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

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2
votes
1answer
339 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
679 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
563 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
84 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
81 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
707 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
467 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
273 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
138 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
386 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 ...
7
votes
1answer
2k 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
47 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
194 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 ...
14
votes
5answers
819 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
320 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
270 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
132 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
459 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
664 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
290 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
119 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
980 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
26 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
188 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
184 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
67 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
1k 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
781 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
258 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
351 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
249 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 ...
3
votes
1answer
685 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 ...
1
vote
0answers
77 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? ...
0
votes
0answers
42 views

Convert a quadratic bezier curve to an f(x) function [duplicate]

Possible Duplicate: Is there an explicit form for cubic Bézier curves? Specifically, on http://matthewlein.com/ceaser/, the ease-out curve. How would I go about getting the f(x) for ...
1
vote
1answer
293 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
votes
0answers
167 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 ...
3
votes
1answer
398 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 ...
0
votes
1answer
120 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 ...
3
votes
1answer
2k 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 ...
0
votes
1answer
471 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 ...
1
vote
2answers
354 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 ...
1
vote
1answer
467 views

How to merge two poly Bézier curves?

I am battling with a programming problem. Given two or more overlapping and interacting Bézier polygons, how can I perform merge (union) operations on a list of input Bézier polygons so as to produce ...
0
votes
1answer
483 views

Newton's Method, and approximating parameters for Bézier curves.

I've been wanting, for quite a while now, to polish up some source code I wrote for approximating arbitrary Bézier curves to given series of points. I managed to accomplish quite a bit, but I hit a ...
0
votes
1answer
159 views

How to fly a curve from one heading to another using only roll and pitch.

I have 3 perpendicular vectors representing an object in 3d space... Heading Right Up ...and I would like to be able to 'fly' this object so that it ends up at a ...
1
vote
1answer
224 views

Detect “Kinks” in Parallel Lines to Bezier Curves [x-post]

I feel like this is just as much a mathematics question as it is a programming question so I figured it couldn't hurt to cross post my question here. Original Post: ...
0
votes
0answers
483 views

Subdividing a cubic Bezier curve at an arbitrary point

I want to subdivide a cubic Bezier curve at a point which may not be the curve's midpoint. I'm using the plain and slow incremental method to plot the curve from its parametric equations.
4
votes
4answers
1k views

Why is the Convex Hull property (e..g of Bézier curves) so important?

Recently I read some course notes and articles on Bézier curves. They all sum up the properties of Bézier curves, like the partition-of-unity property of the basis functions (Bernstein polynomials), ...
2
votes
2answers
833 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
291 views

Application to draw/edit/export 2d bezier-curve over image

I am searching for an application (window preferred but linux, mac are possible too). The application would be used to redraw a bezier-curve over an hand-drawn image. I have multiple images with 2 ...
1
vote
0answers
74 views

How to approximate a trigonometric curve by Bezier curves?

Let me ask how to approximate a trigonometric curve by Bezier curves? Is there any known algorithm? Thank you in advance.