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

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Bezier curvature

I'm trying to understand quadratic Bézier curves but I cannot get pass one thing. Please, what is a "curvature" and how can I calculate it? I'm asking because I found for instance this and this. I ...
2
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4answers
3k views

Can a rational Bézier curve take exactly the same shape as a part of the sine function?

I'm wondering whether a rational Bézier curve could take exactly the same shape as a part of the sine function. The best way to check this seems like this: Find a part of the sine function such that ...
1
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1answer
14 views

How to find a Bezier curve without control points?

Let's say someone created a cubic Bezier curve using software and rasterised it. However, the original equation of the Bezier curve was not noted. Since we have the image of the Bezier curve, we can ...
0
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0answers
26 views

Applying distortion to Bézier surface

I am trying to simulate the image warp effect, that is used in Adobe Photoshop. The rectangular image is warped according to a cubic Bézier surface (in 2D, all Z components are 0). Having any Bézier ...
3
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6answers
8k views

Casteljau's algorithm - practical example

I have a dataset with about 50 points (x,y) and I would like to draw a smooth curve that can pass as closer as possible on those points. I have heard about Casteljau's algorithm for splines but after ...
0
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1answer
15 views

From Bezier Curve basis to B Spline basis functions

Bezier basis functions can be determined using recursion: $B_{i,p} = (1-t)B_{i,p-1}+tB_{i-1,p}$ So for a quadratic bezier basis, we get: $1-2t+t^2$ $2t-2t^2$ $t^2$ So for a quadratic bezier ...
1
vote
1answer
2k views

Calculate intermediary control points in Cubic Bezier Curves

I need to programatically generate two-dimensional circles of various dimensions, knowing only their radius and position. The circles will be drawn by employing 4 cubic Bezier curves. How should I ...
3
votes
4answers
2k views

Find control point on piecewise quadratic Bézier curve

I need to write an OpenGL program to generate and display a piecewise quadratic Bézier curve that interpolates each set of data points: $$(0.1, 0), (0, 0), (0, 5), (0.25, 5), (0.25, 0), (5, 0), (5, 5)...
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0answers
16 views

Derivation of B-Spline basis function recursion formula

Can anyone explain the logic behind the derivation of the seemingly magical b-spline basis function recursion formula (deBoor-Cox) $N_{i,0}(u)=1 $ if $u_i\leq u < u_{i+1}$ otherwise, $=0$ $...
2
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2answers
38 views

Fast Rational Bézier Surface Evaluation Problem

I am currently writing a NURBS ray tracer. What I do is convert the NURBS into rational Bézier patches and then perform the intersection test using Newton's method. To do this fast (the ray tracer ...
11
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3answers
14k views

Arc Length of Bézier Curves

See also: answers with code on GameDev.SE How can I find out the arc length of a Bézier curve? For instance, the arc length of a linear Bézier curve is simply: $$s = \sqrt{(x_1 - x_0)^2 + (y_1 - ...
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0answers
15 views

Computing a percent down on a bezier curve when a control points position is moved

I have a line segment drawn as a percent down on a bezier curve, lets say which has 3 control points. I need to calculate the new percent position of the line segment when one of the control point is ...
0
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1answer
16 views

Compute intersection between bezier curve and a line

Is there ready analytical solution of a set of two equations describes intersection between bezier curve and line
1
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1answer
43 views

High Degree Bezier Curve For Curve Fitting

I have the feeling that I'm way out of my element here, and that maybe this question will be obvious to most of you. Nonetheless, here goes: I have an example set of 22 two-dimensional points, ...
4
votes
2answers
1k views

Find value of '$t$' at a point on a cubic Bezier curve

I have a cubic Bezier curve, and I need to divide it and create same curve between point on the original curve and the end point of the original curve. From my research, I found the DeCasteljau ...
0
votes
1answer
21 views

How to efficiently sample $y$ in intervals of $\Delta x$ in an “ascending” cubic Bézier curve?

For a cubic Bézier curve defined by control points $\boldsymbol{P_0}$, $\boldsymbol{P_1}$, $\boldsymbol{P_2}$ and $\boldsymbol{P_3}$ with the formula $\boldsymbol{B}(t) = (1 - t)^3\boldsymbol{P_0} + ...
0
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1answer
13 views

Are there any cubic bezier curve that cannot imitate by multiple quadratic bezier curve?

I want to make a line curve system with bezier curve. And I want to use only quadratic bezier curve so it can be extend and control easily, it can add control point anywhere and more intuitive But I'...
0
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1answer
26 views

What curves have a closed-form formula for projecting a point onto them in multiple dimensions?

