#parametric-curve

When thinking about serious and specific topics, I suddenly find myself looking back at the basics again. Catenary curves, are not the most beautiful one among those mathematical forms found in nature. However, it still tells us a lot about structures. The name apparently comes from catenaria, the Latin word for “chain”. This Grasshopper code was actually an example from our book 10 years ago (link here). It seems catenary curves could potentially be used in a type of dome…

The Fourier Transform is a powerful mathematical technique that allows us to analyze the different frequency components within a signal or shape. Its discrete version, the Discrete Fourier Transform (DFT), is used when working with numerical data. I studied the Fourier Series here before. One of the most fascinating aspects of the DFT is that it can represent a signal or shape using rotating circles (or vectors). Each circle corresponds to a specific frequency, amplitude, and phase. When connected in…

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 for i in range(int((t1-t0)/dt)+1)]] works by first generating a list of t values from t0 to t1 with an increment of dt using the inner comprehension. The outer comprehension then iterates over these t values,…

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 harmonic motion. Lissajous curve functions can help in understanding the…

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 placing control points, dividing segments, and connecting points in a…

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 0 and 1. I will divide the segments, so I…

I have been studying the Bezier-De Casteljau algorithm for several years. Last week, I finally managed to understand and implement B-Spline Construction in Rhino and Grasshopper Python code, using the Cox-De Boor algorithm. This algorithm calculates the effect of control points on a B-Spline with many control points. After this, I am not limited to the degree+1 control points. This animation shows both algorithms working together. This Rhino Python code in a Grasshopper definition can handle three tasks. The first…

This is the continuation of the previous post on parametric curve equations. In this new version, the script picks a NURBS curve from the user. Then, it analyses the curve’s degree and control points. Unfortunately, only the curves with degree+1 number of control points can be processed. In the future, I hope that I will be able to extend this script to include multi-span curves with more than degree +1 control points. Finally, the script creates the three functions for…

The parametric curve equations are good examples to demonstrate the bridge between computer-aided design and mathematics. Although useless and pointless, it is a good exercise to extract the curve equations. In this Rhino Python code, I present a generalized equation extractor for Rhino. Rhino curves are good examples de Casteljau and Bézier curves. You can see the mathematical underpinnings of Rhino curves with this exercise: This code asks the user the degree of the curve. By default, Rhino’s curves are…

Yes, interesting topics started to reveal themselves, when I dig into the function curves, especially in the parametric representation. The first interesting application is the “derivative”. Nobody in high school told me that the derivative of a function at a given point gives the slope of the graph at that point. Moreover, it is possible to convert the slope value into an angle in degrees, showing the angle of the tangent line to the x-axis at that point. They are…