Tagged Questions

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

5k views

What equation produces this curve?

I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple - a gentle curve which begins and ...
11k 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 (...
3k 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 ...
11k 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 B-...
14k views

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, ...
980 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, ...
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?
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 ...
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?
4k 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), ...
543 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 ...
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 ...
229 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 ...
6k 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.
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 ...
592 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 ...
1k 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 ...
65 views

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 $... 1answer 676 views Understanding the Spiro Spline My name's Wray. This is my first time here. Firstly, I like curves. I've been keeping a pet project for a long time that would implement a delightful new curve-interpolation algorithm named the Spiro ... 1answer 1k views Bézier approximation of archimedes spiral? As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic ... 2answers 353 views Get$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 ... 1answer 103 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,\ n}(t)\vec{\... 1answer 274 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 T,U,... 1answer 232 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 http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.... 6answers 8k 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 ... 2answers 106 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 ... 3answers 333 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) ... 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 ... 1answer 760 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 ... 2answers 46 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 ... 2answers 628 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 ... 2answers 1k 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 ... 1answer 249 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 ... 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 ... 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 x-... 1answer 243 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 ... 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 =$$\... 2answers 1k views Bezier curvature I'm trying to understand quadratic Bézier 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 this and this. I ... 4answers 2k views Find control point on piecewise quadratic Bézier curve I need to write an OpenGL program to generate and display a piecewise quadratic Bézier curve that interpolates each set of data points:$$(0.1, 0), (0, 0), (0, 5), (0.25, 5), (0.25, 0), (5, 0), (5, 5)... 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 ... 1answer 56 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 ... 2answers 130 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 ...
79 views

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 ...