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

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4
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1answer
42 views

Drawing an approximation to a circle in isometric projection

A circle viewed from from the side is an ellipse. A common approximation can be found on the web (eg do a google image search for isometric circle). This produces something like (with arc centers ...
4
votes
1answer
157 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 ...
3
votes
1answer
66 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 ...
2
votes
1answer
26 views

Determining whether two 2D polynomial curves are everywhere close to each other

Let's say we have two curves $P(t), Q(t): [0, 1] \to \mathbb{R}^2$. $P_x(t), P_y(t), Q_x(t), Q_y(t)$ are all polynomials of some degree $n$. We can further restrict this to Bernstein basis polynomials ...
2
votes
1answer
120 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 ...
2
votes
1answer
220 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 ...
2
votes
1answer
135 views

Turning real roots into curves (for visualisation)

One can obviously map a set of real numbers $x_1, x_2, \ldots x_N$ to a curve in 2-D via $y=(x-x_1)(x-x_2)\ldots(x-x_N)$. Thinking about data visualisation, one can portray a set of $N$ observations ...
1
vote
1answer
54 views

Find Quadratic Bezier curve equation based on its control points

If the 3 control points of the quadratic Bézier curve are known, how do you calculate algebraically the equation of that curve (which is an y=f(x) function)? Let's say I have.. P0 (x,y) - startPoint ...
1
vote
1answer
30 views

Bezier curve and deceleration

I have a question regarding calculation of a cubic Bezier curve. I'm programming an app where in there's continuous straight line motion of a vehicle at a constant speed. (Let's call it $u$). When the ...
1
vote
1answer
90 views

Analytical expression for the intersection of a Bézier curve and a line segment

I'm interesting in trying to solve the intersection points for a cubic Bézier curve with a line segment. Background A point on a cubic Bézier curve is given by, $$ P_b(t_b) = \left[ ...
1
vote
1answer
29 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
1answer
36 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
77 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
1
vote
1answer
200 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} ...
1
vote
1answer
140 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 ...
0
votes
1answer
27 views

How can I get a continuous piecewise polynomial curve, with a turning point (not differentiable)?

I would like to make a curve which has turning point(x,y). y= x^2*2 for x<= 0.5 y= 1-(1-x)^2*2 for x> 0.5 and still have a smooth S-shaped curve, where ...
0
votes
1answer
16 views

How to insert a knot in NURBS if it coincides with the first knot?

I want to insert a knot to the knot vector. Currently I use the algorithm from the NURBS book, but it has an assumption that U={0,...0,u_{k},u_{k+1}...,1,...1}, the first knot and the last knot repeat ...
0
votes
1answer
141 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 ...
0
votes
1answer
89 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)?
0
votes
1answer
51 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 ...
0
votes
1answer
73 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: $[x,y,z] = A*s^2 + B*t^2 + C*u^2 + D*2st + E*2tu + F*2su$ ...
0
votes
1answer
83 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
1answer
84 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) = ...
0
votes
1answer
100 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 ...
0
votes
1answer
132 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 ...
0
votes
1answer
27 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
1answer
154 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 ...
0
votes
1answer
61 views

Subdividing a Bézier patch

I have a tensor-product Bézier patch and I want to subidivide this adding a curve inside the patch, which creates two rectangular subpatches. I found that the following statement holds: "if we ...
0
votes
1answer
41 views

I have a function which depends on four parameters and a target value, how can I discover the value for the four parameters that hits my target value?

So I have an equation: $$F(s,t,u,v)=A$$ Where $A$ is some given value. Is there an iterative method to discover the four parameters that will obtain my given $A$? If it helps, my function $F$ is a ...
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
0answers
253 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
2
votes
0answers
324 views

Can elliptic arc be represented by quadratic Bezier curve?

Can elliptic arc (defined as part of an ellipse, with extent not greater than $90˚$) be represented by quadratic Bezier curve?
1
vote
0answers
33 views

Moving a control points on a bezier curve to best fit a moved end point

I have a bezier curve, which I am wanting to manipulate in a certain way. So that it is clear what part of the curve I am wanting to adjust, here is an illustration that labels the parts of the curve ...
1
vote
0answers
67 views

How to find XYZ Coordinates of the Major and Minor Axis end-points in an orbit?

To give some context to this problem, I'm attempting to convert an orbit into a Cubic Bezier Spline, by first plotting four points around the Orbits Ellipse and then computing the Control points of ...
1
vote
0answers
75 views

How can I invert/reverse a curve/ease function?

I have a range of values that represents a curve. This in turn is applied in programming to an interface - rotatable knobs to be precise. Let's say you have a knob that represents a value from 1-20. ...
1
vote
0answers
34 views

Bézier curves as portions of algebraic curves

Can every Bézier curve of any degree be defined as the algebraic (polynomial) curve of which it is a part and it's endpoints? If some Bézier's (such as those of degree $n$ or greater) cannot be ...
1
vote
0answers
541 views

Relation between Hermite interpolation and Bezier curve

I will really appreciate if someone can explain me the relation between Hermite interpolation and Bezier Curve. For example, $p(0)=1,p(3)=2,p'(0)=1,p'(3)=1$, how do we do the Hermite interpolation? ...
1
vote
0answers
63 views

I came across a paradoxical situation when applying Casteljau algorithm

The example is like this: Given 3 points $p(0)=2,p(1)=1,p(3)=1$.The question asks us to apply the Casteljau algorithm to evaluate the Bezier curve b(u) for the given Bezier polygon at $u=2$. I did the ...
1
vote
0answers
143 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
0answers
30 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
0answers
74 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 ...
1
vote
0answers
102 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? ...
1
vote
0answers
84 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.
0
votes
0answers
10 views

Get parameters for given point on quadratic bezier triangle

I have a 2 dimensional quadratic bezier triangle described by the position of its corners $v_0$, $v_1$ and $v_2$ and a handle for each side $h_0$, $h_1$ and $h_2$. The parametric equation with the ...
0
votes
0answers
15 views

Bernstein polynomial

I need some help in the following task. The i-th Bernstein polynomial of degree n on the interval [a,b] is $B_{i}^{n}(x;a,b) = (b-a)^{-n}\binom{n}{i}(b-x)^{n-i}(x-a)^{i}$ Show: The control points of ...
0
votes
0answers
94 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 ...
0
votes
0answers
51 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 ...
0
votes
0answers
58 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 ...
0
votes
0answers
51 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
0answers
45 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) ...