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

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3answers
56 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
919 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 ...
10
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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 ...
3
votes
1answer
60 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
19 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
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2answers
75 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 ...
-1
votes
1answer
377 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
2
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1answer
20 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 ...
1
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2answers
23 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 ...
0
votes
1answer
306 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 ...
1
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0answers
34 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 ...
0
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1answer
32 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 ...
9
votes
1answer
6k 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 ...
1
vote
1answer
29 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

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 ...
0
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1answer
557 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
votes
1answer
31 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 ...
1
vote
2answers
56 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
votes
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
votes
2answers
64 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, ...
1
vote
0answers
38 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. ...
0
votes
1answer
27 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 = ...
0
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1answer
60 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
votes
1answer
67 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
85 views
1
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1answer
90 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 ...
1
vote
1answer
58 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
votes
1answer
116 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 ...
1
vote
2answers
85 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 ...
1
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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 ...
0
votes
2answers
95 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
107 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
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1answer
52 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
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2answers
224 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
58 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
vote
1answer
184 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
votes
4answers
905 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 ...
0
votes
1answer
37 views

Understanding de Casteljau algorithm

I have a problem understanding the de Casteljau algorithm. For example, let these be the given Beziér nodes \begin{align*} d_0 = (0,2)^T && d_1 = (0.5,1)^T && d_2=(1,3)^T \end{align*} ...
0
votes
1answer
44 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
0answers
86 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
1answer
142 views

How to clip Bézier curves using Casteljau's algorithm?

I am attempting to approximate intersections of Bézier curves using iterative clipping. This common method is described here and here. It basically works like this: Find bounding lines outside one ...
2
votes
2answers
668 views

Offseting a Bezier curve

I searched this site and I read that in general it is not possible to calculate offset of a Bezier curve. But is it possible to calculate the offset in some special cases? Obviously, if the Bezier ...
4
votes
1answer
134 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 ...
1
vote
1answer
283 views

Detect “Kinks” in Parallel Lines to Bezier Curves [x-post]

I feel like this is just as much a mathematics question as it is a programming question so I figured it couldn't hurt to cross post my question here. Original Post: ...
3
votes
1answer
142 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 ...
7
votes
3answers
8k 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 ...
0
votes
1answer
173 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 ...
-1
votes
3answers
235 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 ...
1
vote
2answers
149 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
26 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: ...