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

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15
votes
5answers
2k views

Polynomial approximation of circle or ellipse

Trying again, with a somewhat simpler sounding question, since my previous one (Generalizations of equi-oscillation criterion) got zero response: Let $F:[0,1] \to R^2$ be a parametric polynomial ...
14
votes
6answers
9k 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 ...
13
votes
1answer
10k 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 ...
10
votes
3answers
12k 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 ...
9
votes
5answers
360 views

Parabolas through three points

We can draw an infinite number of parabolas that pass through three given points $A$, $B$, $C$ (in that order). For each such parabola, we take the tangent lines at $A$ and $C$, and intersect them to ...
9
votes
1answer
3k views

Find points along a Bézier curve that are equal distance from one another

I'm trying to figure out a generic way of determining a series of points on a Bézier curve where all the points are the same distance from their neighboring points. By distance I mean direct distance ...
8
votes
1answer
5k views

How do I find a Bezier curve that goes through a series of points?

When someone has the 4 control points P0, P1, P2, P3 of a 2D cubic Bézier curve, that person can calculate a series of hundreds of points along the curve that start from P0 at t=0 and end at P3 at t=1 ...
7
votes
3answers
3k views

Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]

Update Alternative Statement of Problem, with New Picture Given three points $P_1$, $P_2$, and $P_3$ in the Cartesian plane, I would like to find the ellipse which passes through all three points, ...
7
votes
0answers
930 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, ...
6
votes
3answers
2k views

How to approximate/connect two continuous cubic Bézier curves with/to a single one?

I subdivide a cubic Bézier curve at a given t value using de Casteljau’s algorithm, which yields two cubic Bézier curves. Afterwards I “scale” the second curve (proportionally). I’d like to reconnect ...
6
votes
2answers
3k views

algorithm to calculate the control points of a cubic Bezier curve

I have all points where my curve pass through, but I need to get the coordinates of the control points to be able to draw the curve. How can I do to calculate this points?
5
votes
4answers
3k views

Why is the Convex Hull property (e..g of Bézier curves) so important?

Recently I read some course notes and articles on Bézier curves. They all sum up the properties of Bézier curves, like the partition-of-unity property of the basis functions (Bernstein polynomials), ...
5
votes
2answers
5k views

How elliptic arc can be represented by cubic Bézier curve?

If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bézier curve? And if yes, how can I calculate control points for that Bézier curve?
5
votes
3answers
524 views

Find sagitta of a cubic Bézier-described arc

I have a situation where I have an arc that was mangled (irrelevant: by c#'s GraphicsPath.AddArc() function). The original arc is guaranteed to be circular, and the new data I have describes the ...
5
votes
1answer
5k 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 ...
5
votes
1answer
196 views

Mathematical definition of Blender's F-Curves

I'm designing software for generating animation curves. I'd like the curves to be based on those found in Blender 3D, which they call "F-Curves." According to the page on the Blender Wiki, they are ...
4
votes
2answers
460 views

What's the shortest distance between two cubic Bézier curves?

This question comes from TeX.SX http://tex.stackexchange.com/questions/183123/whats-the-minimum-distance-between-two-bezier-curves (From typography; TeX) We are trying to find minimum distance ...
4
votes
2answers
2k views

Control points of offset bezier curve

If I have a cubic Bezier segment specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined ...
4
votes
2answers
52 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 ...
4
votes
1answer
562 views

Understanding the Spiro Spline

My name's Wray. This is my first time here. Firstly, I like curves. I've been keeping a pet project for a long time that would implement a delightful new curve-interpolation algorithm named the Spiro ...
4
votes
1answer
1k views

Bézier approximation of archimedes spiral?

As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic ...
4
votes
2answers
316 views

Get $t$ of ascending Bézier curve from $x$

I have an ascending cubic Bézier curve. ($x_0 \leq x_1 \leq x_2 \leq x_3$) Considering this property, there is always one and only one $y$ value per $x$ value. The point ($x, y$) along the curve is ...
4
votes
1answer
90 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,\ ...
4
votes
1answer
167 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
196 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
4answers
5k views

Is it possible to build a circle with quadratic Bézier curves?

i'm searching for a curve type with a minimum of functionality and maximum of usability. I run into quadratic Bézier curves and i wonder, if its possible to draw a circle with it.
3
votes
2answers
76 views

Smoothest function which passes through given points?

