Questions tagged [bezier-curve]

Questions on Bézier curves, which are used for numerical analysis with applications in computer graphics.

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Bezier curve get X based on T? [closed]

If I have the following points in my Bezier curve: (4,8), (7,9), (4,4) How to calculate what is X when T = 1?
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58 views

Pythagorean-Hodograph curve's control points

In the Farouki's book "Pythagorean-hodograph curves: algebra and geometry inseparable" it is said that control points of the cubic Bezier curves with Pythagorean-hodograph qualities are set ...
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Maximum distance of Bezier curve

I have a Bezier curve given by four points: a start point ($P_0$), an end point ($P_3$), and two control points ($P_1$ and $P_2$). The points lie in a certain way. $P_1$ and $P_3$ lie on the x axis. $...
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Can a segment of a Cartesian Oval be exactly represented as a NURBS curve?

I've tried searching for answers to this question, both here and on Google, but have come up empty. The equation for the Cartesian oval I'm using comes from this journal paper, with shape parameters ${...
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45 views

Cusps on quadratic Bezier curve

A quadratic Bezier curve is given in parametric form by: $$C(t) = (1-t)^2P_0 + 2(1-t)tP_1 + t^2P_2.$$ My points are: $(1,1)$, $(2,2)$ and $(3,3)$. How do I show that this curve has cusps? Best regards,...
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Second derivative of Bézier curve segments $\vec{x}_1(t)$ and $\vec{x}_2(t)$

For a specific excerise, I had to illustrate the algorithm of Casteljau on the curve $\vec{x}$ below with $\vec{x}(c)$ and $c=2/3$. Now the question is: If $\vec{x}_1(t)$ and $\vec{x}_2(t)$ are two ...
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2D orthographic projection of a spherical lune (geometry of a crescent moon's terminator)

I am interested in drawing a crescent moon in a vector drawing program. Our moon is a sphere illuminated by the sun in a certain direction, and viewed from Earth in a rotated direction. Our view of ...
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24 views

Traversing a Composite Bézier Path

Does anyone know if there is a way to traverse a composite bezier path (a path made of multiple bezier curves) with a parameter $t$ such that $0 \le t \le 1$? Basically I want to traverse an entire ...
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1answer
53 views

Is this system of geometric points solvable?

Apologies for the vague title, but I am not sure how otherwise to explain my question with a title. I have a set of 3 known points (x, y), and a set of 3 unknown points and a few known relationships ...
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Spline terminology: Can a spline be represented by a Bézier curve?

In this question it was stated that a B-spline is just a representation of a Bézier curve. I wonder if this also holds true for Bézier curves. Splines are defined piecewise by polynomials and so are ...
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2answers
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Is it possible to find the control points of a quadratic bezier curve, given the starting, ending and stationary point?

I know from this question on SO that it is possible to get the stationary point of a bezier curve given the control points, but I want to know wether the opposite is true: If I have the start and end ...
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Get a point for part of Quadratic Bezier Curve

Given a Quadratic Bezier Curve $a$ defined with three points $A$, $Q$, and $B$. Then given a point $P$ such that $P\in a$. The task is to find such a Quadratic Bezier Curve $b$ that is defined by ...
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Proof? Two intersecting Bézier curves always have a BASE-BASE or BOUND-BOUND intersection

Let's be more precise: Suppose you have two quadratic Bézier curves in 2D. Let's label the control points $C, C'$ and the 'base' points $A, A'$ and $B,B'$. Is it true that if these two curves ...
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Bezier clipping - optimal way to handle multiple intersections

I am implementing bezier clipping in 2d to find the intersection points of 2 bezier curves of arbitary degree. I am familiar with the method of bezier clipping using fat lines but I wanted to find out ...
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1answer
23 views

Thoughts on Bézier curve intersection and reliability of the test

Suppose we have two planar parametric curves f(t) and g(t), which are guaranteed to be second- or third-degree Bézier curves, which means they can be placed in "polynomial" form: $$ \textbf ...
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Computer aided geometric design

For two bicubic Bezier surfaces $P(u,w)$ and $Q(u,w)$, $u$ and $w$ being local parameters defining the respective surfaces, joined to create a $C^2$ continuous composite surface, find out the ...
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1answer
113 views

