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

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2
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5answers
2k views

Is it possible to build a circle with quadratic Bézier curves?

i'm searching for a curve type with a minimum of functionality and maximum of usability. I run into quadratic Bézier curves and i wonder, if its possible to draw a circle with it.
0
votes
1answer
52 views

algorithm for Bezier curve with eleven control points

I would like to know the algorithm/ polynomial equation for a Bezier curve with eleven control points. Thanks in advance.
0
votes
1answer
321 views

Intersection of cubic bezier curve and circle

Let $B$ be a cubic Bézier curve with control points $P_0,P_1,P_2,P_3 \in \mathbb{R}^2$, and $C$ be a circle with center $P_C$ and radius $r$. How can I find all intersections of $B$ and $C$? Is ...
1
vote
2answers
146 views

What's the best way to calculate all of the points for a curve given only a few points?

I've been reading up on curves, polynomials, splines, knots, etc., and I could definitely use some help. (I'm writing open source code, if that makes a difference.) Given two end points and any ...
0
votes
1answer
80 views

Link points on a bicubic bezier patch

A bicubic bezier patch is defined by 16 control points. Given two points both lying on the patch boundaries, I think that if you link the two points you will end up with a cubic bezier curve in 3D. Is ...
1
vote
0answers
28 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
931 views

Convert a B-Spline into Bezier curves

I have a B-Spline curve. I have all the knots, and the x,y coordinates of the Control Points. I need to convert the B-Spline curve into Bezier curves. My end goal is to be able to draw the shape on ...
1
vote
1answer
151 views

What is the general formula for NURBS curves?

Give me the general mathematical formula for NURBS curves, with special cases (B-spline and Bézier curves)
0
votes
1answer
92 views

Solving a Cubic Function

Can someone help me find my solution(s) to this cubic equation? x = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3 Where: ...
1
vote
1answer
61 views

Bezier curve, X position of reference points outside [Xstart, Xend]

Not quite sure whether this belongs here or on stackoverflow, but considering it's about the formula itself and not the implementation I'm placing it here. I'm required to implement bezier curves in ...
1
vote
1answer
611 views

Finding a point on a bezier surface using De Casteljau's algorithm

Given $16$ control points $(x,y,z)$ of a bicubic bezier patch, how do I use De Casteljau's algorithm to generate a point $(s,t) = (0.5, 0.2)$ on the surface? As far as I understand, this kind of ...
3
votes
1answer
199 views

Curve through four points — simple algebra??

The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to ...
0
votes
2answers
512 views

Forcing Bezier Interpolation

I found this very informative site that discusses forcing bezier interpolation and the site gives formulae for calculating the control points so that the curve goes through a set of four points, y0, ...
1
vote
0answers
33 views

Equation for bezier curve [duplicate]

I have a cubic bezier curve ; whose 1st anchor-point is (a,b) 1st control-point is (c,d) , 2nd control point is (e,f) and 2nd anchor-point is ( g,h ); Now I want an equation in x and y format; so ...
0
votes
1answer
943 views

Is it possible to convert a B-Spline into a Bezier curve?

If so, do I lose any feature of the curve?
1
vote
0answers
313 views

Relation between Hermite interpolation and Bezier curve

I will really appreciate if someone can explain me the relation between Hermite interpolation and Bezier Curve. For example, $p(0)=1,p(3)=2,p'(0)=1,p'(3)=1$, how do we do the Hermite interpolation? ...
1
vote
0answers
57 views

I came across a paradoxical situation when applying Casteljau algorithm

The example is like this: Given 3 points $p(0)=2,p(1)=1,p(3)=1$.The question asks us to apply the Casteljau algorithm to evaluate the Bezier curve b(u) for the given Bezier polygon at $u=2$. I did the ...
1
vote
1answer
237 views

Relation between Bezier Curve, Bernstein polynomial and control polygon

I am currently learning numerical method and I am somehow confused about the Bezier Curve, Bernstein polynomial and control polygon. For example, if we have a curve $p(u)=-12u^2+12u+1$,and we want to ...
3
votes
1answer
231 views

Understanding the Spiro Spline

My name's Wray. This is my first time here. Firstly, I like curves. I've been keeping a pet project for a long time that would implement a delightful new curve-interpolation algorithm named the Spiro ...
2
votes
1answer
408 views

Intersect a line with a bicubic Bezier Surface Patch.

This question mentions Bezier surfaces, but doesn't go into any detail. How do you going about finding the intersection between a line, $E_{pos} + E_{dir}*t$ and a Bezier surface patch, $P = $ ...
1
vote
1answer
968 views

Length of bezier curve with Simpson's rule

I wish to approximate the length of a Bézier curve in a 2D plane. I could do this iteratively – and may – but I want to experiment using Simpson's rule. I am not sure how to set this up and could use ...
0
votes
1answer
709 views

Convert segment of parabola to quadratic bezier curve

How do I convert a segment of parabola to a cubic Bezier curve? The parabola segment is given as a polynomial with two x values for the edges. My target is to convert a quadratic piecewise ...
1
vote
1answer
90 views

How to combine bezier curves to a surface?

My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of bezier curves of different orders. But this algorithm is defined in a two dimensional space for ...
2
votes
2answers
96 views

Duplicate quadratic Bézier curve with new start point?

I have Bézier curve as shown by the wikipedia gif here: I would like to create a new curve that is a segment of the old one. For example, in this gif (from the same article): .. if I wanted B to ...
2
votes
1answer
900 views

Approximate arc length of cubic bezier curve?

