Questions tagged [bezier-curve]

Questions on Bézier curves, which are used for numerical analysis with applications in computer graphics.

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That is the math behind interpolating circle using Bezier curve?

This is a basic circle build by a graphic editor using Bézier spline. The X here is 0.552125R: But how this value had been gotten? I mean reverse engineering, the mathematical equation which results ...
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Rational Beziér Surface isocurves

In Variational "Design of Rational Bezier Curves and Surfaces" Georges-Pierre Bonneau's PhD thesis it's stated that isocurves (which are curves obtained fixing one parameter of a ...
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Conic section and Bézier curves

I'd like to prove or disprove the following: given a compact arc of a conic section $\mathcal{T} \subset \mathbb{R}^2$ such that for any $P \neq Q \in \mathcal{T}$ the tangent vectors $\tau_P, \tau_Q$ ...
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Why Bezier Curve is not undefined at t=0 and t = 1? [duplicate]

Sorry,this might be a dumb question, but I couldn't really understand it. If we define Bezier Curves as: $B(t) = \displaystyle\sum_{i = 0}^{n} P_i\binom{n}{i}t^i(1-t)^{n-i}$ when t and i are zero it ...
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Rational Bézier Curves are projectively invariant

I want to prove that a Rational Bézier Curve is not only affine invariant but also a projective invariant. By affine invariance i mean that applying an affine map to the curve is the same as applying ...
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Variation diminishing property

in the book "CAGD" by Farin he gives a proof of the variation diminishing property for piecewise linear interpolation. Given a continuous curve $c$ in $\mathbb{R}^3$ we define a piecewise ...
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Cubic bezier speed curve solve for t for a given d

I am trying to currently achieve an animation as following ...
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Equation to find the Principal Unit Normal of a Bezier curve at $t$ ($0\leq t\leq1$)?

I have recently been working on Bezier curves and have come across a very interesting snippet of code (don't worry its almost entirely mathematical) relating to finding the principal normal of a ...
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Tensor product surface

I'm studying Bézier surfaces and they are a special case of a more general construction, tensor product surfaces. I'd really like to know why this name. I am familiar only with the abstract (algebraic)...
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corners of a rectangular bended sheet in 3d

I write a mesher for special geometric forms to train Computer Aided Engineering with them. I need to display a rectangular with rounded corners in 3d. It is a Stadium Shape/Obolong/Capsule Shape, ...
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Scaling Bezier curve first and second derivative?

I'm trying to use a cubic Bezier curve for ease-in-ease-out movement in 1 dimension. The output is position, first derivative is velocity, and second derivative is acceleration to control a motor. But ...
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Shape invariant of a rational Bézier curve

I'd like to prove that $c_1=\frac{w_0w_2}{w_1^2}\,$ is a shape invariant for a quadratic Bézier curve, i.e. if the weights $w_i$ are modified but $c_1$ is kept constant, then the shape of the curve ...
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Slope of bézier curve, with t not between 0 and 1

I work with so called "animation curves" in Unity which are basically bézier curves. For an algorithm I need to know the slope of the curve at point t. I already found some solutions on the ...
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Measure for the number of curves in a shape in $\mathbb{R}^2$

I am looking for a measure of the following form: Say we have some geometric ribbon-like "shape/curve' in $\mathbb{R}^2$. Example: How can we model the number of "curves" (twists, ...
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How to quantify the error of explicit methods of merging Bezier curves

I came across this paper here describing a method of creating an n_th order Bezier curve that approximates multiple other Bezier curves that are connected. The method is based on minimzing the error ...
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When to choose Bezier curve over B-Spline curve?

I am reading about Bézier curves and B-Spline curves. I have understood both mathematically and intuitively that the big difference between these two kind of curves is that when dealing with Bézier ...
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Joining a straight line to a Bezier curve

Suppose that we have a Bezier curve of order n represented in the matrix notation as $P(t) = T(t) \Lambda P_c$ (where P(t) is a point on the Bezier curve, $\Lambda$ is a lower triangular matrix ...
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Join two bezier o that the radius of curvature is smooth

I need to join more Bezier curves in order to obtain a smooth trajectory for a vehicle simulation. Let' say the curve 1 has the points P0, P1, P2, P3, and the curve 2 has the points Q0, Q1, Q2, Q3. I ...
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Equation of a cubic Bezier curve

For a quadratic Bezier Curve defined by points $A, B, C$, with point $M$ on the curve interpolated by $i$, When points $A, M, C$ and angle $\alpha$ are given, $i$ and $B$ are: $$i^2 = \frac{(y_A-y_M)\...
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Most efficient method for creating a lookup table for a 4 point Bezier Curve?

