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

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8
votes
3answers
9k views

Arc Length of Bézier Curves

I posted this on gamedev.stackexchange.com earlier, but I didn't get any answers... and only a few views. EDIT: I've got two answers there (as of now), and I'm investigating them now. How can I find ...
12
votes
6answers
7k 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 ...
15
votes
5answers
2k 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 ...
5
votes
2answers
3k views

algorithm to calculate the control points of a cubic Bezier curve

I have all points where my curve pass through, but I need to get the coordinates of the control points to be able to draw the curve. How can I do to calculate this points?
3
votes
4answers
4k 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.
2
votes
2answers
1k 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 ...
2
votes
4answers
2k views

Find control point on piecewise quadratic Bézier curve

I need to write an OpenGL program to generate and display a piecewise quadratic Bézier curve that interpolates each set of data points: $$(0.1, 0), (0, 0), (0, 5), (0.25, 5), (0.25, 0), (5, 0), (5, ...
2
votes
2answers
747 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 ...
1
vote
1answer
883 views

Calculate control points of cubic bezier curve approximating a part of a circle

I'm not mathematically inclined, so please be patient with my question. Given $(x_0, y_0)$ and $(x_1, y_1)$ as the endpoints of a cubic Bezier curve. $(c_x, c_y)$ and r as the centerpoint and the ...
8
votes
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 ...
9
votes
1answer
2k views

Find points along a Bézier curve that are equal distance from one another

I'm trying to figure out a generic way of determining a series of points on a Bézier curve where all the points are the same distance from their neighboring points. By distance I mean direct distance ...
8
votes
5answers
312 views

Parabolas through three points

We can draw an infinite number of parabolas that pass through three given points $A$, $B$, $C$ (in that order). For each such parabola, we take the tangent lines at $A$ and $C$, and intersect them to ...
7
votes
0answers
874 views

How can I tell when two cubic Bézier curves intersect?

I'm working a little program that converges on vector-based approximations of raster images, inspired by Roger Alsing's genetic Mona Lisa. (I started on this after his first blog post two years ago, ...
3
votes
2answers
4k views

How elliptic arc can be represented by cubic Bézier curve?

If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bézier curve? And if yes, how can I calculate control points for that Bézier curve?
2
votes
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 ...
5
votes
3answers
492 views

Find sagitta of a cubic Bézier-described arc

I have a situation where I have an arc that was mangled (irrelevant: by c#'s GraphicsPath.AddArc() function). The original arc is guaranteed to be circular, and the new data I have describes the ...
3
votes
1answer
185 views

inflection point of cubic bezier with restrictions

Say you have this type of cubic Bézier curve: The 4 control points A,B,C,D have restrictions: A & B have the same Y-axis coordinate C & D have the same Y-axis coordinate B & C have ...
3
votes
1answer
291 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 ...
2
votes
1answer
51 views

Polynomial root finding: Bernstein vs Budan

Budan's and Vincent's theorems can be used to isolate the real roots of a real polynomial. I have read papers which compared it favorably to other root finding methods. However, roots can also be ...
2
votes
3answers
99 views

Convert quadratic bezier curve to parabola

A quadratic Bézier curve is a segment of a parabola. If the $3$ control points and the quadratic Bézier curve are known, how do you calculate the equation of the parabola (which is an $y=f(x)$ ...
2
votes
1answer
624 views

Find value of '$t$' at a point on a cubic Bezier curve

I have a cubic Bezier curve, and I need to divide it and create same curve between point on the original curve and the end point of the original curve. From my research, I found the DeCasteljau ...
2
votes
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
votes
1answer
213 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
1answer
561 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 ...
2
votes
1answer
609 views

Determining the result of Boolean shape operations on closed Bézier shapes

Given two closed shapes made up of Bézier curves (and/or straight lines), I'm looking for an efficient way of calculating the resulting shape of the following Boolean operations: union difference ...
2
votes
4answers
999 views

Changing a bezier curve by dragging a point on the curve itself rather than a control point

I'm developing an iPhone app that allows users to cut out part of an image from its background. In order to do this, they'll connect a bunch of bezier curves together to form the clipping path. Rather ...
1
vote
1answer
18 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 ...
1
vote
1answer
22 views

Explicit Bezier Curves: Lerping between curves same as lerping control points?

Let's say that you have two explicit (one dimensional) quadratic Bezier curves: $f(t) = A(1-t)^2+B(1-t)t+Ct^2$ $g(t) = D(1-t)^2+E(1-t)t+Ft^2$ Where $A, B, C, D, E, F$ are scalar constants. Then, ...
1
vote
1answer
271 views

maximum curvature of 2D Cubic Bezier

Given a 2D cubic Bezier segment defined by P0, P1, P2, P3, here's what I want: A function that takes the segment and outputs the maximum curvature without using an iterative approach. I have a ...
1
vote
2answers
113 views

Fastest way to obtain the parametric value t of a bezier curve, for a given set x coordinates.

The problem is the following: Having a bezier curve B(t) we have coordinate x from the curve, and we need to obtain the y values from it, hence we need to compute the t values. What is the fastest ...
1
vote
3answers
666 views

Formula to get a control point closest for a given point what belongs to this quadratic curve

We have a quadratic bezier curve, with control point A (red start), control point B (red end), and yellow point X what belongs to the curve and what you actually "drag" - so it should be the closest ...
1
vote
1answer
573 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 ...
1
vote
2answers
870 views

bézier to f(x) polynomial function

I've got a 2D quadratic Bézier curve which, by construction, is a f(x) function : no loops, a single solution for each defined x. Is there a common mean to convert this curve to a 3rd degree ...
0
votes
2answers
148 views

Is it possible to generate a circle with a Bezier curve?

I am designing an algorithm that generates shapes of bezier curves. Each output are control points for a single curve. In some cases, it should return a circle. Which control points does the ...
0
votes
1answer
216 views

Intersect Ray (Line) vs Quadratic Bezier Triangle

I'm trying to find the intersection between a line segment and a quadratic bezier triangle for my OpenCL real time raytracer. The main recomendations I've seen are to try subdivision, or tensor ...
0
votes
1answer
227 views

Degree elevation of weighted Bezier curve

I'm having difficulty understanding the derivation of the formula for degree elevation of a weighted Bezier curve given here. The only information that's given is to project a Bezier curve info affine ...
0
votes
1answer
642 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 ...
0
votes
2answers
1k 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, ...
0
votes
2answers
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 ...
0
votes
1answer
670 views

Finding two Bézier control points given three points

My apologies if this is asked in the wrong spot, I believe that this problem has a fairly simple solution... but it is beyond me. Given three points (A,B,C) drawn at random, how do you figure out the ...
-1
votes
1answer
442 views

Calculating points on the curve

I want to get the x and y coordinates of a curve..How can i do that... In the above image.Is it possible to calculate the intermediate points(one side) by knowing starting and ending point