# 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 ...
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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 ...
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1 vote
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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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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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 ...
1 vote
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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 ...
1 vote
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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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### 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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### 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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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 ...