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Questions tagged [bezier-curve]

Questions on Bézier curves, curves that are frequently used in computer graphics.

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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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32 views

How to get area between two beziers? [closed]

Figure A. Let's say I have a blue path (bezier), then I have the red path (which is just a copy of blue path moved to a location (in this case just shifted down). If I join starting points and ending ...
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3answers
69 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
18 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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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
47 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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40 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
113 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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22 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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23 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
42 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
43 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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59 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
51 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
22 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 ...
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2answers
30 views

Bézier Curves: where the 3 (or 1/3) constant comes from when moving from Hermite curves?

I'm learning Bézier Curves, and I'm stuck at the reason why they use 3 (or 1/3) constant when moving from Hermite curves? Like in this, this, and this source. e.g. why t0 = 3(q1 - q0) ? or why v1 =...
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50 views

Cubic Bézier curve arc length parametrization reversal: find t given a length

I am following this paper Approximate Arc Length Parametrization, M. Walter & A. Fournier, 1996 and have succesfully implemented the direct solution, as in finding the length $s(t)$ given $t$. ...
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1answer
28 views

Calculate quad curve control point from on-curve point

I'm trying to calculate the control point of a quad curve (I know the start and end points) so that it passes through a given point. Here is an image to help you see what I'm doing: https://i.stack....
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1answer
77 views

C2 continuity of 5th Bézier curve

I'm trying to fuse multiple Bézier curves and get a C2 continuous curve. Every document or book that I read says that imposing: $B_1(1) = B_2(0)$ $B_1'(1) = B_2'(0)$ $B_1''(1) = B_2''(0)$ But this ...
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3answers
72 views

NURBS circle without all the double knots?

I've been looking at various examples of a circle parametrized as a degree-2 NURBS curve, e.g.: NURBS circle example on wikipedia Philip Schneider's "NURB Curves: A Guide for the Uninitiated" David ...
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2answers
66 views

Drawing De Boor's algorithm

Assume we have 4 control points $[c_0, c_1, c_2, c_3]$ and uniform knot sequence $[0,1,2,3]$ If we were to draw an quadratic bezier we would be forced to use only 3 of the control points and then we ...
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1answer
47 views

Drawing bezier curve from a parabola

I'm not a math guy, sorry. I read posts on the subject but couldn't find the answer to my problem (or didn't understood the answers). I'd like to get a simple answer. I know a generic parabola ...
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1answer
43 views

arc length reparameterization of a cubic Bezier, in parts

While there is no closed form solution arc length reparameterization for cubic Bezier curves, is there a set of solutions that taken together cover all possible classes of cubic Bezier curves, such ...
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17 views

Conditions for two B-Splines to represent the same curve

What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space? Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($...
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30 views

how to calculate the value of “t” for the highest point in a quadratic bezier curve?

I need to find a the value of t exactly at the point where the curve stops going up an start going down, But I have no idea how to get this value I have the 3 control points values, the blue dot on ...
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1answer
48 views

why a Bezier curve is guaranteed to lie within the convex hull of its control points?

why if Bernstein basis polynomials are non-negative ($ B_{k,n}(x) \geq 0 $) and also due to the Partition of Unity/sum up to one ($ \sum_{k=0}^n B_{k,n}(x) = 1, for\ all\ x \in [0,1] $) implies Bezier ...
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1answer
68 views

What is a basis polynomial?

Can anyone explain in a simple approach, what basis polynomials are? for example, what is Bernstein basis polynomials when we are saying that The $ n+1 $ Bernstein basis polynomials of degree $ n $, ...
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48 views

Rational bezier curves and pdfs

My goal is to write out the 2D projection of a 3D bezier curve to a pdf document. I am told that the 2D perspective (conic) projection of a 3D bezier curve is a rational bezier curve. There are ...
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30 views

Symmetric bezier curve with unsymmetric control points

I'm solving this problem about Cubic Bezier Curve which have $4$ control points $C_0,C_1,C_2,C_3$. $$ C_0 = (0,0),\: C_1 = (x_1,y_1),\: C_2=(x_2,y_2),\: C_3=(1,0) $$ where $0 < x_1 < x_2 < 1$ ...
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73 views

formula for representing a logarithmic spiral as of cubic bezier curve segments

I need to write an algorithm that approximates joined cubic bezier curve segments into a logarithmic spiral curve. How can I achieve this?
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3answers
148 views

How do I calculate the maximum velocity of a CSS Bezier curve?

