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 a Gerstner Wave generator, but a much simpler one. It combines many user-fed wave components and calculates the resulting single wave. I played with this […]
Posts categorized under Curves
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 than the regular planar cycloids, epicycloids, and hypocycloids. This curve is believed to have been studied first by Jean Bernoulli in 1732. The interesting and […]
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 the space into two intertwining and symmetrical volumes. By making one of these volumes solid, I left the other void. So, in this tutorial, I […]
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 tangent circle that has the same curvature as the curve at point P. The larger the circle, the more “flat” the curve is. An infinitely […]
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. To construct the fractal, you start with an equilateral triangle and then repeatedly remove smaller equilateral triangles from its interior, leaving holes. Each iteration involves […]
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 oscilloscopes to visualize the phase relationship between two oscillating signals. Similarly, they are also useful in mechanical engineering for analyzing and designing mechanisms that involve […]
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 any number of control points. By examining knot points and dividing segments appropriately, the B-Spline curve can be manually subdivided into Bezier curves. This involves […]
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, I connect them by a polyline in order. I explode the polyline into the segments. The parameter of my point must be a number between […]
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 fuse art and mathematics, seamlessly sculpting curves and splines with technical finesse, creating a harmonious symphony of numerical intricacies in computational geometry. ChatGPT is so […]
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 on the mathematical background of this curve, I implemented a Python code into Grasshopper Python. However, I wanted to explore more variations by playing with […]
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 fractal tree by using iteration. I did this with the help of the Anemone add-on. The interesting and original part of this definition is group […]
Today’s fractal is the Julia Set, the amazing simplicity of chaos. There are lots of applets and articles on the internet about this fractal. You can generate this with the iteration of a basic function many times and placing points on the complex plane. I developed a Grasshopper implementation in 2012. Also, this was my first study on complex numbers. At each iteration, the detail level increases. I utilized a […]
A Moebius strip, also known as a Moebius band, is a fascinating mathematical object and a type of non-orientable surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in the 19th century. To visualize a Moebius strip, imagine taking a long, narrow strip of paper and giving it a half twist before connecting its ends to form a loop. The result is a […]
I developed this code 13 years ago while learning the fundamentals of Visual Programming in Grasshopper. I was studying the ways of NURBS curve geometry. The animation shows the construction process of several Bezier Curves. In 2024, I optimized the code and added the thickness. The Flow Earring project showcases the beauty of parametric curves. The Grasshopper definition displays the animated construction process and the variations. The flow of the […]
Since the mid-20th century, the hyperbolic paraboloid surface has been one of the most popular mathematical forms for architects. Named Hypar in short, this is the Quadric Surface equation of the Hyperbolic Paraboloid. Erik Demaine summarizes several examples from architecture such as the roof of the Girls’ Grammar School in London (designed by Chamberlin, Powell, and Bonn), the Philips pavilion at the 1958 Brussels exhibition designed by Le Corbusier, and […]