Curves

The Hilbert Curve is one of the results of David Hilbert’s vision of mathematics as a network of symbolic systems. Defined in 1891, this curve is a fractal that fills a two-dimensional plane with a one-dimensional line in the limit. It exhibits self-similarity. It emerged…

A spiral is a curve that begins at a central point and continuously moves outward, either getting larger (in an outward spiral) or smaller (in an inward spiral). It often revolves around a central point or axis. Spirals are fascinating because they appear in many…

A fractal is a complex geometric shape that we can split into parts, each of which is a reduced-scale copy of the whole. This property is self-similarity. Fractals often exhibit patterns that repeat at different scales. We can find fractals in nature, such as in…

In mathematics, “curve” describes one-dimensional objects or line shapes, regardless of their curvature. Straight lines, polylines, and curved lines all fall under the category of “curves.” You may remember working with equation graphs in high school math classes. For instance, a first-degree equation produces straight-line…

The Flowsnake, or Gosper curve is a space-filling fractal. It is also known as the Peano-Gosper curve. There are other similar space-filling fractals such as the Dragon curve, or the Hilbert curve. A space-filling fractal is a special type of curve, that fills a plane…

In computer-aided design (CAD), a polyline is a series (or a chain) of straight lines. Each straight section of a polyline is a “segment,” and the points where the segments connect are “vertices.” If a polyline’s starting and ending vertices coincide, it is a “closed…

Below is the first paper of my son, Mete Yazar. It is about a mathematical and geometric exercise of calculating the surface area of an arbitrary shape (a classical guitar’s body panel). He did a good job in utilizing Bezier/de Casteljau curves and generating the…

Here is the shortest possible way of generating quick parametric curves in Rhino Python. So, you may change the f, g, and h functions to test any function curve. In this Python code, the list comprehension [(f(t), g(t), h(t)) for t in [t0 + i*dt…

Here is a wave generator code I developed using Grasshopper and Python. While searching for a solution to the realistic water simulations, I came up with the Gerstner Waves. I tried to implement it. However, I came up with this final result, which is not…

Today’s beautiful curve is the spherical cycloid. It is a cycloid, rolling on a 3d circular path rather than a straight and 2d one. There are algebraic explanations of this curve. Therefore, I find it interesting to experiment with them, since it is more interesting…

Gyroid is a popular triply-periodic minimal surface. Although it is a mathematical entity, designers and architects like its approximations very much. We used an interpretation of Gyroid in the rammed-earth structure: “Common-action Wall” in 2017. In that project, we utilized its spatial quality of dividing…

Curvature can be roughly described as how much a curve is “turning” at point a P. We place two “very” close tangents and measure the difference between them. The closer these tangents are, the more precise our approximation would be. An osculating circle is a…

Today’s computational curve is the beautiful Sierpinski Triangle. It is a fractal named after the Polish mathematician Waclaw Sierpinski, who described it in 1915, though it had been previously described by other mathematicians. It is a self-replicating pattern that arises from a simple recursive process….

Lissajous curves, named after the French physicist Jules Antoine Lissajous are a family of curves that emerge from the interaction between two harmonic oscillations. They have applications in various fields including physics, engineering, and signal processing. They are commonly used in electronic devices such as…

This is a short video tutorial on the B-Spline decomposition I studied earlier here. This tutorial demonstrates how to decompose a B-Spline curve into Bezier curves using Rhino. Despite the original Bezier-de Casteljau algorithm requiring degree+1 control points, Rhino allows drawing a degree-3 curve with…

In this short tutorial, I am going to show you how to locate a parametric point on a Bezier curve. This will be a third-degree cubic Bezier curve. So, I start by placing four control points. I name these points from P0 to P3. Then,…

De Boor’s algorithm, a maestro of basis spline refinement, meticulously navigates through knots, unraveling the intricacies of B-splines with mathematical precision. Meanwhile, De Casteljau, the geometric orchestrator, takes center stage in the Bezier ballet, elegantly guiding control points through a recursive dance. Together, these algorithms…

The Hilbert Curve, also referred to as the Hilbert space-filling curve, was initially introduced by the German mathematician David Hilbert in 1891. It is a continuous fractal curve, presenting a variation of the space-filling Peano curves uncovered by Giuseppe Peano in 1890. After a study…

This week’s fractal is the famous Dragon Curve. Dragon Curve exhibits self-similarity, meaning parts of the curve resemble the overall shape, regardless of scale. It’s fascinating because a relatively simple construction process generates a complex and visually captivating fractal pattern. The Dragon Curve is often…

The Animated Tree Growth is an interesting study for Grasshopper. First, I developed a regular tree generation definition similar to those I studied earlier, here, here, and here. In component group 1, I develop an initial generator arc. Then, in group 2, I generate the…