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

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1answer
19 views

How to plot a smooth curve function from given points

I have several points, how can I plot a smooth curve that pass through those points? Is there any function that I can create or formula that I can use to get all points in the curve? I have read ...
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1answer
17 views

Calculating control points of Cubic Bézier curve

I'm trying to draw different arcs with a Cubic Bezier curve and my problem is that after reading different blogs that explain it, I can draw only a 90degree arc using this article. Is it possible if I ...
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1answer
286 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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2answers
25 views

maximum of a 5th order bezier curve with restrictions

Say you have a Bézier Curve of the 5th order with restrictions on the Control points: P0 & P1 are on a horizontal line P2 & P3 are on a horizontal line P4 & P5 are on a horizontal line ...
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3answers
38 views

What's the technical term for “ternary interpolation”?

While researching how to render 2D bezier curves given the control points, I found a simple formula and the resource where I found this marked this iterative process as a ternary interpolation and ...
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2answers
25 views

How can I limit the amount of curvature of a bezier curve?

I've been creating bezier curves in a program according to a user clicking where the endpoints ought to be with success. Now, I wonder if there is a way to restrict the shapes of the beziers such that ...
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0answers
13 views

n-th degree Bezier curve-bernstein polynomial

I want to make a code that will draw n-th degree Bezier curve which would be calculated through Bernstein polynomials.My problem is not related to code writing,but its math kind.Reason I have ...
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1answer
50 views

Calculating originally arc approximated by cubic bezier curve

I have an cubic bezier curve, which is representing an arc by an approximation. The approximation was calculated with the kappa constant: $$ \\k = \frac43*(\sqrt{2}-1) $$ This means, that the ...
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1answer
20 views

Continuity of composite Bezier curves

The composite curve S with pieces where c0 = (−1, 1), c1 = (−1, 0), c2 = (0, 0), and d0 = (0, 0), d1 = (1, 0), d2 = (2, 1). What is the order of continuity of s at (0, 0)?
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2answers
56 views

How to find the N control point of a bezier curve with N+1 points on the curve

I have a the set of points my curve has to pass through, 2 of those are the start and end points. I'm looking for a way to find the control points of my bezier curve (mostly quadratic and cubic) by ...
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2answers
41 views

Number of Curvature Maxima of a 2D Cubic Bezier curve

I am trying to prove that a standard cubic Bezier curve can only have at most 2 curvature maxima over $t \in [0,1]$. Assuming that no 3 adjacent control points are colinear, the curvature will either ...
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1answer
64 views

B-spline: compute control points given equations and knots?

Assuming a cubic or higher-order 2-D B-spline: if all piecewise polynomial equations for the final spline (and thus the knot vector as well) are already known, is there a relatively "streamlined" ...
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4answers
803 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 ...
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1answer
26 views

Understanding de Casteljau algorithm

I have a problem understanding the de Casteljau algorithm. For example, let these be the given Beziér nodes \begin{align*} d_0 = (0,2)^T && d_1 = (0.5,1)^T && d_2=(1,3)^T \end{align*} ...
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1answer
18 views

Calculating bezier path when target is moving. (And calculate total travel time)

As a response to another question I asked here (2d spaceship movement + eta) someone suggested to use a bezier curve. This is not answering the question, but it can provide the effect I am looking ...
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0answers
67 views

Real roots of a quintic polynomial with constraints

This is a question on the edge of math and programming. I pondered about the best way to state the problem: should I provide context, or get straight to the point of the question? Given various ...
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1answer
55 views

How to clip Bézier curves using Casteljau's algorithm?

I am attempting to approximate intersections of Bézier curves using iterative clipping. This common method is described here and here. It basically works like this: Find bounding lines outside one ...
2
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2answers
567 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
82 views

Distance from point to parabola (quadratic bezier)

I'm trying to draw quadratic bezier curve (as line). I approximate quadratic bezier curve as parabola ($y=x^2$), according to this document ...
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1answer
264 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: ...
3
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1answer
79 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 ...
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3answers
7k 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 ...
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1answer
118 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 ...
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3answers
129 views

Finding parametric distance on quadratic curve from given $(x,y)$ point

I want to get the parametric distance (the "$t$" value) at a location on a quadratic Bezier curve, given the "$x$" and "$y$" coordinates of the point. I have start point, end point and control point ...
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2answers
136 views

How do I find equation of this curve?

