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

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1answer
81 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 ...
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1answer
934 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 ...
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1answer
247 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 ...
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2answers
949 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, ...
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0answers
39 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 ...
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1answer
1k views

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

If so, do I lose any feature of the curve?
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0answers
479 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? ...
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0answers
61 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 ...
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1answer
344 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 ...
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1answer
394 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 ...
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1answer
684 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 = $ ...
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1answer
2k 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 ...
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1answer
1k 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 ...
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1answer
120 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 ...
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2answers
116 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 ...
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1answer
1k 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 ...
2
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1answer
1k 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 ...
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1answer
592 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 ...
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1answer
202 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 ...
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2answers
662 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 ...
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1answer
4k 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 ...
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1answer
50 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 ...
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1answer
249 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 ...
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5answers
1k 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 ...
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1answer
491 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 ...
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3answers
382 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 ...
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1answer
170 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 ...
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2answers
750 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 ...
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5answers
902 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)\\ ...
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1answer
536 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 ...
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0answers
138 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 ...
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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 ...
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0answers
30 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 ...
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1answer
256 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 $[-1,0]$): ...
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2answers
318 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 ...
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0answers
73 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 ...
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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, ...
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1answer
2k 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 ...
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3answers
372 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 ...
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1answer
509 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 ...
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1answer
304 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 ...
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1answer
995 views

Bézier approximation of archimedes spiral?

As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic ...
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0answers
94 views

How can I calculate all possible Bézier handle points in order to make the curve to a given length?

Given two anchor points and a handle point of a cubic Bézier curve, how can I calculate the other handle point in order to make the curve length to a fixed value? What kind of orbit will it be? ...
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1answer
520 views

Finding point of inflection on a Bézier Curve

I need to determine the first point of inflection on a Bézier curve, if it exists, for a computer graphics application. My original idea was to iteratively walk the curve, evaluating 2nd derivatives ...
2
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1answer
210 views

Motion on a parametric surface

Please excuse what will surely turn into a long rambling question, full of incorrect terminology. I'm trying to figure out the mathematics of moving on a parametric surface - that is, for some ...
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1answer
741 views

Finding Y given X on a Cubic Bezier Curve?

I just asked this in the Computing sections but they sent me here: "So I've been looking around for some sort of method to allow me to find the Y-coordinate on a Cubic Bezier Curve, given an ...
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1answer
184 views

How can a Bézier curve be periodic?

As I know it, a periodic function is a function that repeats its values in regular intervals or period. However Bézier curves can also be periodic which means closed as opposed to non-periodic which ...
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1answer
3k views

How to derive the equation for a bézier curve

So, I remember a while back there was a maths competition and we were given a curve that we needed to write an equation for. I just skipped the question since I didn't even know where to begin. I ...
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1answer
654 views

A function that can smooth out line joining three points

I want to know if there is a function with which we can smooth out line joining two( or more )points . I've read we can do it with Quad and cubic splines or something but I am not clear with them. i ...
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2answers
426 views

Shifting a quadratic Bézier curve

I have a quadratic Bézier curve. I want to make a path that runs along that curve, whose width is 'w'. (So at any point in time along the exact center of that path, you will be also on the original ...