I understand as discussed here that one cannot easily get a $y(x)$ curve from a Bezier equation as Beziers are solved as $y(t)$ and $x(t)$. Thus to get around this, I have created a lookup table with ...
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Calculate length of a cubic spline, given it's vertices in 3d space?

Given a cubic spline defined by $n$ number of vertices in a 3d space, how would one calculate the length of this spline? (Attached picture is just for illustration, to explain what I mean by vertices, ...
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What is $\mathbb{E}^d$?

In a paper (chapter 3, the paper is in Italian) I'm reading I found: A Bézier curve of degree $n$ is a parametric polynomial curve $X:[0;1]\to\mathbb{E}^d$ defined as follows: I'm not an expert in ...
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How to make a looping (Bézier?) curve?

I want to display the path traced by a point, $p_1$ rotating around another, moving point, $p_2$. The point $p_2$ moves parallel to an axis on a plane, and the distance between the points is fixed. ...
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2 votes
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Classification of quadratic rational Bézier curves

My teacher months ago gave me a few hints on a method that can classify quadratic rational Bézier curves as different conic sections (arcs of those). As I recall, it starts with such a curve given ...
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How do you fill a composite Bézier curve composed of a list of cubic Bézier curves?

According to Wikipedia: "A composite Bézier curve is a series of Bézier curves joined end to end where the last point of one curve coincides with the starting point of the next curve." I'...
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Creating Bézier curves with a maximum curvature

I am creating $5$-point Bézier curves. Sometimes, these curves are too sharp, see for example the below curve: there is a sharp cusp on the left. Is there any way that I can generate Bézier (or ...
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Are there curves similar to Bézier curves, but with a fixed length?

Bézier curves have some nice properties, such as starting at $P_0$ and ending at $P_n$ (for an $n$-degree Bézier curve). I am looking for a class of (curvy) curves, but with the additional property ...
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Matching Cubic Bézier to match Cubic Polynomial

I have a series of cubic polynomials that are being used to create a trajectory. Where some constraints can be applied to each polynomial, such that these 4 parameters are satisfied. -Initial Position ...
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1 answer
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Equation for a surface given N points in 3d space?

In 2D it is fairly well established how to generate a smooth curve from arbitrary points à la Bézier curves. Is there an equivalent to this for a smooth surface with arbitrary 3d points?
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How to use Bezier curve to model one to one functions?

I recently have been looking at Bezier curves. I thought it would be useful in modeling the continuous function across a series of plotted points. So I started with $f(x)$ and $g(x)$ and real ...
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2 votes
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Find the control point of a quadratic bezier curve with known maximum curve value

I have a quadratic Bézier curve which starts at $P0(5,10)$ and ends at $P2(7,0)$. I know that the maximum x-value of the curve is supposed to be at $x = 10$ and that the y-value of the control point $...
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Curve scheme that is not affine invariant

In chapter 4 of "Curves and surfaces for CAGD" by Gerald Farin it is asked to find a curve scheme (i understood it as a procedure to produce a curve since the topic was De Casteljau ...
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Find a continuous bijective function with the given form and its inverse

I am looking to find a function $g:\mathbb{R}^2\rightarrow\mathbb{R}^2$ and its inverse $g^{-1}$, such that $g$ and $g^{-1}$ are continuous, smooth, and bijective, such that: $$ g(x,0)=\begin{pmatrix}...
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Minimum (or mean?) radius of a cubic bezier curve

I'm trying to calculate the minimum radius in a cubic Bezier curve (in C#). I know this question is around on StackExchange, and have thoroughly browsed the answers and tried different implementations....
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How do you move multiple points with a set distance (linear) between them on a non-linear spline?