I need to find equation of the curve as shown below, for which, I need to find equation for upper part. lower part is half circle. upper part is a constant distance from circle with line passing ...
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1answer
24 views

Prove monotocity of cubic Bezier's curve under certain restrictions

Suppose I have a cubic bezier curve with the points $(x_0, y_0); (x_1, y_1); (x_2, y_2); (x_3, y_3)$. I want to show that the resulting function is monotonic for $x$ for the following restrictions: ...
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1answer
36 views

Describing Bézier surfaces

I'm having some trouble with Bézier surfaces and I was hoping someone could help me. Question is rather simple: lets say we have 2 Bézier curves with control points: P00,P10,P20,P30 and second ...
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2answers
89 views

Deformable circle from a cubic Bezier approximation

I plan to draw approximate circles using a piecewise cubic Bezier representation. The representation should use four Beziers and be defined by four interpolating control points (let us call them ...
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1answer
194 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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0answers
29 views

Find curve passing through n points

I'm currently trying to find a method to interpolate a curve and find its control points such as the curve passes through n points that I have computed earlier. What I'm trying to do in fact is find ...
2
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1answer
131 views

Generating bezier handles based on constraints

I want to try to emulate what this application can do: Given the red round dots (from the mouse) it is able to solve for the bezier handles given that the tension of the curve is set to $0.6$. How ...
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0answers
38 views

Pick the right control point of a cubic bezier curve to form a part of a sinusoid

A,B,C,D are the control points of a Cubic Bézier Curve with approximately this shape: How do you pick point D (the last one, on the right) so that if you mirror the segment J-D of the curve ...
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1answer
40 views

How to take derivative of Bezier function?

I am trying to figure out how to take the derivative of the following quadratic Bezier equation, with respect to 't' for the set of numbers between $0$ and $1$. I understand how to take the derivative ...
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2answers
91 views

Retrieve the initial cubic Bézier curve subdivided in two Bézier curves

I have a cubic Bezier curve subdivided to two cubic Bezier: Assuming that "t_cut" is the t value where this initial Bezier is cut: example of function subdivision(BezierCurve initialCurve, ...
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3answers
26 views

How to find 2 data point in Bezier satisfy the condition the Chord Length Method?

Suppose bezier curve have 4 control point $P0$, $P1$, $P2$, $P3$. How to find 2 data point $D1$, $D2$ satisfy the condition the Chord Length Method? The Chord Length Method : ...
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1answer
29 views

Question regarding Bezier Curve

A Bezier curve $Q$ has control points $P_0 = (0,0,0), P_1 = (0,1,0), P_2 = (1,1,0) and P_3 = (2,0,0)$. What point is $Q(\frac12)$?
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1answer
29 views

At what extent I can use trigonometric functions and properties with parametric curves?

I have a know-how and a library about trigonometry and trigonometric operations, I would like to know if I can possibly rely on trigonometry for parametric curves too and how the trigonometry from the ...
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1answer
41 views

What basis and coordinate system is used in this quadratic Bézier triangle equation? $[x,y,z] = A*s^2 + B*t^2 + C*u^2 + D*2st + E*2tu + F*2su$

I have the following equation for a quadratic Bézier triangle, but I'm having a lot of trouble understanding how to describe it: $[x,y,z] = A*s^2 + B*t^2 + C*u^2 + D*2st + E*2tu + F*2su$ ...
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1answer
192 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 ...
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2answers
35 views

what is the parametric function of the new Bezier curve?

The cubic Bezier curve can be given in matrix form as If a cubic Bezier curve is rotated by an angle 30 around x-axis what is the parametric function of the new Bezier curve?
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1answer
36 views

Formula to derive angle and radius from Bezier circular curve control points

I know the x,y coordinates for the 2 endpoints and the 2 control points for a Bezier circular curve that is less than 180 degrees. I do not know the radius of the circle or the angle of the curve. ...
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1answer
47 views

Creating a surface from a path of 3D cubic bezier curves

I have a list of cubic bezier curves in 3D, such that the curves are connected to each other and closes a cycle. I am looking for a way to create a surface from the bezier curves. Eventually i want ...
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1answer
892 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
98 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 ...
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2answers
71 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 ...
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1answer
75 views

Mathematical definition of Blender's F-Curves

I'm designing software for generating animation curves. I'd like the curves to be based on those found in Blender 3D, which they call "F-Curves." According to the page on the Blender Wiki, they are ...
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1answer
30 views

Points interpolation for tracking

I have set of points for ex. $A_0 (0,0); A_1 (1,2); A_3 (3,3);$ I need an object to travel between these points during some period of time. I was able to construct this trajectory with Bezier curve ...
3
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1answer
312 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
57 views

Bezier curves, control points & reparameterization

Given a Bezier curve $\gamma$(t) defined by 3 control points P0 = (-1,4), P1 = (0, 0), P2 = (1, 0) such that the curve lies on the parabola $\ y = (x-1)^2 $. Reparameterize to $\alpha$(t) = ...
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0answers
17 views

Bezier Surface evalution

So the problem I'm having at the moment, is a thinking problem. I can draw a bezier surface (parametric surface) with 16 control points and if I evaluate S(u, v) I get a coordinate in the 3D space. ...