly underneath that is the view as seen from the top, which looks very similar to the one in the front view, but is distinct. Since a helix is like a round coil, it looks like a circle if viewed from a side as shown in bottom left. Only in the perspective view seen on the right is the coil shape readily apparent. The blue helix has a fixed distance equal to 1/2 part (4 units) from the axis and it makes 8 full turns along the entire length of the axis, meaning it repeats with a period (or wavelength) of 192/8 = 24 units or 3 parts. This is crucial: For a 3 strand braid, the period must be a multiple of 3. Duplicate the blue helix and double it in size while centered on the origin. Then, shift the larger helix by 1/4 of period (6 units) to the left. This larger helix is shown in magenta. These helices are still in 3D. Make sure that the front, top, and right views look like what's shown here. Otherwise, the next step won't work.

Splines (@[email protected])

#3StrandBraids #Braids are the last of the #decorative elements on the #IonicScroll, but like #EggsAndDarts, they are not specific to the #IonicOrder. Braids are a popular design motif that find wide currency in modern #hairstyles, #fashion, and fashion accessories like #belts and #bracelets. Braids come in infinite varieties with varying number of strands, thickness of strands, roundness or flatness of strands, and how tightly or loosely they are wound together. Here, I focus on the 3-strand variant mentioned in #Vignola's book and previewed in https://pixelfed.social/p/Splines/792015485979791089. The image here is brightly colored to draw attention to the 3 strands. The geometry of braid strands is not at all obvious despite how familiar they look. Also, a braid strand is the only feature in the entire iconic order whose geometry cannot be captured with straight lines and circular arcs. Instead, a strand geometry must be defined in a series of steps starting with a basic #sinusoidal curve. A sinusoidal curve or #sinusoid is a wave form whose function belongs to a family of functions known as #transcendentalFunctions that also include #logarithmic and #exponential functions. I mentioned #logarithmicSpirals in https://pixelfed.social/p/Splines/792499765146596723, and in a future post I will show how to construct one and compare it with the #spiral used in our implementation of #IonicVolute. They are called transcendental functions because they transcend the math of finite algebraic polynomials and go beyond geometry into trigonometry. Fortunately, we don't have to go there. Few #CAD tools have a direct primitive for a sinusoid, but almost all have a primitive for a 3-dimensional round coil shape called a #helix which we can use to create the sinusoids we need for a braid strand. To create a sinusoid, all we need to do is #project a helix on a flat surface to convert it into a 2D waveform.

Splines (@[email protected])

The #Capital is the last essential component of the complete #IonicOrder. The column #flutes remain, but they are #decorativeElements, and I will cover them later when I cover the decorative elements of the capital like the #EggsAndDarts motif on the #ovolo and the #3StrandBraid on the ribbon or belt around the middle of the smooth #scrolls. The Ionic capital is complex, but not unapproachable. We will systematically construct everything in this draft rendering using just straight lines and arcs as promised in https://pixelfed.social/p/Splines/789956327130679640, with the exception of the #cymaReversa near the top and the 3-strand braid on the ribbon. In this rendering, the cyma reversa near the top is made using a flattened half-turn of a #helix, but it can also be constructed using elliptical arcs as I described in earlier posts. The braid is a #periodic shape with infinite variety and is also based on a helix. You can vary the number of strands, their thickness, pitch, and so on, none of which are essential to the Ionic Order itself. They're only a jumping point for further exploration. The eggs in the 'eggs and darts' motif can have different shapes. They can be convex like real eggs or concave as shown here, but the top is almost always sliced off. The total depth of the convex or concave shapes can vary, but only within a range of 1 part, or 8 units. The #volutes in the front and back of the capital are based on #spiral shapes, of which there are many different kinds. Some have #continuous curvature changes, while some do it in #discrete steps, like #fibonacci spirals that can approximate #logarithmic spirals seen in nature, e.g., nautilus. When curvature changes are discrete, the spiral arms can diverge in #arithmetic, #geometric, or some other sequence. We will construct all of these, and most notably the smooth, sweeping surface of the scrolls using just straight lines and arcs, and let the #CAD software deal with delicate #NURBS curves and surfaces.

How to make sinusoid wobbly functions – Piter Pasma's Pages