# Tagged Questions

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

14k views

2k 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 ...
538 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 ...
897 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 ...
543 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 ...
409 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 ...
249 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 ...
335 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 $y=f(x)$ ...
760 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 ...
1k 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 ...
736 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 ...
1k 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 ...
100 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 ...
271 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 ...
2k 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 ...
218 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 ...
2k 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 ...
805 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 ...
45 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, ...
744 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 ...
72 views

### Computing the coordinates of a Bezier Curve

I just started messing with Bezier Curves over the past couple days and I'm trying to get some of the basics down. I have this problem. ...
62 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 ...
350 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 ...
2k 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, ...
297 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 ...
324 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 ...