Self-intersection of a cubic Bezier, interpretation of the solution

I am trying to understand the form of the determinants that I get when I try to calculate the self-intersection. Since I might have made some mistakes, I'm including the whole derivation: Let $\vec{P}(...
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2answers
47 views

Cubic bezier get $y$ as a function of $x$ (not $t$)

I'm trying to write a program to display curves. I have 4 control points $ A, B, C $ and $ D $ and a value for $x$. From that, I want to find the value for $y$. Where I've gotten so far We can use De ...
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2answers
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Is it possible to do an integration over a Bézier curve “area”?

I have a shape defined by a Bézier spline that has a width, and I want to give it an attractive force. Is such a thing even doable without approximating it? Edit: To integrate the area, I think I ...
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1answer
79 views

Calculating the circle between two quadratic curves

I have two quadratic curves sharing one control point $A$. The control points for the first curve are $A, B, C$ and for the second curve: $A, D, E$. The two quadratic curves are defined as: $P(t) = (...
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1answer
31 views

Which Bézier curve rises first?

Given two second order (quadratic) Bézier curves which both start at the origin $[0, 0]$ and end at two arbitrary points, and both of whose control points lie at the positive portion of the X-axis ($y ...
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1answer
41 views

How do you ensure that a third-degree Bezier curve is a straight line segment?

Suppose that we have a Bezier curve of degree $3$. How can I ensure that the curve will be a straight line? So as an example take this figure I think that we can guarantee that the red line is a ...
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1answer
42 views

Is there a way to find the point of intersection of two quadratic Bezier curves?

Each Bezier curve have two quadratic parametric equations to express the curve. Can these be utilised to find the point of intersection with the other curve? If so, how? I am really confused.
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1answer
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Smoothing asymptotic behavior in the curvature of a cubic bezier

I have two cubic bezier curves. Here are their control points P0,P1,P2,P3: ...
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1answer
80 views

How were Bézier curves invented?

In trying to understand basics of CAGD I stumbled upon Bézier curves. Their presentation usually starts with the definition using Bernstein polynomials, and then the determination of their properties. ...
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1answer
43 views

How can I find coordinates of points along a curved path? [closed]

I have a very basic grasp of the math concepts behind this question so I need a very basic explanation if at all possible. The question is in regards to a game program where a player is traveling ...
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39 views

How do I get the control points of a bezier line?

I do have a set of N points. How can I fit a Bezier curve of order three through these points? I know the formula to get a Bezier line from three control points, ...
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1answer
99 views

2D cubic Bezier curve. Point of self-intersection

I have a 2D cubic Bézier curve defined by a set of control points A, B, D and C. How can I find a point of self-intersection P (two parameter values t)?
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2answers
177 views

Convert cubic spline to Bézier curve and get control points

There is a cubic spline represented by the standard equation: $$ f(x) = a + b (x - x_0) + c (x - x_0)^2 + d (x - x_0)^3 $$ and 2 endpoints: $P_0~ [x, y]$ - starting point $P_1~ [x, y]$ - end ...
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1answer
32 views

How to move the control points of the cubic Bézier curve, to keep the curve invariant?

I have 3 cubic Bézier curves with different control points: https://cubic-bezier.com/#.17,.8,.77,1 https://cubic-bezier.com/#.18,.59,.5,1 https://cubic-bezier.com/#.12,.41,.41,1 They look similar ...
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How to decide if a point of a Bézier is at a strong curvature or not?

For work I need to decide if a random point is at a "strong" curvature of a cubic-2D-bezier or not. I'm testing with the Bezier below. I got the 1st 2 derivatives from Wikipedia. In the ...
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Common tangent to two Bezier quadratics [duplicate]

I am trying to find the tangent that is common to two quadratic Bezier curves. Concretely, I want to draw the corner of a bending piece of paper, like this: Theoretically it looks simple: Given a ...
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29 views

