Bézier curves - a primer

Preface In order to draw things in 2D, we usually rely on lines, which typically get classified into two categories: straight lines, and curves. The first of these are as easy to draw as they are easy to make a computer draw. Curves, however, are a much bigger problem. They're named after Pierre Bézier, who is principally responsible for getting them known to the world as a curve well-suited for design work (working for Renault and publishing his investigations in 1962), although he was not the first, or only one, to "invent" these type of curves. So, what if you need to program them yourself? —Pomax (or in the tweetworld, @TheRealPomax) Note: virtually all Bézier graphics are interactive. This page uses interactive examples, relying heavily on Bezier.js, as well as "real" maths (in LaTeX form) which is typeset using the most excellent MathJax library. This book is open source. This book is an open source software project, and lives on two github repositorites. Questions, comments: and Related:  CODING

math - How to remove a node of a Bezier curve so that the shape of the curve does not change NYU Media Research Lab | Projects | Multiresolution Surfaces | Booleans Subdivision representations of surfaces are convenient for many common tasks such as rendering, topology-preserving editing and simulation. However, some important editing operations modify the topology of surfaces, for example, computing a union, intersection or difference of solids represented by closed surfaces. In general, it is impossible to represent the resulting surface as a subdivision surface precisely. In this talk, we describe how to approximate the result of a boolean operation applied to a pair of solids represented by closed subdivision surfaces. Our algorithms aim to minimize the size of the new control mesh, and to control the valence of the newly created vertices. Presentation H. SIGGRAPH 2001 Paper: H. Examples Many important modeling operations can be expressed as boolean operations on closed solids. Boolean operations may change the surface topology. Our algorithms proceed in three main stages: remeshing, parameterization and fitting.

Ruby on Rails Tutorial: Learn Rails by Example book and screencasts by Michael Hartl B-spline/NURBS Curves: Knot Insertion The meaning of knot insertion is adding a new knot into the existing knot vector without changing the shape of the curve. This new knot can be equal to an existing one and in this case the multiplicity of that knot is increased by one. Because of the fundamental equality m = n + p + 1, after adding a new knot, the value of m is increased by one, and, consequently, either the number of control points or the degree of the curve must also be increased by one. Although knot insertion looks not very interesting, it is one of the most important algorithms for B-spline and NURBS curves since many other useful algorithms will be based on knot insertion. The left figure below shows a clamped B-spline curve of degree 4 with uniformly spaced knots. Inserting a Single Knot Given a set of n+1 control points p0, p1, ..., pn, a knot vector of m+1 knots U = { u0, u1, ..., um } and a degree p, we want to insert a new knot t into the knot vector without changing the shape of the B-spline curve.

PythonOCC and SMESH | pythonOCC The SMESH module that is included in the PythonOCC distribution has a lot to offer in terms of advanced CAD meshing techniques and therefore deserves some special attention. With this blog post I will try to shed some light on SMESH’s capabilities and the merits it has to offer for the PythonOCC CAD platform. I will mainly focus on prism meshing techniques, simply because this is my current field of interest. But also because it is a fruitful topic for demonstrating SMESH’s capabilities. PythonOCC’s SMESH is a standalone version (created by Fotios Sioutis) of the SMESH module that comes shipped with the Salome Platform. Together with these updates fully functional Netgen (version 4.9.13) and Tetgen (version 1.4.3) meshing plugins will be available to enjoy high quality unstructured grid generation of complex CAD models directly from within PythonOCC. SMESH mesh framework: batteries included Viscous Layers: Prismatic boundary layers Defining algorithms and hypotheses Mesh export

[Site] 99 Bottles of Beer Fundamentals of Transportation/Horizontal Curves Horizontal Curves are one of the two important transition elements in geometric design for highways (along with Vertical Curves). A horizontal curve provides a transition between two tangent strips of roadway, allowing a vehicle to negotiate a turn at a gradual rate rather than a sharp cut. The design of the curve is dependent on the intended design speed for the roadway, as well as other factors including drainage and friction. These curves are semicircles as to provide the driver with a constant turning rate with radii determined by the laws of physics surrounding centripetal force. Fundamental Horizontal Curve Properties Physics Properties A Horizontal Curve in Oslo, Norway When a vehicle makes a turn, two forces are acting upon it. could serve as a countering force, but it generally would provide very little resistance. To deal with this issue, horizontal curves have roads that are tilted at a slight angle. The allowable radius Where: Application of Superelevation and .

