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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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42 views

Is there a relationship between the length of the inner and outer edges of a curve with a given width?

If you have an arbitrary line, say a bezier curve, that has a width, is there a relationship between the length of the curve at the inner and outer edges? Given the width of the curve at the ...
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0answers
12 views

Finding the length of control points to obtain a specific curve length

I want to begin by letting you know that my math skills don't go far beyond basic Algebra. I can understand most concepts after a walk-through, but this one is, so far, beyond my capabilities. What ...
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2answers
34 views

Sample points between two Bezier curves

I am looking for a way to randomly pick points between two Bezier curves (red and greenhttps://drive.google.com/file/d/1aRIOFf6-zzInadZg3yusyAroXGsAGz4P/view?usp=sharing). I also have a set of points ...
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13 views

balistic: aiming at a point on a bezier curve, given velocity of the bullet and velocity of the moving target

Given a moving point E with know velocity and no acceleration moving on a Bezier spline composed of a set of bezier path, joined continuously, which are approximated by line segments. a static point ...
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1answer
33 views

How to manipulate the $x$-Axis of a Cubic-Bézier-Curve?

I have a Cubic Bézier-Curve a la: $$ f(x) = A (1-x)^3 + 3 B (1-x)^2 x + 3 C (1-x) x^2 +D x^3 $$ with my control constant A,B,C,D which are just constants. x ist running from 0 to 1. What I ...
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1answer
20 views

Find continuous curvature approximation for going from a straight line into a half circle

I have the problem that I want to steer a 4 wheel robot. It should move on a straight line, and for turning to go on a circle with a defined radius, as in this picture: The problem I have is that if ...
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0answers
34 views

Adjust a one dimensional Bezier-Curve?

I really need to adjust my one dimensional Bezier-Curve, but I don't know how. I got this equation: $$ y = A(1-x)^3 + 3B(1-x)^2x + 3C(1-x)x^2 + Dx^3 $$ with the constants A,B,C,D which represent ...
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10 views

Why Are Cubic Bezier Curves Used In Animation Transitions On The Web?

CSS allows defining animations using a timing function along a cubic bezier curve. In this case the curve is defined by four control points P0, P1, P2, and P3, where P0 and P3 are fixed as [0,0] and [...
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3answers
38 views

Cubic Bezier curve - how to make height always the same as a given vertical line

I have a cubic bezier curve that is always drawn from a source to a destination. but its not always drawn on a horizontal line. anyway ideally it looks like this: but the PEEK of the curve should ...
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0answers
46 views

Is a generalization of Bézier curves to 3 dimensions possible?

Curiosity of the application of groebner bases to problems involving finding the envelope of a family of curves drove me to verify the following: Given a quadratic Bézier curve with control points $(...
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14 views

How to sketch Bezier Curves

I am reviewing for a numerical analysis exam, and I came across a question on drawing Bezier curves. "Draw on 3 separate figures planar Bezier curves $$ c(t)=\sum_{k=0}^3p_k B_{3,k}(t)\hspace{5mm} t\...
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13 views

Determining closest (smoothed) point on a spline

I have a spline defined by a list of Vector3. I have an object that roughly follows the path defined by the spline [but drifts to either side], and I have a goal that precedes the object by n points. ...
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1answer
38 views

Subdividing a Coons Patch

Given a Coons Patch, how could I subdivide the patch to create two seperate patches? So far I've been able to subdivide the effected bezier curves using the Casteljau algorithm but I am unable to ...
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1answer
55 views

Given a Length, Start Point, and End Point, how can I determine the Control Point of a Quadratic Bezier Curve? [closed]

I have a start point, an end point, and a length of the curve. How do I determine the control point? In my scenario, the control point most be equidistant between both end points. I'm aware that ...
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1answer
17 views

Finding a point in a Bezier closed path

Consider a close path given by a set of Bezier points as 0,0 1,1 1,1 0,1 0,1 -1,-1 -1,-1 0,0 How can we check if a given point is within the closed path or ...
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1answer
20 views

Visualizing directed edges without crossing

I am given a directed graph with set of nodes with fixed positions (Map of cities) and want to add connections to them, so that I have as few crossings as possible. For simplicity, let's consider only ...
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2answers
576 views

Difference between bezier segment and b-spline

i am currently learning about bezier curves and splines in computergraphics. Where is the difference between a b-spline curve and a curve that consists bezier curves as segments. I have read in a lot ...
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0answers
25 views

Calculating $\max\{\min\{||B(t)-p_k||, t \in [0,1]\}, p_k \in p\}$ or at least bounding it

I'm trying to calculate the maximum distance from a control point $p_m = \begin{bmatrix}a_m\\b_m\end{bmatrix}$ defining a Bezier curve $B(t)$ to the closest point of the Bezier curve. For $m=0$ and $m=...
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1answer
40 views

Proof that a point in a triangle is inside a quadratic Bezier curve.