What curves have a closed-form formula for projecting a point onto them in multiple dimensions? For example, give a simple, straight line $$ c(t) = v t $$ where $v\in\mathbb{R}^m$ and $c:\mathbb{R}\...
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1answer
53 views

B-Spline approximation deviates a lot while increasing the number of control points???

I'm dealing with a problem to approximate some data points with B-Spline. I follow the method and implemented the algorithm from this site: Curve Global Approximation. 1) The first step is to ...
21
votes
11answers
5k views

What equation produces this curve?

I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple - a gentle curve which begins and ...
0
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0answers
31 views

How to prove that Bezier(t) polynomial lies in convex hull of points (i/n,ai) for i from 1 to n

I think i should prove firstly that: Bn,$x(t)$ for t between $0$ and $1$ lies inside the convex hull of the points $(k/n, xk)$. I know only that$ k/n$ = max between $0$ and $1$ and i found that Bezier ...
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vote
2answers
43 views

How to smooth a very narrow quadratic bezier curve with a very low number of points?

I am a software engineer working on a whiteboard application for iOS. One of the features we have is a drawing tool. This tool gathers x,y coordinates and other information like the applied pressure, ...
3
votes
2answers
626 views

Cubic Bezier curve and a straight line intersection

Suppose that two ends of a cubic Bezier curve is connected by a straight line. Is there a simple way to find out whether this straight line intersects the Bezier curve (apart from the endpoints)? If ...
0
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1answer
34 views

how to generate Bezier curves from 3D mesh?

after generating 3D mesh (car chassis) By : RGB-D camera (Like : Kinect - Intel Real Sense etc ... ) and extracting feature lines on the surface of the 3D mesh. I need to generate the Bezier curves ...
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3answers
51 views

Bezier Curve Problem, finding missing control point

Given the two sets of control points: A: $(1, 2)$, $(2, 3)$, $(a, b)$, $(4, 2)$. B: $(4, 2)$, $(c, d)$, $(5, 5)$, $(6, 4)$. Find values for the control points $(a, b)$ and $(c, d)$ so that the ...
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1answer
60 views

Equivalent operations on Bezier curve points as control points?

In this question Explicit Bezier Curves: Lerping between curves same as lerping control points?, it shows that linearly interpolating between the result of evaluating two explicit bezier curves is the ...
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votes
3answers
2k views

Control points of offset bezier curve

If I have a cubic Bezier segment specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined ...
1
vote
1answer
56 views

How can I draw a Bézier Curve through a set number of points?

For high school Mathematics Pre-Specialist, I have been given the task of writing a mathematical investigation based on the following three questions: Quadratic Bezier curve enables a smooth curve to ...
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vote
2answers
56 views

Computing Bezier curve of a high order

I have a set of ten points that much be used to compute a bezier curve. As you are probably aware, computing a bezier curve of order 9 is a very strenuous activity. I need it in polynomial form. I ...
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2answers
112 views

NURBS Curves to Interpolate Points and Derivatives on a Surface of Revolution

Problem in Prose My starting point is a set of conic segments on a plane. Each of these conic segments interpolates between three points and known slopes on the two outer points. I want to find a ...
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1answer
25 views

Bezier curves expressed parametrically as products of matrices.

Express the point P(t) on the Bezier curve on the control points $P0 = (5, 3)$, $P1 = (1, 8)$, $P2 = (7, 4)$ a product of three matrices. Formula: $$A(kj) = (-1)^{k-j} \frac{k!}{j!(k-j)!} \frac{ L!}{...
0
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1answer
38 views

Addition of two B-spline curves

Suppose I have two B-splines, both with the same degree, $p$, and uniformly distributed knots, but with different numbers of knots and control points. Is it possible to sum the two splines to obtain ...
8
votes
3answers
74 views

Can different control points lead to the same Bézier curve?