Calculating the two outer Bezier Curves of a thick Bezier curve

I'm creating a Bezier curve using 4 control points (left photo). If the same curve was 100px thick (right photo) is it possible to calculate the control points needed to create the 2 red curves using ...
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1answer
46 views

go from an extended polynomial to a Bezier polynomial

What is the analytical expression to do a change of basis from the extended polynomial form i.e. $$ \mathbf {B} (t)=\sum _{j=0}^{n}{t^{j}\mathbf {C} _{j}} $$ to a Bezier form $$ \mathbf{B} (t) =\sum ...
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1answer
20 views

Find the cubic bezier control points from end points and tangents

If i have 2 end points and two unit vectors as tangents at the two end points is it possible to find the cubic bezier curve control points that make the curve ? Is there one solution or many solutions ...
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2answers
49 views

Finding coordinate of control point in Bezier Quadratic Curve

I have both the endpoints of the Bezier Curve also I know the length of the curve and x' coordinate of the control point. So how can I find the y' coordinate of the control point? I went through ...
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1answer
19 views

Slice 3D curve into series of Bézier curves

I've been stuck on this for 2 weeks and have not had much luck asking else where either. I have a curve, which is an arc, at each end of the arc there is tangent vector which is sloped in some ...
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93 views

Closed form expression for parallel transport / Bishop Frame along a cubic Bezier curve?

I would like to define a rotation minimizing frame (also known as a Bishop Frame) for a curve. This should be equivalent to parallel transporting a unit normal vector along the curve and then building ...
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77 views

How to apply Bézier curve to a linear function?

I have a function that linearly transforms ratio to a value within a range: ...
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1answer
32 views

Converting old database with recorded 3D motion data into SVG Bezier curve

I'm tasked with reverse-engineering and re-writing as a web app of some old application that draws 2D projections of some object's motion. The new application should draw the 2D projection into SVG ...
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1answer
32 views

Ideal spline type for software optimized well-defined interpolation over time domain

I'm working on a software program which currently utilizes cubic Bézier splines for generating continuous well-defined function output (only 1 output value per input) in the time domain with varying ...
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31 views

I need a spline type that interpolates through knot points and produces a single-valued y=f(x)

I have a set of $(t, f(t))$ points that describe a function of time. I need to interpolate those with a spline that goes through all the given points, and doesn't have any loops, so it creates a ...
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2answers
39 views

Bezier Cubic Curves for six points

Given six points in $\Bbb R^2$: $(-1,0),(-1,1),(-1,2),(1,0),(1,1),(1,2)$. How to prove that there is not Cubic Bezier curve that cross all these points? The picture of the points
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Derive control points for bezier curve from a bounding box

Given that I start with a bounding box of 4 (x,y) vertices representing the corners. And given that I have 2 of the 3 control points to plot a quadratic bezier curve, how do I derive the 3rd control ...
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3answers
140 views

Plotting a parabola based on data points

I am trying to draw a parabola inside a chart which I am developing using D3.Js library and using SVG paths to draw the curve. I have a set of 5 points for drawing the parabola: ...
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Is there a way to find the range and domain of a cubic spline?

Wolfram alpha is telling me there isn't. I guess this reduces down to finding the inflection point in some way and testing against points A and D. Here's the equation I'm using. $$ x=(1-t)^3*A_{x}...
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3answers
73 views

Finding the formula for a parabola from the control points of a bezier

I'm trying to write a program that can detect whether the mouse is inside the parabola defined by three Bezier control points. I first tried doing this by looping through each line in the "string art"...
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1answer
34 views

Which kind of curve is this

Basically I'm currently using a software that edit curve and I was wondering which kind of interpolation it was using. You have on point at (0,0) and another at <...
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1answer
26 views

Can Bézier curves be generalized to any number of dimensions or are they purely 2D?

Can you generalize a Bezier curve to a function $\acute e : [\mathbb R^n] \to C_n$ where $C_n$ represents a curve in $\mathbb R^n$ and $[x]$ is a list of $x$?
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1answer
64 views

Points with given curvature on cubic Bézier curve

I am looking for all points on cubic Bézier curve that have a specific curvature $\kappa_1$. The curvature $\kappa$ is caluclated as $$ \kappa=\frac{x'y'' - y'x''}{\sqrt{({x'}^2+{y'}^2)^3}} $$ ...

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