Given 3 control points, there is a theorem that states that for a point to be inside the region enclosed by the curve and the triangle, the baryxentric coordinates of the point must obey: $(\frac{s}{...
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41 views

A rank puzzle, or finding the implicit form of a Bézier curve without resultants

I've recently completely reworked the code in my Kinross library that finds the intersection of Bézier curves. I've come a long way from first developing my own method and asking for better ones, then ...
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1answer
61 views

Explain the Tiller and Hanson Bezier Curve Offset Algorithm

I am having great trouble finding a resource that adequately explains the Tiller and Hanson Bezier Curve Offset algorithm. I know this question has been asked, I have read the answers to to the very ...
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12 views

Simpler function that's similar to 2D cubic Bezier spline (x-monotonous)

2D cubic bezier curve can be expressed as a pair of cubic function of additional parameter $t$: $$(b3x(t), b3y(t))$$ If $b2x(t)$ is monotonous, then the 2D bezier curve is a function of $x$: $$b3(x) =...
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1answer
38 views

get 2nd and 3rd control points of a Cubic Bezier Curve fit in a rectangle

I would like to fit a Cubic Bezier Curve in a rectangle, and wondering how to get the 2nd and 3rd control points's Y value, illustrated below: (Sorry the rectangle is bit distorted.) Basically, ...
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2answers
44 views

Model Bézier curve of N control points with more Bézier curve of n-1 points.

Given a Bézier curve defined by N control points. Is it possible to model that curve using a finite amount of Bézier points with less than N control points? eg. Model a 4 control point Bézier with 2, ...
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1answer
54 views

Can the $~t~$ of a quadratic Bezier curve be found given a $~x~$ or $~y~$ (using the quadratic formula)?

Given an $y$-coordinate and the three control points of a quadratic Bezier curve, can you calculate $~t~$ in the following way ?: $ y_{p0} - y + 2(y_{p1} - y_{p0})t + (y_{p0} - 2y_{p1} + y_{p2})t^2 = ...
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1answer
76 views

Cubic Bézier radius of curvature calculation

I modelled a racetrack using cubic Bézier curves. I understand that the representation is parametric: x and y vary with t. I have successfully created the cartesian coordinates thanks to this: https://...
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2answers
31 views

How to prove axis of symmetry for Bezier-Bernstein curve?

I have a planar bezier curve of degree $\ n=3 $ with the control points: $\ p_{o}=[0, 0]^T $ $\ p_{1}=[1, 1]^T $ $\ p_{2}=[2, 1]^T $ $\ p_{3}=[3, 0]^T $ Suppose I have the bernstein polynomials. ...
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1answer
25 views

How to prove if all points of Bezier-Bernstein belong to the same plane?

I have a deggree $\ n=3 $ Bezier - Bernstein curve with the following control points: $\ p_{o} = [0, 0, 1]^T $ $\ p_{1} = [1,2, 2]^T $ $\ p_{2} = [2,2,3]^T $ $\ p_{3}=[3,4,4]^T $ I have found the ...
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1answer
26 views

Algorithm to Identify if B-spline Surface is a Surface of Revolution

I'm trying to reach an algorithm to determine if a given a general B-spline/Bezier surface (could be rational) is a surface of revolution around the Z axis. I tried to solve it analytically and ...
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31 views

Confusion on the formula of Rational Bezier curve in Duncan's book

Duncan Marsh's book Applied Geometry for Computer Graphics and CAD stated I'm really confused about this formula. First of all, why do we replace $w_i\mathbf{b_i}$ with $\mathbf{b_i}$ when $w_i=0$? ...
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1answer
52 views

How do I produce a Bézier curve with 4 points and without just using a website?