A cubic Bézier curve is a polynomial $$F(u) = \sum_{i=0}^{n} \mathbf{b}_i^n P_i \;\;\;\text{ with } u \in [0,1], P_i \in \mathbb{R}^2, n=3 \text{ and } \mathbf{b}_i^n = \begin{pmatrix}n\\i\end{...
2
votes
2answers
46 views

Bézier curve approximation of a circular Arc

I would like to know how I can get the coordinates of four control points of a Bézier curve that represents the best approximation of a circular arc, knowing the coordinates of three points of the ...
1
vote
1answer
64 views

Fitting a straight line and a curve (hypocyloid) with C2/C1 conitinuity (problem at joints)

(Kinldy have a look at the link of the picture in the link) I am joining a straight line, a hypocycloid curve (in between), and a straight line again, which are joined arbitrarily. At the point of ...
0
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1answer
40 views

Bounding the difference between a function and a line connecting its endpoints by Taylor's Theorem

I am unsure how the acquire the following result of the Lemma from using Taylor's Theorem. How exactly would I go about proving this? Thank you in advance.
0
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1answer
58 views

Find the control point of quadratic Bezier curve having only the end-points

Sorry for naive question but don`t have any idea. How to explicitly find the control point $C_0(x_0,y_0)$ of quadratic Bezier curve if I have only its end-points $C_1(x_1,y_1)$ and $C_2(x_2,y_2)$? ...
0
votes
0answers
34 views

Square root of Bezier curve via deconvolution

I calculate the product of two Bezier curves via convolution as described in Sanchez-Reyes 2003. I would also like to calculate the square root of a Bezier curve (I have not seen this published ...
0
votes
1answer
39 views

Fit $n$ Bézier paths to coordinates

I have a some coordinates $(X_i, Y_i)$ and I have to fit exactly $4$ cubic Bézier-paths to them (in other words, I have to find the 4 best fitting Bézier-paths, and by best fitting I mean that the ...
9
votes
1answer
6k views

How do I find a Bezier curve that goes through a series of points?

When someone has the 4 control points P0, P1, P2, P3 of a 2D cubic Bézier curve, that person can calculate a series of hundreds of points along the curve that start from P0 at t=0 and end at P3 at t=1 ...
17
votes
6answers
11k views

Is there an explicit form for cubic Bézier curves?

(See edits at the bottom) I'm trying to use Bézier curves as an animation tool. Here's an image of what I'm talking about: Basically, the value axis can represent anything that can be animated (...
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vote
1answer
279 views

Continuity of composite Bézier curves

The composite curve S with pieces where c0 = (−1, 1), c1 = (−1, 0), c2 = (0, 0), and d0 = (0, 0), d1 = (1, 0), d2 = (2, 1). What is the order of continuity of s at (0, 0)?
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2answers
83 views

Reliable test for intersection of two Bezier curves

Is there a test which reliably decides whether two Bezier curves intersect or not? I don't need to know how many intersections there are or at what parameters they appear at. I just would like to ...
0
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1answer
23 views

Control vertices of nonparametric Bézier curve $y = 2x –2x^2$

My teacher solved this problem, but I don't know how he get that the: $$y_0-2y_1+y_2 = -2$$ $$-2y_0 + 2y_1 = 2$$ $$y_0 = 0$$ Here is the full example with solution, step by step: $$y=2x-2x^2$$ $$y=...
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1answer
38 views

Bézier curves as portions of algebraic curves

Can every Bézier curve of any degree be defined as the algebraic (polynomial) curve of which it is a part and it's endpoints? If some Bézier's (such as those of degree $n$ or greater) cannot be ...
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1answer
19 views

Simplify trigonometric expression in finding bezier control point

I'm trying to fit integral (non-rational) quadratic Bézier curves to circular arcs. $$ B(t) = (1 - t)^2 P_0 + 2 t (1 - t) P_1 + t^2 P_2 \tag{1} \label{1} $$ Let the angle of the arc be $2\theta$. ...
2
votes
3answers
49 views

Bezier curve coefficients intuition

I understand that the coefficients for a Bezier curve falls easily from its recursive definition. However, looking at the polynomial unto itself, I'm struggling to understand why we need the ...
3
votes
2answers
106 views

Smoothest function which passes through given points?

I am trying to interpolate/extrapolate on the basis of a known collection of (finitely many) points. I'm wondering if there is a way to formalize this intuitive notion: find a 'smoothest' function ...
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0answers
37 views

Power Function as a Cubic Bezier Curve

With: A Power Function $f(x)=x^n$, where $x\in[0,1]$ and $n\ge0$. A Cubic Bezier with points $P_0, P_1, P_2, P_3$ such that $P_0=(0,0)$ and $P_3=(1,1)$. The Cubic Bezier function is $B(t)=(1-t)^3P_0+3(...
0
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1answer
36 views

How to display character drawn by the Bezier curve

How can I draw/display a character based on Bezier equations? I have the plot equations: x(t)=3t-3t^2 y(t)=2-3t^2+2t^3 x(t)=3t-3t^2 y(t)=1-3t^2+2t^3 and ...