I want to divide a cubic bezier curve, with 4 points, start, end and 2 control points, into segments that are not bigger then a certain distance. So, am looking for a computationally quick way to ...
0
votes
1answer
636 views

Find coordinates of equidistant points in Bezier curve

I have to find points (say 10 points) in Bezier curve with 2 control points such that they are at equidistant positions in the curve. Currently I am using the following formula which gives me points ...
1
vote
1answer
369 views

Bezier Curves and Acceleration

So I'm working on a program that graphs a bezier curve by manipulating the control points. This curve represents the velocity of something over time; I also want the option manipulate it all in terms ...
2
votes
1answer
158 views

Equation for subsection of Bezier curve

Say I have a cubic Bezier curve, with a starting point s, an ending point e and control points ...
2
votes
2answers
448 views

Offseting a Bezier curve

I searched this site and I read that in general it is not possible to calculate offset of a Bezier curve. But is it possible to calculate the offset in some special cases? Obviously, if the Bezier ...
8
votes
1answer
3k 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 ...
0
votes
1answer
49 views

What $t$ coefficient should I choose for a Bezier curve?

For a cubic Bezier curve, I have this formula: $$\mathrm{B}(t)=\mathrm{P}_0(1-t)^3+3\mathrm{P}_1t(1-t)^2+3\mathrm{P}_2t^2(1-t)+\mathrm{P}_3t^3,\ t\in[0,1]$$ Now about $t$ I only know that is ...
0
votes
1answer
213 views

B-spline curve fitting with conditions on derivatives

I have 5 data points. I'm trying to make a b-spline that passes through these points. At each data point I also have a derivative. The b-spline must meet this condition. Anyone that has an idea of how ...
15
votes
5answers
998 views

Polynomial approximation of circle or ellipse

Trying again, with a somewhat simpler sounding question, since my previous one (Generalizations of equi-oscillation criterion) got zero response: Let $F:[0,1] \to R^2$ be a parametric polynomial ...
2
votes
1answer
354 views

Bezier unit tangent

What is the explicit formula for a unit tangent vector to a Bezier curve? I.e. if the formula for a Bezier curve is $\mathbf{B}(t) = \sum_{i=0}^n\binom{n}i(1-t)^{n-i}t^i\mathbf{P}_i$, what is its unit ...
1
vote
3answers
303 views

Understanding cubic bezier curve

I do not have experience of Mathematics past a-level, so please excuse the incorrect terminology. I am trying to better understand the fundamentals of how a cubic bezier curve works. I am going by ...
1
vote
1answer
152 views

Approximating an algebraic curve using cubic bezier splines

Suppose I have an algebraic curve in its implicit form, i.e. described as the set of points $(x,y)$ where some polynomial $P(x,y)$ becomes zero. All of this is in the real Euclidean (or with minor ...
3
votes
2answers
526 views

How to calculate the square area under a Bezier curve?

I did search at Google and this website before asking this question, so sorry if this somehow has already been answered and I didn't notice. BTW I'm a humanities scholar and not a trained ...
1
vote
5answers
727 views

Rough y(x) approximation for simplified Cubic Bezier curve

I need to get a very rough (and fast) $y(x)$ approximation of a simplified Cubic Bezier curve to use in my animation code, where there's only one control variable: $$ P_0 = (0, 0)\\ P_1 = (0, 0)\\ ...
1
vote
1answer
354 views

Finding the mid-point of a B-spline curve.

As part of my computer graphics, we've learnt about Bezier and B-spline curves. I'm going over some questions in preparation for my exam and I've come across a past exam question. The question is "A ...
1
vote
0answers
126 views

Approximating Bezier curves

I would like to approximate one cubic Bezier curve with two quadratic ones. In other words, I would like to split a cubic curve at some parameter t and approximate ...
1
vote
1answer
1k 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 ...
1
vote
0answers
27 views

Is it possible to change a piece of curve's interpolation type of a B-Spline via modifying knots?

I am going to implement a curve editor based on (cubic) B-Spline. Sometime the user may change a piece of curve's interpolation type, that is, use linear/constant value between two consecutive ...
1
vote
1answer
206 views

An almost straight curve with infinite curvature?

I played around with computing the curvature of some curves, and found this weird example that is driving me nuts. Consider the following (Bézier) curve (on a plane, the first point is $[-1,0]$): ...
2
votes
2answers
210 views

Can an involute gear profile be modeled with a Bézier curve?

In the context of a game, I want to draw gears. The most common curves available on the platforms I'm using are third degree Bézier curves. Is there an exact representation of the involute gear ...
1
vote
0answers
70 views

How can the equation of a Bézier curve be transformed from a Bézier basis function to a bivariate function?

Several nights ago, I was researching the problem of identifying self-intersections in arbitrary curves, particularly Bézier curves. (The reason being is that I want to write a program that inserts ...
5
votes
1answer
2k 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, ...
0
votes
1answer
1k views

Cubic Bezier Curves - Calculate Y for any given X [duplicate]

Possible Duplicate: Is there an explicit form for cubic Bézier curves? I want to calculate Y for any given X of a bezier to help me chart a graph. X represents time and Y represents ...
2
votes
3answers
289 views

How can I find out 2 unknowns in a cubic equation?

I need to give a bit of a background first, so please bare with me. I have a set of values that represent servo motor position values. By default I end up with a large set of values and I'd like to ...
0
votes
1answer
395 views

Function to represent curve between two 2D points

I need a (simplest) function that interpolates values in range from predefined point $A$ to $B$ with rules: it must be smooth curve direction near $B$ must be the same as predefined $D$ vector ...
0
votes
1answer
268 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 ...