I have these for points, (1,-2), (4,3), (12,3) and (15,-3), and I was wondering how do I make a model out of this, and how would this relate to polynomials and Pascal's triangle? I have this formula: ...
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1answer
43 views

approximating a b-spline with a single bezier curve of any degree

suppose we have b-spline composed of several bezier curves with known control points and nodes and is not necessarily smooth. Is there a way to create a single bezier curve of any degree that will ...
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1answer
25 views

Let $P(t)$ be a Bézier curve of degree $N$ with control points $P_0, \ldots, P_N$ . Show that $P'(0) = N(P_1−P_0)$ and $P'(1) = N(P_N − P_{N−1})$

I am trying to learn more about Bézier Curves and I am following an old professor's course online, and I can't seem to see how to go about this problem. I understand the format of finding $x(t)$ and $...
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2answers
76 views

Convert section of an equation to a cubic Bezier-curve

I have the equation $$f(x) = \frac{0.25x}{1.25 - x}$$ that I would like to turn into a cubic Bezier-curve in the window $[0, 1]$. I have tried to find an answer but I can only find sources on how to ...
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0answers
25 views

Creating a curve at a fixed distance from a cubic Bezier curve

Let $P(t)$ be a cubic Bezier curve. I want to create a second curve $P_r(t)$, in which each point is a fixed distance $r$ from a point in $P$. In other words, add the vector of magnitude $r$ that is ...
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1answer
64 views

calculate cubic equation from two points and two slopes, variably [closed]

I'm trying to variably calculate the cubic equation between two points and the slopes at said points. (Ultimately to get an approximation of a bezier spline where i can calculate the y value from a ...
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1answer
54 views

quadratic bezier to parabola matrix equation

I'm trying to follow the matrix equation solution presented here: Convert quadratic bezier curve to parabola by @robjohn I'm assuming his solution can be used for any quadratic coordinates. My goal ...
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3answers
120 views

Bezier curve, constrained shortest path and minimum time: is the optimal curve always of minimal degree?

In the unconstrained case, the shortest path between two points in arbitrary dimension is given by a straight line, which is also a Bezier curve of degree one, with two control points corresponding to ...
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1answer
20 views

Is the sum of cubic Bezier curve segment arc lengths the arc length of the whole spline?

I have created a spline made of several cubic Bezier curve segments. I have been able to calculate the arc length of each of these cubic segments. I would like to know in the sum of all the individual ...
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0answers
15 views

Bézier curve for some nodes

How do I write the Bézier curve with the nodes $(x_1,y_1),(x_2,y_2)$and control points $(x_3,y_3),(x_4,y_4)$? I know how to write it as parametric form with $x(t)$ and $y(t)$, but I have no idea how ...
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3answers
62 views

How can I create a seam for an ellipse?

I want to create a seam for an ellipse. (Mathematically I think this means that there are constant-length normal lines between the ellipse and the curve that creates the seam, but I'm not 100% sure ...
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0answers
86 views

Algorithm to calculate the positions of bezier curve control point handles to transform from circle to straight line

From the image linked below (in A, I don't have enough points to add it directly) I have approximated two bezier cubic curve segments to a circle. I know how to move the anchor points AP1 and AP3 ...
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1answer
147 views

Bounding rectangle of ellipse

In a stale branch of the code base I'm working on, I found an interesting algorithm to go from the SVG definition of an elliptical arc, to Bézier curves. This is a question about a small but crucial ...
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0answers
41 views

Bézier with fewest inflection points

I have an algorithm to automatically generate control points for smooth Bezier curves which pass through given nodes. It's inspired by what Inkscape appears to do when you make nodes "auto-smooth". In ...
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0answers
31 views

How to find shortest bezier curve with N control points passing though N target points?

I'm working on a pattern making software (clothing). I have a set of N target points and I need to find the shortest bezier curve with N target points that goes through all the N target points. My ...
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1answer
388 views

How to perfectly split a bezier curve into two curves of unequal length

I couldn't find a title that didn't seem duplicate, but my question is very different from every other I could find on this site. I have a Bezier curve, defined by 4 points. Two of those points (...
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1answer
121 views

The good pseudo-half-circle-like curves that quadratic bezier curves can create

Since quadratic bezier curves can only approximate circles, I'm wondering what they can actually do well. That is, the sort of half-circle-like shapes that quadratic curves can make. For instance, in ...
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2answers
205 views

Understanding Bezier Curves vs. Circular/Elliptical/Other Arcs

From what I've read, you cannot construct an elliptical or circular arc with a single bezier curve (though I read maybe you can if the arc is less than 1/4 of a circle or something small like that, or ...
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1answer
60 views

General way of modeling Bézier curves and circles

So it turns out that you can't totally model circles with Bézier curves: How to create circle with Bézier curves? I'm wondering if there is a mathematical system or construction that unifies circles,...
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1answer
76 views

Relation between a Bezier curve and B-Spline curve

While the ideas behind Bezier curves are rather straight forward, I'm really struggling trying to understand B-Splines. I really researched quite a lot about it and still can't figure it out. I ...