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

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3
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1answer
235 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 ...
1
vote
1answer
33 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: ...
1
vote
2answers
192 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 ...
1
vote
1answer
63 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 ...
-1
votes
3answers
502 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 ...
2
votes
2answers
292 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 ...
1
vote
1answer
59 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 ...
1
vote
2answers
600 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, ...
0
votes
1answer
98 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: $[x,y,z] = A*s^2 + B*t^2 + C*u^2 + D*2st + E*2tu + F*2su$ ...
3
votes
1answer
2k 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 ...
1
vote
1answer
54 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 $Q(\frac12)$?
0
votes
1answer
144 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. ...
0
votes
3answers
69 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 : ...
0
votes
2answers
96 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?
0
votes
1answer
41 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 ...
2
votes
1answer
180 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 ...
1
vote
1answer
473 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 ...
5
votes
1answer
216 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 ...
0
votes
1answer
35 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 ...
1
vote
2answers
192 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 ...
0
votes
1answer
115 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) = ...
1
vote
1answer
222 views

Resample Bézier Curve with curvature and number of points constraints

I have an algorithm that implements an uniform resample process throughout a Bézier curve. This is done using a chord parametrization process. However, the results achieved do not accomplish my ...
0
votes
3answers
69 views

what does “$t$” represent in De Casteljau's algorithm?

Hi everybody I need your help. My question is: what does "$t$" represent in De Casteljau's algorithm? We have the following formula to calculate the point $Q$: $Q=(1−t)P_1+tP_2,\;t\in[0,1]$ But ...
4
votes
2answers
565 views

What's the shortest distance between two cubic Bézier curves?

This question comes from TeX.SX http://tex.stackexchange.com/questions/183123/whats-the-minimum-distance-between-two-bezier-curves (From typography; TeX) We are trying to find minimum distance ...
1
vote
1answer
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. ...
0
votes
0answers
86 views

draw a circle using beizer curve and co-ordinate of control points

I want to draw a circle of radius R centered at the origin using Bezier Curve Segments. I have to draw the circle using four Bezier Curve segments - one for each quadrant as shown in the following ...
0
votes
1answer
91 views

Bézier curve limits

Can be any curve of any shape (without sharp edges) described by Bézier curve with unlimited (but finite) number of control points? The answer to the question above would probably be no, because I ...
0
votes
2answers
284 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 ...
1
vote
1answer
69 views

How can I apply Newton's method with boundaries?

I am trying to use Newton's method to minimize the distance between a line segment and a bezier curve. The distance function $f(x, t)$ that I'm minimizing is only defined for $x_1 \le x \le x_2$ and ...
1
vote
1answer
339 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
111 views

What are some alternative ways of describing n-dimensional surfaces using control points other than Bezier surfaces?

I'm interested in problems involving geometric constraints and curve subdivision. I noticed that most of these problems describe the curves/surfaces using the Bezier form. I wanted to know if there ...
2
votes
1answer
265 views

Bézier curves and optimization

I have a very peculiar problem. Assuming that you know how B-Splines or Bézier Curves work, you may also know that if we assume the result of the function, let's say tri-dimmensional, as a position in ...
0
votes
1answer
206 views

keeping c1 continuity in joining several bezier curve

I have some complex curves, I separate the long curves to smallest one to be able to fit them with Bezier curve. However, my Bezier curve has no C1 continuously, if I force C1 continuously, my curves ...
1
vote
1answer
95 views

Reparametrize of cubic bezier curve in arclength

I am looking for a way to re-parametrize the cubic Bezier curve in t domain to cubic bezier curve in S (arclength) domain. Thanks
0
votes
1answer
249 views

Given a control polygon, how do you find a tangent vector at a given point (CAGD, Bezier Curve)?

I am having a hard time with my Computer Aided Graphic Design class. I am presented with the bezier control polygon (in a Mathematica statement): P = Table[{x^2, x^3, 0}, {x, 4,7}] And the ...
1
vote
1answer
251 views

Gradient of a rational Bezier curve

I'd appreciate help working out the gradient of a rational Bezier curve $C = (\,x(t) \,, \,C_y(t) \,)$. I know that the gradient $g$ of a the parametric curve is $$ g(t) = \left( \frac{dy(t)}{dt} ...
0
votes
2answers
74 views

How to create a nice sky route

I'm trying to find a nice algorithm to trace a sky route between 2 points of a planet. Here is where I am : https://dl.dropboxusercontent.com/u/17657227/migrationGlobe/index.html (or here ...
3
votes
1answer
76 views

Is there anything interesting about this figure constructed from a set of points and their barycentre?

Playing with the TikZ package for (La)TeX, I made a nice figure. Well, I think it is nice, anyway. You can ignore the distracting colours and the concentric circles, they are not important for this ...
0
votes
1answer
31 views

Degree elevation of weighted Bezier curve to an arbitrary degree

Following on from a past question about degree elevation of a rational Bezier curve, of degree $n$ by one to $n + 1$, I am now looking to derive a single expression for degree elevation by an ...
0
votes
0answers
20 views

Draw a parallel bezier curve [duplicate]

This may be a duplicate of Control points of offset bezier curve but I am not quite able to say so. Also, the answers linked there are just one level more abstract than is helpful to my limited ...
1
vote
1answer
321 views

How to apply perspective transform to Bezier curve?

I found that both Bezier curves and B-splines are described with a formula $p(t)=\sum\limits_{i=0}^d B^i_m p_i$ but in the case of B-splines $B^i_m$ are B-spline blending functions, while for Bezier ...
0
votes
0answers
51 views

Spline interpolation problem akin to Bezier spline

Given three pairwise distinct points $p_1, p_2, p_3 \in \mathbb{R}^2$, I'd like to find a function $f: \mathbb{R} \to \mathbb{R}^2$ with at least $f \in C^1$ such that $f(0) = p_1, f(1) = p_3, f'(1) ...
3
votes
1answer
1k 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 ...
0
votes
1answer
69 views

Showing that Bezier curve length is less than its control polygon

This is a homework and pardon me for the huge gap of my Mathematics knowledge. After thinking and referencing for a few days I came up with something like following, appreciate help to comment whether ...
0
votes
1answer
2k views

how to calculate the normal vector for a bezier curve

Say we have a cubic Bezier curve (so 4 control points) named Q. I understand how to calculate the tangent at by taking the derivative of Q and substituting but i'm not sure how to calculate the normal ...
0
votes
1answer
288 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 ...
9
votes
5answers
379 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 ...
1
vote
3answers
798 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
760 views

Find angle at point on bezier curve

I have two end points and two control points. I am using these points and this link. i have found a point on bezier curve. Now i would like to find angle at this point on bezier curve. Is there any ...
0
votes
1answer
171 views

X-axis coordinates of outer control points (only) for a Quadratic Bézier curve through 3 points

I am interested in the distance between the 2 outer (left & right: P0 & P2) control-points of a quadratic Bézier curve that goes through 3 data points. The curve's non-equidistant control ...