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Questions tagged [bezier-curve]

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

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Optimal Value t for Subdivision of Cubic Bézier Curve and How to Calculate It

In Gabriel Suchowolski’s paper, “Quadratic bezier offsetting with selective subdivision”, he explains how the midpoint—or better said, a parameter $t$ of 0.5—is often not the optimal* point on a ...
Avana's user avatar
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How to Prove Division of One Bezier Curve into Two using de Casteljau Algorithm

$\newcommand{\brstbasis}[2]{b^{#1}_{#2}}$ $\newcommand{\posintset}{\mathbb{Z}^{+}}$ $\newcommand{\intset}{\mathbb{Z}}$ $\newcommand{\realset}{\mathbb{R}}$ $\newcommand{\domain}[1]{\operatorname{dom}\...
Ziqi Fan's user avatar
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Solving a bézier curve backwards

So usually to create a bézier curve; you'd set your origin, control point(s) and target. However the problem I am facing is that exact process but inverted. Problem: For a circle with a given radius, ...
Dennis Solomon's user avatar
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Determine Intersection Point Quadratic Bezier Curve and Plane

I need to compute the intersection point between a quadratic bezier curve and a plane. Thereby i have to solve the equation for the parameter t. The normal vector is in the dot product to the square ...
LuaBoss's user avatar
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Approximation Error on Arc Length of Quadratic Bezier curve

Given a quadratic Bezier curve defined by: $$ B(t) = (1-t)^2P_0 + 2t(1-t)P_1 + t^2P_2 $$ The arc length $ s(t) $ from $0$ to $ t $ is: $$ s(t) = \int_0^t |B'(τ)| dτ. $$ It's known that the arc length ...
cxh007's user avatar
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I want to understand how to determine NURBS control points to fit some curve.

I've been trying to find the control points for a NURBS curve that I want to trace the first octant (1/8) of a unit circle using a second-degree rational bezier. I'm not entirely sure whether such is ...
idk's user avatar
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Proving that a rational b-spline is equivalent to some conic section.

I'm pretty sure that we can trace a conic section like a circle exactly using a NURBS curve. The first quadrant of a unit circle, for example, can be traced by: $C(t)=\frac{(1-t)^{2}P_{0}+2t(1-t)P_{1}+...
idk's user avatar
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Understanding $\mathbb{P}^2$ and rational Bézier curves

I've never taken a projective geometry course, and I'm trying to understand the real projective plane $\mathbb{P}^2$ and its description using homogeneous coordinates, and how these relate to rational ...
bubba's user avatar
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Implicit equation of all points that a circle that traces along a 2d parametric curve.

I want to find an implicit equation that contains points that fall within a circle that has an origin that follows a 2d parametric curve, which would look like you painted a circle along that curve. I ...
Allan J.'s user avatar
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Combining multiple Bézier curves

Consider the situation where we have two [cubic] Bézier curves with the following properties: They share one common point (end of curve 1 = start of curve 2) They have the same direction at the point ...
OJW's user avatar
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Finding the ideal B-spline through data points using Euler-Lagrange: is it just too hard to do?

I am not even sure I have a question anymore (I will just give up)... in the past month or so I have been researching cubic Bézier curves. The idea was to find a fit through data points, using ...
Carlo Wood's user avatar
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1 answer
86 views

Find the control points in 3D space of a 2D Bézier mapped on the parametric space of a (3D) Bézier patch

I was able to solve this for a Bézier curve of order 1 on a bicubic patch (it is a Bézier curve of order 6 Image here ) But for higher degree curves I couldn't find anything. The question is too long ...
Romain Guimbal's user avatar
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Fitting a curve between 2 poses (position and orientation) when the length of the curve is known

I have two visual aruco markers on a flexible line and a camera. I can calculate the pose (position and orientation) of each marker. I have measured the distance between the markers when the line is ...
Wesley's user avatar
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Semicircle as cubic rational bezier curve [closed]

I know how to express a circle as a quadratic rational Bezier curve. Now I need to do it for a cubic one. I'm not sure how to choose the weights. Also I haven't found any online resources so I'm ...
Magne Seier's user avatar
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Defining a Quad Spherical Cube Tile as a Uniform NURBS Surface?

I am trying to create NURBS surface that perfectly fits one face of a Quadrilateralized Spherical Cube (QSC) [also called a Cobb sphere in some contexts, I believe]. I have seen some visualizations of ...
Chaosoahc's user avatar
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2 answers
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Reparametrization of rational Bézier curve

I am trying to solve the following task Using rational Bézier curve find the control points and weights of one sixth of a circle $c_1$, such that $$c_1(0)=\{3,0\},c_1(1)=\{\frac{3}{2},\frac{3\sqrt{3}}{...
Weyr124's user avatar
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Aligning Cubic Bezier Curves (with a width) [duplicate]

I have a cubic Bezier curve which has a width, and I am trying to align another cubic Bezier curve around the outside of it. This is hard to explain in text, so please see the following image: ...
alex's user avatar
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1 answer
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How to represent sequential points from a function more economically as a Bézier curve or else?

