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

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22
votes
11answers
4k 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
votes
0answers
17 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 ...
1
vote
2answers
34 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
438 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
votes
1answer
30 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 ...
0
votes
3answers
45 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 ...
1
vote
1answer
50 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 ...
4
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
25 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 ...
1
vote
2answers
48 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 ...
1
vote
2answers
80 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 ...
0
votes
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{ ...
2
votes
1answer
26 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 ...
0
votes
1answer
31 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
71 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 = ...
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
53 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
votes
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
votes
1answer
50 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
23 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
35 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 ...
15
votes
6answers
10k 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 ...
1
vote
1answer
255 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)?
2
votes
2answers
77 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 ...
2
votes
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 ...
0
votes
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: $$y=2x-2x^2$$ ...
1
vote
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 ...
1
vote
1answer
18 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
48 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
95 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 ...
0
votes
0answers
31 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 ...
0
votes
1answer
32 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 ...
0
votes
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 ...
3
votes
2answers
41 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 ...
1
vote
1answer
31 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 ...
0
votes
2answers
40 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). ...
7
votes
3answers
3k views

Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]

Update Alternative Statement of Problem, with New Picture Given three points $P_1$, $P_2$, and $P_3$ in the Cartesian plane, I would like to find the ellipse which passes through all three points, ...
4
votes
2answers
64 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 ...
0
votes
1answer
345 views

Reconstruct Control points in a Bézier Curve?

I have a curve that I know is a (non-periodic) Cubic Bézier Curve (because I constructed it as such). I stored each ordered pair in the curve, but not the control points. Is it mathematically ...
2
votes
2answers
1k views

Find Quadratic Bezier curve equation based on its control points

If the 3 control points of the quadratic Bézier curve are known, how do you calculate algebraically the equation of that curve (which is an y=f(x) function)? Let's say I have.. P0 (x,y) - startPoint ...
0
votes
0answers
59 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 ...
0
votes
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 ...
0
votes
1answer
43 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... ?
3
votes
1answer
49 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 ...
1
vote
1answer
21 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.
2
votes
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 ...
0
votes
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 ...
1
vote
1answer
33 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 ) $, ...