I am trying to interpolate/extrapolate on the basis of a known collection of (finitely many) points. I'm wondering if there is a way to formalize this intuitive notion: find a 'smoothest' function ...
3
votes
3answers
214 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 $y=f(x)$ ...
3
votes
1answer
2k views

Calculate control points of cubic bezier curve approximating a part of a circle

I'm not mathematically inclined, so please be patient with my question. Given $(x_0, y_0)$ and $(x_1, y_1)$ as the endpoints of a cubic Bezier curve. $(c_x, c_y)$ and r as the centerpoint and the ...
3
votes
1answer
898 views

Find value of '$t$' at a point on a cubic Bezier curve

I have a cubic Bezier curve, and I need to divide it and create same curve between point on the original curve and the end point of the original curve. From my research, I found the DeCasteljau ...
3
votes
4answers
7k views

Casteljau's algorithm - practical example

I have a dataset with about 50 points (x,y) and I would like to draw a smooth curve that can pass as closer as possible on those points. I have heard about Casteljau's algorithm for splines but after ...
3
votes
2answers
36 views

Are Bezier curves invariant under conformal mapping?

I've spent quite a bit of time on google trying to find information on whether or not Bezier curves are invariant under conformal mapping (i.e. a conformal mapping of all points on the curve is the ...
3
votes
1answer
3k views

Convert a B-Spline into Bezier curves

I have a B-Spline curve. I have all the knots, and the x,y coordinates of the Control Points. I need to convert the B-Spline curve into Bezier curves. My end goal is to be able to draw the shape on ...
3
votes
1answer
1k views

Finding Y given X on a Cubic Bezier Curve?

I just asked this in the Computing sections but they sent me here: "So I've been looking around for some sort of method to allow me to find the Y-coordinate on a Cubic Bezier Curve, given an ...
3
votes
1answer
146 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 ...
3
votes
1answer
214 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 ...
3
votes
1answer
1k views

Intersect a line with a bicubic Bezier Surface Patch.

This question mentions Bezier surfaces, but doesn't go into any detail. How do you going about finding the intersection between a line, $E_{pos} + E_{dir}*t$ and a Bezier surface patch, $P = $ ...
3
votes
2answers
760 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: ...
3
votes
2answers
1k views

How to calculate the square area under a Bezier curve?

I did search at Google and this website before asking this question, so sorry if this somehow has already been answered and I didn't notice. BTW I'm a humanities scholar and not a trained ...
3
votes
1answer
43 views

Translating Equations to Algorithms

I can't understand equations. But I'm a software engineer. I think the brevity of the equation is confusing to me where a program spells it all out. Trying to translate the equation for a bezier ...
3
votes
2answers
110 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 ...
3
votes
1answer
73 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 ...
3
votes
1answer
384 views

Curve through four points — simple algebra??

The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to ...
3
votes
0answers
282 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
5answers
1k views

Rough y(x) approximation for simplified Cubic Bezier curve

I need to get a very rough (and fast) $y(x)$ approximation of a simplified Cubic Bezier curve to use in my animation code, where there's only one control variable: $$ P_0 = (0, 0)\\ P_1 = (0, 0)\\ ...
2
votes
2answers
918 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 ...
2
votes
1answer
237 views

Equation for subsection of Bezier curve

Say I have a cubic Bezier curve, with a starting point s, an ending point e and control points ...
2
votes
4answers
2k views

Can a rational Bézier curve take exactly the same shape as a part of the sine function?

I'm wondering whether a rational Bézier curve could take exactly the same shape as a part of the sine function. The best way to check this seems like this: Find a part of the sine function such that ...
2
votes
1answer
693 views

Determining the result of Boolean shape operations on closed Bézier shapes

Given two closed shapes made up of Bézier curves (and/or straight lines), I'm looking for an efficient way of calculating the resulting shape of the following Boolean operations: union difference ...
2
votes
4answers
1k 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 ...