I have written a graphics library that has a module that plots function lines. Internally, software goes from a starting x-value to an ending x-value in steps that equal a pixel. This produces ...
DRD's user avatar
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Generalize Finding $y$ value of Bezier curve given a $x$

A quadratic Bezier curve is a parametric curve. Its $x$ value can be represented by the equation: $$(t² × (x_0 - 2 × x_1 + x_2)) + (t × (2 × x_1 - 2 × x_0)) + (x_0 - x) = 0$$ For $0<t<1$ If I ...
Nobu Nishimura's user avatar
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2 answers
84 views

Fitting a modified Bézier curve

I am connecting various vectors with a Bézier curve, using the De Casteljau algorithm. These vectors have a variety of lengths and directions, and when they are equal and orthogonal the curve (purple) ...
Weather Vane's user avatar
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Approximate quadratic Bezier by a $1.0-\sqrt{x^2 + y^2}$ distance

Summary: I have a triangle with points at $A=(0, 100); B=(100, 0); C=(100, 100)$ each point also has an $F$ value, this value is linearly interpolated between every other point and used to get the ...
not-a-real's user avatar
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How to draw circle using Quadratic Bézier curves

I am trying to draw a circle using 4 Quadratic Bézier curves. By referring https://www.degruyter.com/document/doi/10.1515/math-2016-0012/html. $$C(t) = (1-t)^2P_0 + 2(1-t)tP_1 + t^2P_2.$$ If the arc ...
JustWe's user avatar
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How to find the inscribed and circumscribed circles from convex region?

Let $\mathbf{P}(t)$ be a piecewise bezier curve of degree $p$, which defines a jordan curve in the plane such the interior region $D$ is convex. Question: How can I find the external $C_e$ and ...
Carlos Adir's user avatar
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1 vote
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NURB curves with interior knot multiplicy higher than curve degree

I have the following question. If I have a NURB curve where one of the interior knots has multiplicity higher than the degree of curve (I did not chose to have such curves, I am writing a code for ...
Donatas Šimeliūnas's user avatar
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Finding Δt for a Bezier curve following a certain speed

I'm writing an algorithm where I would like a body to move along a Bezier curve at variable speed. My bezier curve So I have a Bezier curve which represents a body's X and Y positions over time. ...
Matt's user avatar
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206 views

How to get enough data to draw an arc or a curve from 3 points?

Note: Although I want to accomplish this in java, I think the question is more suitable for this site since it is mostly mathematical. I am in the following scenario. I want to draw a curve and I have ...
Dante S.'s user avatar
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Numerical instability in solution of perpendicular points for High-order Bezier Curves near t=1

I have implemented code in Python that attempts to find the perpendicular points of a higher order bezier curve of order $n$ (where $n$ is around 25 control points) parametrized in standard form on $t ...
Gary Allen's user avatar
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1 answer
128 views

On implicit equations of rational quadratic Bézier curves

Rational quadratic Bézier curve with control points $\boldsymbol{B}_0 = [x_0 : y_0: w_0], \boldsymbol{B}_1 = [x_1 : y_1: w_1], \boldsymbol{B}_2 = [x_2 : y_2: w_2]$ in homogeneous coordinates of $\...
cxh007's user avatar
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Parametrization relationship between rational Cubic Bezier curve and two curves resulting from subdivision.

I have a following situation: Say I have a rational cubic Bezier curve $B_0$. I then project it into 4D, to make it non-rational, and then use De Casteljau's algorithm to subdivide it into two ...
Donatas Šimeliūnas's user avatar
2 votes
1 answer
56 views

Finding best 2D curve passing by waypoints

For a robotic application, I am looking for a mathematical tool for finding the best curve passing by or near (up to a certain radius) from waypoints as shown below. The pink circles are obstacles ...
nowox's user avatar
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1 answer
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What is a general form cubic-spline approximation of an arc?

Say I have an arc from $P_0$ to $P_3$, that has unit tangents $t_0$ and $t_3$, respectively. Let $L$ denote the distance between $A$ and $B$. If $P_0 = (0, 1)$, $P_3 = (1, 0)$, $t_0 = (1, 0)$ and $t_3 ...
Matthew's user avatar
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60 views

Restricting maximum curvature of cubic bezier curve

Is there any way to reasonably restrict control points of cubic bezier curve so it's oscilating circle will never have radius smaller than r? Bezier curve with it's ...
Crimsoon's user avatar
1 vote
1 answer
60 views

Evaluating a 2D cubic Bézier curve with interval coefficients with interval arithmetic

I would like to know how to evaluate 2D cubic Bézier curves at an interval when the Bézier coefficients themselves are intervals. If the coefficients are not intervals, evaluating a Bézier curve on an ...
Will's user avatar
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1 answer
113 views

best high order interpolation method for set of points

what is the best way to interpolate a set of points in 2d, such that there is only one parameter to indicate position on the curve (like is the case for a Bézier curve)? one thing I know is that we ...
Hassan Ali's user avatar
1 vote
1 answer
159 views

Calculating this integral along a Bezier curve in code

I'm doing some graphics code work and need to solve this integral in code: $$ \int_{0}^{1} f(P(t)) dt $$ Where $P(t)$ is a quadratic or cubic Bezier curve, and $f(p)$ ($p$ is a point) is defined as: $$...
Udit Dey's user avatar
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2 answers
308 views

Find the 2 coordinates of the 2 control points of a Bezier curve which is an arc of a circle.