I'm doing some things related to spline code, and I've run into a problem. Please take a quick look at the picture below. Black = spline (bezier), red circles = points on spline Imagine that the ...
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2 votes
2 answers
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What are the conditions for the union of two Elastica curves to be an Elastica curve as well?

An Elastica curve is defined as one that minimises the bending energy, i.e. the curvature squared. Suppose I have two Elastica curves, and I decide to join them end to end. Then what are the ...
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How to represent a circular arc with a rational Bézier curve with prescribed weights of the endpoints?

I would like to construct a rational Bézier curve that represents a circular arc of sweep angle less than $180^\circ$. It is clear to me how to construct the control points ($\mathbf{P}_0$ and $\...
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2 votes
1 answer
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Is it possible to check if two Bézier curves intersect based only on their control points? [duplicate]

By defining a set of control points $P_i$ a smooth Bézier curve can be constructed. Imagine we have two sets of control points $A={P_i}$ and $B={Q_i}$ that each define a different curve, can we ...
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1 vote
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Sphere and cubic Bézier curve intersection points

I am trying to find a sphere and a cubic Bézier curve intersection points. Sphere: $$ \begin{align} x &= x_0 + r \sin \theta \; \cos\varphi \\ y &= y_0 + r \sin \theta \; \sin\varphi \qquad (...
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What kind of cubic "interpolation" is this?

I recently came across an "interpolation" scheme of the form $$ P_1\frac{(1-x)^3}{6} + P_2\left(\frac{x^3}{2} - x^2 + \frac{2}{3}\right) + P_3 \left(-\frac{x^3}{2} + \frac{x^2}{2} + \frac{x}{...
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2 votes
1 answer
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Is there an easy way to convert a 2d Bezier curve into 1d (with time as another dimension)?

I was trying to do some curve fitting for 2d points, and I immediately found this piece of code here: https://github.com/erich666/GraphicsGems/blob/master/gems/FitCurves.c But after some research I ...
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3 votes
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Getting the most accurate bezier curve that plots a sine wave

I would like to draw a bezier curve that looks the most like a sine wave that has a single wave length of 1000 pixel and an amplitude of 1, which is 159.15 pixels high (wave length / 2𝜋). Here are ...
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Number of Quadratic Bezier Curve-Ray Intersections

Given some quadratic bezier curve $B(t)$ and some ray $R$ is there an equation to calculate the number of intersections between the two. (For my application I only need to consider 2d space).
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Relaxed Uniform Cubic B-Spline in knot vector representation?

I am curious if its possible to represent relaxed uniform cubic b-spline curves in knot vector form. With relaxed uniform cubic b-spline curves, a curve with four control points represents 3 bezier ...
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Convert Bezier casteljau bezier curve to control point bezier curve

I have a Bezier curve that was constructed using the CastelJau algorithm - if I'm understanding this algorithm correctly, you input 4 points, and it gives you a curve that will roughly pass through ...
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1 answer
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What kind of animation interpolation curve is this?

I am working on reverse engineering the animation system of a video game. I initially assumed it used a Bezier curve to interpolate between keyframes, but I've since worked out the curve it actually ...
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Eliminate method for getting implicit function of biquadratic bezier surface

The bezier surface is the function of parameter like x = f(u,v). I have the function of biquadratic bezier surface which contains 9 control points x1 to x9. $$((1-u)^2)((1-v)^2)x_1+2u(1-u)((1-v)^2)x_2+...
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Can a cubic Bézier curve be accurately represented by a series of $f(x)$ functions (i.e. traditional non-parametric)?

Please forgive my naive assumptions, if these come across as such -- I am not a mathematician by the standards of this community. Allow me to elaborate: for what's it worth, intuitively I feel ...
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How can I find the point(s) where the tangent to a Bézier is horizontal?

I am working on cubic Béziers, and need to find the points where the tangent to the curve is vertical or horizontal. So far I've managed to: convert the Bézier to a cubic polynomial find the ...
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1 vote
1 answer
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Unnecessary first and last knots of B-splines?

Given any B-spline with sorted knots $t_0,t_1,\dots,t_{n+1}$ and degree $p$, the domain of the curve is $[t_p,t_{n+1-p}]$. Outside of this domain the basis functions don't sum up to $1$. Inside the ...
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