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

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2answers
69 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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0answers
7 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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2answers
42 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
15 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
20 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
27 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
22 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
12 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 ...
2
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1answer
24 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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2answers
400 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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1answer
313 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
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 ...
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3answers
78 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
66 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
955 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
6k 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
62 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
21 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
106 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
398 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
23 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
25 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
44 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 ...
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1answer
44 views

Approximating a cubic Bézier curves with a collection of quadratic ones

I need to approximate a cubic Bézier curve with a minimal collection of quadratic ones given a maximum acceptable error. Trying to read up on this problem, it seems like there are about as many ...
10
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1answer
7k views

What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline?

I am reading a book about computer graphics. It is confusing about the various splines and their algorithms. What is the difference between natural cubic spline, Hermite spline, Bézier spline and ...
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1answer
37 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[ ...
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1answer
34 views

Calculating values for cubic Bezier curve

I'm trying to use cubic bezier curves for some non-linear animations in my iOS app. Let's say I'm animating position of some element on the screen. I'm using this curve from cubic-bezier.com for ...
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1answer
574 views

Point projection on curve

Point projection on Bézier curves can be easily accomplished using Newton Iteration to try to minimize the dot product between the vector connecting the point P and its projection on curve C and the ...
3
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1answer
38 views

Bezier curve polynom coefficients

How can I calculate coefficients for bezier polynom? I can do this manually on the paper, but I need to plug this into program, where degree of polynom can be higher than 3 ( more than 4 control ...
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2answers
67 views

Smooth curve between 2 points with given gradient at first point?

I'm trying to create a smooth curve between 2 given points with a given gradient/tangent at the first point and any gradient at the last. The idea being to be able to join these to create a smooth ...
5
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1answer
3k views

How to derive the equation for a bézier curve

So, I remember a while back there was a maths competition and we were given a curve that we needed to write an equation for. I just skipped the question since I didn't even know where to begin. I ...
2
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2answers
73 views

Bézier curve - higher order - detect “sharpness” (serpentine or cusp) in curve

I have high order Bézier curve (n > 5). I would like to detect points of self intersection or too pointy ones. In lower degrees, I could use derivative of curve equation and solve roots for valeu = 0, ...
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0answers
44 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. ...
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1answer
30 views

Generate easing function from hash table/value pairs

I have a hash-table/value-pair list consisting of, what I call, linear control value paired with curved/eased real values. Something like this: 0 = 0 1 = 0.0010000000000000002 2 = ...
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1answer
66 views

Improvement over gamma correction for brightening images?

I'd like to brighten one of my own images for printing purposes, using a program I made. When I use the formula: pixelBrightness^0.6 to brighten an image (0.6 being an example, and where ...
4
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1answer
74 views

Why does the Bezier Curve work?

Recently I've been looking at Bezier curves and trying to understand how they work. I know that a general Bezier curve is given by the equation $$ \vec{\mathbf{B}}(t) = \sum_{k=0}^n{b_{k,\ ...
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2answers
95 views
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1answer
111 views

Find cubic Bézier control points given four points

What I need is to generate an SVG file while having a series of (x,y) ready. P0(x0,y0) P1(x1,y1) P2(x2,y2) P3(x3,y3) P4(x4,y4) P5(x5,y5) ... I need to make a ...
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1answer
67 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 ...
0
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1answer
158 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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2answers
106 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
42 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
103 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 ...
0
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1answer
116 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
62 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
303 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 ...
3
votes
2answers
64 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 ...
1
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1answer
208 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" ...
2
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4answers
923 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 ...