I've been searching all day on this topic and I could not figure it out. Hopefully someone is able to dumb it down to my very practical level. I want to draw an arc. I know the following information: ...
FMaz008's user avatar
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4 votes
1 answer
232 views

How would a better mathematician than I complete this half-finished definition of what it means for a curve to be smooth?

Once upon a time, I was taught how to play connect the dots. Some years later, I was given pseudo-code for an algorithm which would compute a polynomial of minimum degree passing through some points. ...
Toothpick Anemone's user avatar
1 vote
1 answer
187 views

Why are Bezier curves numerically less stable for a larger number of control points?

I think the question is quite straightforward. Why are Bezier curves with more control points numerically more unstable. Can someone give me clear substantiated reason(s)? And with this the notion of ...
Math98's user avatar
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2 votes
1 answer
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Find inverse of matrix with bezier coefficients

Question: Let $\mathbf{A}$ be a square matrix of side $(p+1)$, and coefficients $$A_{ij} = \binom{p}{j} \left(1-\dfrac{i}{p}\right)^{p-j} \left(\dfrac{i}{p}\right)^{j} \label{1}\tag{1}$$ Is there an ...
Carlos Adir's user avatar
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2 votes
1 answer
444 views

Proving that Bernstein polynomials are basic B-splines

I want to prove that if we use the knot vector $t_0=\dots=t_n=0, t_{n+1}=\dots =t_{2n+1}=1$ then $N^n_i=B^n_i$ on $[0,1)$. I have the following definitions: $N^0_i=1$ on $[t_i,t_{i+1})$, $0$ ...
QED's user avatar
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1 vote
0 answers
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Placing dots on quadratic Bézier curve at distance

I need to place points on a quadratic Bézier curve at length intervals l. Found a pretty good resource at quadratic Bézier curve length and used it to calculate the ...
Igor Shmukler's user avatar
0 votes
1 answer
88 views

How do I find the optimal control points to fit a cubic Bézier curve to a known function?

I have the function $f(x) = \frac{x^3 − x^2 \sqrt{2 x^2 + 2} + 4 x} {x^2 + 2}$. I want to find the optimal control points $\{P_0, P_1, P_2, P_3\}$ for a cubic Bézier curve $B(t) = (1 − t)³ P_0 + 3(1 − ...
Lawton's user avatar
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2 answers
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Why $B_{-1}^n(t)=B_n^{n-1}(t)=0$ where $B$ are the Berstein polynomials?

i was reading a book about Bezier curves and Berstein polynomials and i have a big doubt. Why $B_{-1}^n(t)=B_n^{n-1}(t)=0$ where $B$ are the Berstein polynomials? Link of the book: http://math.aalto....
rcoder's user avatar
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1 vote
1 answer
60 views

Solving Bezier cubic derivative for t

Getting the derivative of the cubic Bezier curve: $P(t)=P_0(1-t)^3+P_13t(1-t)^2+P_23t^2(1-t)+P_3t^3$ Produces the following: $P'(t)=3(-P_0-2P_1)+6t(P_0+P_1+P_2)+3t^2(P_3-P_0)$ Assuming P'(t)=0, is ...
Darkreaper's user avatar
2 votes
0 answers
156 views

Compute signed area enclosed inside a NURBS curve

I need to compute the signed area enclosed inside a closed 2D - planar NURBS curve $\gamma(t)$. One way to accomplish this task is to use the Green Theorem. By appropriately choosing the functions of ...
aSpagno's user avatar
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1 vote
1 answer
434 views

Derivatives of a Linear and Cubic Bézier curve

Thanks to the answer here: how to calculate the value of "t" for the highest point in a quadratic bezier curve? I know that the derivative of the quadratic bézier curve of $P(t) = P_0(1−t)^...
Darkreaper's user avatar
0 votes
1 answer
142 views

Algorithm for a Bezier Curve approximation on a Cartesian Grid

Given any bezier curve, I would like to find a set of lines such that: a) the lines are all connected in series b) the start point and end point of the series of lines are the start point and end ...
jaxoncreed's user avatar
1 vote
2 answers
251 views

Aligning Bézier curve

I'm trying to calculate the tight bounding box of a Bézier curve. The curve is in 3D space and can have any amount of points. These articles are pretty much the only resources on the internet: https://...
Vaschex's user avatar
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0 votes
1 answer
188 views

cubic Bezier curve interpolation that looks like piecewise continuous function

One popular method that I came across for interpolation of a set of points is by using cubic Bezier curve segments with $C^1$ and $C^2$ continuity conditions at the junction point (or node) between ...
Hassan Ali's user avatar
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1 answer
217 views

Getting appropriate normals for 3d bezier curve, when the normals at the start and end are also known

Before I explain the rest of my post, I have tried both Frenet Frames, and Rotation Minimising Frames, and some others, but neither produces the desired result I am looking for. Consider there are two ...
tukars's user avatar
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