Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [bezier-curve]

Questions on Bézier curves, which are used for numerical analysis with applications in computer graphics.

Bézier curves are widely used in computer graphics to model smooth curves by using control points. Affine transformations of the curve correspond to affine transformations of the control points. The curve is contained in the convex hull of its control points. On computers, these points are often graphically displayed and used to manipulate the curve by dragging the control points.

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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 (position, scaling, color, basically any numerical…
subb
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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-spline?
user960456
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Arc Length of Bézier Curves

See also: answers with code on GameDev.SE How can I find out the arc length of a Bézier curve? For instance, the arc length of a linear Bézier curve is simply: $$s = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$$ But what of quadratic, cubic, or…
Mateen Ulhaq
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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 ends horizontal. Is there a simple equation for…
Giffyguy
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How do I find a Bezier curve that goes through a series of points?

When someone has the 4 control points $P_0$, $P_1$, $P_2$, $P_3$ of a 2D cubic Bézier curve, that person can calculate a series of hundreds of points along the curve that start from $P_0$ at $t=0$ and end at $P_3$ at $t=1$ (but, in general, those…
David Cary
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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 curve of degree $m$. I want to adjust the coefficients…
bubba
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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 get a point, $P$, which is called the "apex" of…
bubba
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Difference between Bezier segment and B-spline

I am currently learning about Bezier curves and splines in computer graphics. What is the difference between a B-spline curve and a curve that consists of Bezier curves as segments? I have read in many sources that B-splines have better properties…
Jag
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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, subject to the following constraints: $P_1$ and…
Todd Lehman
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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 an html5 canvas element. The B-Spline is coming…
TimSum
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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 between the points not distance along the curve.…
Eric Anastas
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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?
Rogach
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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 radius of a circle. $(x_0, y_0)$ and $(x_1, y_1)$…
markE
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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 Spline. The details of the Spiro are laid out in…
Wray Bowling
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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 or approximate the two curves to/with a single…
david
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