myGrain | Régis Frias [Sem versão em português por ora, amigos.] This post was originally written as a requirement for a course at my Masters studies at Aalto University‘s Media Lab and describes the process of creating an audio visualizer in openFrameworks. The name of the application is an homage to my friend Can’s song, which he kindly allowed me to use on this video. The project’s source code can be found here, along with my course colleagues’ projects. Inspiration (Magnetosphere, by Robert Hodgin): I believe this was not supposed to be a tutorial or anything, but some people were curious to know how I did this exercise. I started this sketch in order to study some behaviors I’ve seen on Robert Hodgin’s work (see video above). The basic formula for the sphere in computer parlance looks like this: separationAngle is the angle between 2 successive points (360/numberOfPoints; more points => smaller angle PI/180 converts degrees to radians (case you’re wondering) And that’s about it for now!

Python Programming Language -- Official Website Draw a Smooth Curve through a Set of 2D Points with Bezier Primitives Download source code - 26.12 KB Introduction From time to time, I am faced with the question: how to draw a smooth curve through a set of 2D points? We have interpolation methods at hand. Cubic spline is a cubic polynomial, but Win32, .NET Forms, or WPF do not provide methods to draw a piecewise cubic polynomial curve. On the other hand, the platform provides us with a suite of Bezier curve drawing methods. Bezier spline is a sequence of individual Bezier curves joined to form a whole curve. I spent some time Googling for a code in any C-like language for a Bezier spline, but couldn't found any cool, ready-to-use code. Here, we'll deal with open-ended curves, but the same approach could be applied to the closed curves. Bezier Curve Representation A Bezier curve on a single interval is expressed as: B(t)=(1-t)<sup>3</sup>P<sub>0</sub>+3(1-t)<sup>2</sup>tP<sub>1</sub>+3(1-t)t<sup>2</sup>P<sub>2</sub>+t<sup>3</sup>P<sub>3</sub> (1) where t is in [0,1], and The first derivative of (1) is: The Code

ScorchCAD ScorchCAD is an OpenSCAD clone written for Android devices. ScorchCAD executes OpenSCAD code and displays/exports the resulting model. ScorchCAD is a clone of OpenSCAD meaning that it aims to provide the same functions as OpenSCAD but it does not share the same source code. Most of the basic features are currently working. - 3D shapes: cube, cylinder, sphere and polyhedron - Transformations: translate, rotate, scale, resize, mirror, multmatrix, hull, color - Boolean Operations: union, difference and intersection - Functions: norm, cross, lookup, str, , version, version_num and mathematical functions - Other: echo, for, intersection_for, if, assign, projection, surface - File Import: DXF and STL (ASCII and binary) - User defined variables and special variables : $fa,$fs and \$fn - User defined functions and Math functions - Modifier character * (disable) OpenSCAD function that are not supported minkowski, search, children, parent_module, Modifier Characters (! Graphic Summary of Features

Inline Inline is an esoteric programming language created by OberoN. Description Every value is 8-bit long and has a suffix: :b for binaries, :h for hexadecimal, :d for decimal and :a for ASCIIs. :r and :m for indexed addressing is also possible, like [0:h:r] is the address pointed by register 0 and [0:h:m] is the address pointed by memory cell 0. You can write [0-1:h:r] which is the address pointed by 0 and 1 registers, treating 0 as high part and 1 as low, and also [0-1-2:h:r] and such. In Inline immediate values are closed in parentheses. All other numbers are treated as registers. Every line is treated as a subroutine/labeled code. List of commands IO ! Inputs a char in ASCII to the memory. *(s) position Outputs a char in ASCII from the position (register or memory). " position Outputs a number from the position (register or memory). Memory register > memory Transfers the register into the memory. register < memory Transfers the memory into the register. register <> memory @ label

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