I tried to do calculate if it's possible to get the top velocity of a co-ordinate point on a CSS Bezier curve. Below is my working process. Calculate the top velocity point in a bezier curve (4 ...
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Constant speed/distance on bezier curve with any number of points [duplicate]

I'm working on a graphics project and I'd need to find a way to get a constant speed/distance for a bezier curve when lerping with T (between 0 and 1). I've found a few answers, but the problem is ...
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1answer
137 views

Convert continuous Bezier curve to B-Spline

Is there an algorithm/process for converting a sequence of bezier curves into a b-spline? I've found much discussion of the reverse, but nothing for this. I'm attempting to make a spline editor in ...
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2answers
77 views

Is a B-Spline always made up of Bezier curve segments?

According to what I have read, a B-Spline curve is made up of segments, with each segment controlled by 'k' control points (where k is the order of the curve). Also, a B-Spline curve can be formed by ...
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84 views

algorithm/formula for finding corresponding control point indexes of a closed path of bezier curves

From the image above, I have 4 bezier curves that form a closed path, numbered from 0 to 11 (number of points - 1). Say the main corners $(0, 3, 6, 9)$ are Anchor Points and the in-between points $(1,...
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59 views

Calculation of bezier curve anchors to fit around points [closed]

I am working on a game that requires more math than I can handle. It's a car game. So the problem relates to handling curves. The problem: Given two points; P1 (car), P2 (exit point of corner) and ...
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Determine if a Bezier surface contains a specific Bezier curve

Let's suppose that we have the following Bezier surface $$ P(t1,t2)= \sum_{i=0}^{3} { \sum_{j=0}^{3} p_{ij} { \varphi _{i}(t1) \varphi _{j}(t2) } } $$ Is there a way to determine if a specific ...
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58 views

Convert multiple Bézier segments to a nurbs curve

I have multiple cubic Bézier curve segments which are contiguous and G1 (they are the result of the fitting of many curve samples). Now I would like to transform these Bézier segments into a single ...
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1answer
92 views

Deriving $t(x)$ for a cubic Bezier curve

I'm having some trouble deriving the function $t(x)$, given a Bezier curve with $x(t) = a(1-t)^3 + 3b(1-t)^2t + 3c(1-t)t^2+dt^3$ for its $x$ component, with further restriction that all coefficients ...
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1answer
102 views

How do I find a value (Y) given time (X) passing through a bezier-curve?

>> Visual picture of my problem << In the attached picture I have this bezier-curve (or easing curve), represented by 4 coordinates: P0 (0, 0) P1 (0.7, 0) P2 (0.3, 1) P3 (1, 1) ...
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2answers
107 views

How to effectively check whether a path is inside a series of bounding boxes?

Given a series of interconnected rectangular bounding boxes, and a path that is a described by a polynomial (or a polynomial spline), how can I check whether the path is entirely inside the bounding ...
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1answer
460 views

The advantage of B-spline compared to Bézier if the number of control points is very small

If the number of control points is n+1, and the degree of the basis function is p If n = p, B-spline is as same as Bézier curve. Suppose I have a chance to increase the number of control points say ...
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3answers
413 views

Getting consistent normals along a 3D (Bezier) curve

I'm trying to get consistent normals along a 3D Bezier curve $B(t)$, where for any point I compute the normal as: $$ \begin{align} \vec{a} &= B'(t) \\ \vec{b} &= B''(t) \\ \vec{c} &= \vec{...
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1answer
356 views

How to calculate C1- and C2-Continuity of Bezier-Curve?

I have the following example I don't understand. Here is a Bezier-Curve over [0, 1] with the following setup Now the example explains how to construct the other Bezier-Curve, which connects to ...
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1answer
42 views

How to prescribe points for Bezier curve to pass through?

Inspired by this question earlier today, I started wondering if it is possible to prescribe points (... or tangents... or higher order derivatives) to a Bezier curve and then to go backwards to solve ...
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2answers
137 views

Are higher-order Bézier curves envelopes?

I only realized from this question and the answers to it that quadratic Bézier curves are the envelopes of the lines used to compute them iteratively. That is, if a quadratic Bézier curve for points $...
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3answers
1k views

What is the equation to produce this Bezier curve?

What is the equation for the first curve in the image? I have this formula: But when I substitute values, I get an image that looks like x^2. It isn't the same shape as the image.
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1answer
161 views

How do you compute the derivative of a matrix algebra expression?

I was working along with http://jimherold.com/2012/04/20/least-squares-bezier-fit/ to see if I could understand each step of the way for fitting a Bezier curve to a set of coordinates, and I ...
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2answers
94 views

How to convert 3D Bézier curve to separate curves

Disclaimer: I am a programmer, not a mathematician, so some math terminology can be completely wrong. Excuse me for that. I will try to illustrate my question with pictures. I am trying to convert a ...
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72 views

Approximation with Bézier curves

I have just completed basic research on Bézier curves, mainly focused on the characteristics of Bernstein polynomials and the de Casteljau algorithm. Now I am supposed to approximate $\sin(x)$ and $\...