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

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1answer
14 views

How do I find a Bézier curve that fulfills a given width and height?

I am building a software application that works with vector graphics and I need to use Bézier curves to draw a heart shape, like this one here which I created in MS Paint: The only information ...
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1answer
324 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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1answer
21 views

When is a quadratic Bézier curve nearest the origin?

Consider a planet moving along a quadratic Bézier curve through points A B C, with $t$ = time: $\qquad \operatorname{curve}( t, A, B, C ) \equiv t' (t' A + t (2B - A)) \ + \ t (t' (2B - C) + t C ) $, ...
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0answers
21 views

Compute outer and inner outlines of graph of curves

Let's have some cubic Bezier curves and straight lines. Some of the Bezier curves and straight lines might have a shared start or end point, some might intersect. Input: A list of cubic Bezier ...
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1answer
35 views

Conditions that guarantee a composite Bezier curve in the cartesian plane represents a function?

Context I am allowing users of my application to control a curve connecting $(0,0)$ and $(1,1)$. There are a finite number of knots that are evenly spaced horizontally. The user can specify the ...
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1answer
25 views

Subdividing a Bézier curve into N curves

NOTE: I am only concerned with quadratic Bézier curves. So, dividing a Bézier curve into two is remarkably easy; just interpolate between start and control points by $t$, and get the end point for ...
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1answer
24 views

Bezier bicubic surface intersection genus

I have two bicubic Bezier surfaces that will intersect. According to this paper: http://nishitalab.org/user/nis/cdrom/cad/CAGD91geometric.pdf At the end of page 1. The general genus of intersection ...
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1answer
37 views

Smaller enclosing shape for Bézier curves

It is well known that a Bézier curve is contained within the convex hull of its control points. This is basically a consequence of the fact that the Bernstein polynomials are non-negative and sum to ...
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1answer
42 views

Calculate Gradient (Partial Derivatives) of Bezier Curve

From this page I know that a Bezier curve of degree $N$ has a derivative which is a Bezier curve of degree $N-1$, and I know how to calculate the control points of it: Derivatives of a Bezier Curve ...
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1answer
36 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 ...
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0answers
23 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 ...
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1answer
25 views

Finding the Control Point in a bezier curve

This is a basic (and probably a stupid) question, math is not my forte and I don't know much about math, in this site: http://www.ams.org/samplings/feature-column/fcarc-bezier in the bezier curves ...
2
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1answer
35 views

Derivative of Bezier Rectangle

From this page Derivatives of a Bézier Curve, I can see that the derivative of a degree $N$ Bezier curve is just a Bezier curve of degree $N-1$ and it explains how to calculate the control points by ...
2
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1answer
56 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 ...
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3answers
22 views

How to move the position of a curve $x,y$ coordinates?

I have some silly problem. I want to know how to move the curve in $x,y$ coordinates which I have some curve. For example, $f(x) = x^2$ and this is originally start at $(0,0)$. But I want to this ...
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3answers
10k 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
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 ...
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1answer
72 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 ...
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2answers
34 views

Optimal step for drawing Bezier curve

Bezier curves are parametric in the sense that for each dimension their polynomials share common parameter $t$ [1]. To draw a Bezier curve on screen one could increment $t$ by tiny step and calculate ...
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1answer
39 views

What is the correct notation for curves?

What is the correct math notation to use is when referring to linear interpolation, curves, and points on curves? For instance, let's say we are talking about a quadratic Bezier curve. The control ...
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0answers
19 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 ...
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1answer
18 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 ...
4
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1answer
58 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 ...
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1answer
50 views

Cubic Bezier curve and a straight line intersection

Suppose that two ends of a cubic Bezier curve is connected by a straight line. Is there a simple way to find out whether this straight line intersects the Bezier curve (apart from the endpoints)? If ...
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0answers
326 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?
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1answer
19 views

Ray intersection with explicit (1 axis) Bezier triangle?

