Questions tagged [bezier-curve]
Questions on Bézier curves, which are used for numerical analysis with applications in computer graphics.
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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 ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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Decomposing NURBS curve into piecewise Bezier segments
I have a question concerning a paper. In it the authors try to approximate a NURBS curve by biarcs and to do so, they first need to find a polygonal approximation of the NURBS curve. They chose a ...
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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 ...
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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 ...
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Convert B-Spline into Bezier surfaces
There is one topic regarding converting B-Spline curves into Bezier curves. (Given knots and control points),
Convert a B-Spline into Bezier curves
I wonder if there is any way to convert(or partition)...
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Speed of Bezier curve
Let's assume, that we're working on 3rd degree Bezier curves only.
$$
B(t) = p_1 \cdot (1-t)^3 + p_2 \cdot 3 \cdot (1-t)^2 \cdot t + p_3 \cdot 3 \cdot (1-t) \cdot t^2 + p_4 \cdot t^3
$$
One way to ...
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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 ...
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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 ...
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What kind of curve is used in Digital Audio Workstations for i.e. Automation?
I'm trying to recreate the curves used in automation clips of DAWs but I can't seem to find the right one. Here is an example of what i mean (specifically the first curve on the left; the right one ...
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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:
$$...
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Find the 2 coordinates of the 2 control points of a Bezier curve which is an arc of a circle.
I've been searching all day on this topic and I could not figure it out. Hopefully someone is able to dumb it down to my very practical level.
I want to draw an arc. I know the following information:
...
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How would a better mathematician than I complete this half-finished definition of what it means for a curve to be smooth?
Once upon a time, I was taught how to play connect the dots.
Some years later, I was given pseudo-code for an algorithm which would compute a polynomial of minimum degree passing through some points.
...
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Why are Bezier curves numerically less stable for a larger number of control points?
I think the question is quite straightforward. Why are Bezier curves with more control points numerically more unstable. Can someone give me clear substantiated reason(s)? And with this the notion of ...
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Find inverse of matrix with bezier coefficients
Question:
Let $\mathbf{A}$ be a square matrix of side $(p+1)$, and coefficients
$$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 ...
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Determine Bezier curves from the cubic interpolation of a data set
I'm currently working on an IT project that plots and draws line charts given data points. As of now, I'm doing simple drawings by connecting data points with straight lines (like a linear ...
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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$ ...
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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 ...
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What is the difference/relation between DMPs and Bezier curves?
I'm studying the Dynamical Movement Primitives (DMPs) for generating some trajectory from demonstration data, but I'm wondering why they are preferred to other solutions like Bezier curves (or also B-...
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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 − ...
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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....
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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 ...
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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 ...
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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)^...
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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 ...
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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://...
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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 ...
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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 ...
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Intuitive Way to Find Maximum Gradient of a Bezier Curve
I'm trying to find if any point on a bezier curve has a slope that is lesser than some predetermined angle, let's say $45^\text{o}$.
For certain cases I can see that the answer is obvious. Like ...
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Calculating the control points of a cubic Bézier curve given extrema
Is it possible to find the 2 missing control points of a Bézier curve given that:
You know in advance that it's a cubic Bézier curve
You know the endpoints
You know the coordinates of the local ...
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Cubic Bezier curve mapping to a segment of a curve that is y = sin(fx)
I've seen [1] Getting the most accurate bezier curve that plots a sine wave
and [2] Can a rational Bézier curve take exactly the same shape as a part of the sine function?
but neither actually ...
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How do I solve a quadratic bezier curve for x and y individually?
I'm looking at specific cases of quadratic bezier curves defined by three points: p1 = [0,0],
p2 = [i,m],
p3 = [d,m], where i,d, and m are positive real numbers and i < d. p1 and p3 are the start ...
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How to find $x$ by $y$ in bezier curve.
I draw the lines connecting the different nodes on the screen with the cubic bezier curve.
These lines must be suitable for interaction. So I need to know how close the user is to the line based on ...
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How to calculate curvature of a 3 dimensional Bezier Curve?
The formula for curvature of a 2D bezier curve is as follows:
$κ(t)=\frac{|B′(t),B′′(t)|}{||B′(t)||^{3}}$
The dividend is the determinant of two joined vectors (2x2 where each vector is a column), but ...
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How to scale bezier handles to preserve curvature
Given a cubic bezier path $P_0P_1 P_2 P_3$, when one handle is scaled how, to scale other handle to preserve same level of curvature at $P_0$?
I tried to find $\Delta$ of $P_1$ using a second ...
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Find configuration where two cubic Bézier curves intersect at 9 points
By Bézout's theorem, we know that two cubic Bézier curves can intersect at 9 points (not counting self-intersections). Is there any way to compute the end points and control points of two cubic Bézier ...
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Limit of argmin
Fix $t \in [0,1]$ and $n \in \mathbb{N}$. Consider, for $r \in \mathbb{N}$, the ratio $\frac{i}{n+m}$ where $i \in I=\{0,1,...,n+r\}$. How can i prove that
$$\lim_\limits{r \to \infty}\frac{i(r)}{n+r}=...
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Calculating arc length of Bezier curves by hand
I was wondering if there was a viable method to calculate arc length of a quadratic Bezier curve without using coding. If I know the points of P0, P1 and P2, would it be possible to calculate the arc ...
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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 Bézier Surface isocurves
In Variational "Design of Rational Bézier 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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