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

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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 ...
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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
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1answer
40 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, ...
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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} + ...
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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'...
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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
36 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 ...
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28 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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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 ...
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1answer
33 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
50 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
43 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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2answers
53 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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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!}{...
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2answers
96 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
31 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 ...
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1answer
35 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 ...
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2answers
41 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, ...
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3answers
73 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{...
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1answer
60 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 ...
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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 ...
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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.
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1answer
53 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 $C_2(x_2,y_2)$? ...
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0answers
25 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 ...
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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 ...
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2answers
82 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 ...
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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
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 $2\theta$. ...
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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 ...
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0answers
36 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(...
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2answers
101 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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1answer
35 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 ...
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1answer
28 views

How to calculate the controls of this Bézier curve?

How to calculate the controls of this curve if I know three points: start, one on the curve and the end? Here is the curve with the coordinates I know: The curve with the points I've never done this ...
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2answers
45 views

Are Bezier curves invariant under conformal mapping?

I've spent quite a bit of time on google trying to find information on whether or not Bezier curves are invariant under conformal mapping (i.e. a conformal mapping of all points on the curve is the ...
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1answer
34 views

Bézier Curve and b spline curves.

Well I am learning about curves. I have come across Bézier and Spline curves. I want to know which one should be learned first? Are their concepts independent? or I need to know about one before ...
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2answers
45 views

How to set control points for spline curves

I've written a program that calculates points on spline curves (including Hermite, Bezier, and B-splines) and plot the curve on the screen (the curve is plotted on an html canvas using javascript). ...
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0answers
61 views

Find value of $t$ at a point on a cubic Bezier curve, part 2

I would like to find the value of parameter $t$ of a cubic Bezier curve for a given point $x, y$ lying on the curve. In other words, I would like to find $t$ which, if the Bezier curve would be ...
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1answer
15 views

How to call this Bezier curve?

With Anchor point inside and with two Handle lines that with different lengths and different angles (i.e. 90 degree between two Handle line). And Handle lines of two Anchors does not cross between ...
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1answer
46 views

Convert Bézier curve to equation

How to convert for example this Bézier curve: cubic-bezier(.65,0,.65,1) (plot) to an equation like f(x) = x... ?
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1answer
53 views

Translating Equations to Algorithms

I can't understand equations. But I'm a software engineer. I think the brevity of the equation is confusing to me where a program spells it all out. Trying to translate the equation for a bezier ...
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1answer
25 views

Is cardinal $B$-spline of order $n$ really piecewise Bezier order $n$ curve?

Is cardinal $B$-spline of order $n$ really piecewise Bezier curve $n$? I think I saw this in some lecture notes, but I can't recall where.
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2answers
19 views

How can I prove in general form that the tangent at the start point of a Bézier curve goes through control point 1?

I need to prove that the tangent to the start point of any Bézier curve goes through the control point. I have proven this for specific Bézier curves but I am struggling to do it in general, thank you....
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1answer
21 views

How do I find a Bézier curve that fulfills a given width and height?

I am building a software application that works with vector graphics and I need to use Bézier curves to draw a heart shape, like this one here which I created in MS Paint: The only information ...
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1answer
35 views

When is a quadratic Bézier curve nearest the origin?

Consider a planet moving along a quadratic Bézier curve through points A B C, with $t$ = time: $\qquad \operatorname{curve}( t, A, B, C ) \equiv t' (t' A + t (2B - A)) \ + \ t (t' (2B - C) + t C ) $, ...
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1answer
107 views

Subdividing a Bézier curve into N curves

NOTE: I am only concerned with quadratic Bézier curves. So, dividing a Bézier curve into two is remarkably easy; just interpolate between start and control points by $t$, and get the end point for $t$...
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1answer
44 views

Bezier bicubic surface intersection genus

I have two bicubic Bezier surfaces that will intersect. According to this paper: http://nishitalab.org/user/nis/cdrom/cad/CAGD91geometric.pdf At the end of page 1. The general genus of intersection ...
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2answers
65 views

Smaller enclosing shape for Bézier curves

It is well known that a Bézier curve is contained within the convex hull of its control points. This is basically a consequence of the fact that the Bernstein polynomials are non-negative and sum to $...
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1answer
278 views

Calculate Gradient (Partial Derivatives) of Bezier Curve

From this page I know that a Bezier curve of degree $N$ has a derivative which is a Bezier curve of degree $N-1$, and I know how to calculate the control points of it: Derivatives of a Bezier Curve ...
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1answer
77 views

Finding the Control Point in a bezier curve

This is a basic (and probably a stupid) question, math is not my forte and I don't know much about math, in this site: http://www.ams.org/samplings/feature-column/fcarc-bezier in the bezier curves ...