This question asks about how to intersect a ray with a bezier triangle: Intersect Ray (Line) vs Quadratic Bezier Triangle What would happen if we had a bezier triangle that had scalars for control ...
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1answer
21 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 ...
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1answer
48 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 ...
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1answer
29 views

What is the difference between a rational spline and a “regular” spline?

I'm pretty comfortable with Bezier curves (not as much with b-splines, nurbs, hermite, catmull rom, etc), such that i know how to generate a bezier curve of any degree using the bernstein polynomials ...
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2answers
75 views

algorithm for Bezier curve with eleven control points

I would like to know the algorithm/ polynomial equation for a Bezier curve with eleven control points. Thanks in advance.
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3answers
1k views

Control Points of Bézier Curve?

Could someone give me a reason/proof why the control points do not lie on the Bézier Curve? Perhaps involving Bernstein Polynomials, if possible? Thanks!
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2answers
66 views

Which side of a 2d curve is a point on?

Given a point $Q$ and $2d$ Cubic Bezier Curve: $$P = A(1-t)^3 + 3Bt(1-t)^2 + 3Ct^2(1-t) + Dt^3$$ Is there a way to know which side of the curve the point lies on? I know that the term "side" is a ...
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1answer
26 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, ...
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1answer
29 views

How to spline together Bezier curves to form a smoth closed curve?

Given $k=m\cdot n$ points: $P_1,P_2,...,P_k$ (all points are two dimensional points), how can I spline together $m$ Bezier curves of $n$ degree to form a smooth closed curve? Denote $B_{i,j}(t)$ to ...
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1answer
39 views

Finding cubic bezier curve endpoints based on relationship between endpoints and a point on the curve.

I have the following information about a bezier curve: The curve begins at $x=0$ and ends at $x=1$. The curve has two control points each at the same height as their closest endpoints, one at ...
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1answer
28 views

How do I fill in points in an equation?

I'm doing research for Bézier curves for school, and I don't really understand how to fill in a point in an equation. I mean, I'd like to represent this on an x,y (or t,y) curve, but what do I need to ...
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1answer
19 views

Do Bezier control points aproximate their curve?

I was just reading here about degree elevation in Bezier curves and I noticed that in the diagrams of the progressively higher degree curve, that the control points began to approximate the curve ...
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2answers
534 views

Bezier curvature

I'm trying to understand quadratic Bezier curves but I cannot get pass one thing. Please, what is a "curvature" and how can I calculate it? I'm asking because I found for instance: ...
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2answers
1k 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 ...
2
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3answers
107 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 $y=f(x)$ ...
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3answers
141 views

What does the 2nd degree derivative of a cubic Bezier curve actually represent?

I have a $3D$ Bezier curve. Each co-ordinate along its path is defined by the equation: $$ f(t) = t^3 \bigl(a_2+3(c_1-c_2)-a_1\bigr) + 3t^2 (a_1-2c_1+c_2) + 3t(c_1-a_1) + a_1 $$ where $a_1, a_2$ are ...
2
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2answers
1k views

Equation for control point distance for fixed-length cubic Bézier path (with specific constraints)

A particular Stack Overflow question asks how to construct a specific cubic Bézier path of constant length. I have experimentally determined the ideal distances of the control points from the nearest ...
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6answers
7k views

Is there an explicit form for cubic Bézier curves?

(See edits at the bottom) I'm trying to use Bézier curves as an animation tool. Here's an image of what I'm talking about: Basically, the value axis can represent anything that can be animated ...
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1answer
69 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 ...
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0answers
45 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 ...
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2answers
171 views

Bezier extrapolation

The red dots are my data: I know that they are on a Bézier curve of order 5 (6 control points). There are extra restrictions on the 6 control points A,B,C,D,E & F: A & B are on a ...
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1answer
459 views

Calculating points on the curve

I want to get the x and y coordinates of a curve..How can i do that... In the above image.Is it possible to calculate the intermediate points(one side) by knowing starting and ending point
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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 ...
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2answers
32 views

Determining whether images of two curves are close to each other

I am coding an implementation of Boolean operations on SVG paths, and need to solve the following problem: Given two sequences of curves, determine whether the distance between their images never ...
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0answers
78 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 ...