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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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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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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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 algorithm helps in doing this. But, to use this…
Vivek Molkar
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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 the other 2 endpoints? The red dots in the…
SummerCode
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How to perfectly split a bezier curve into two curves of unequal length

I couldn't find a title that didn't seem duplicate, but my question is very different from every other I could find on this site. I have a Bezier curve, defined by 4 points. Two of those points (first and last) are "real", and the other two are…
Kaito Kid
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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 it intersects then what will be the corresponding…
John Railman
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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), variation diminishing property of the curve (the…
Ailurus
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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 hours searching on google I was not able to find…
Duck
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What is the equation to produce this Bezier curve?

What is the equation for the first curve in the image? I have this formula: But when I substitute values, I get an image that looks like x^2. It isn't the same shape as the image.
user117903
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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 mathematician (probably if I were I wouldn't be asking this…
jamadagni
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Intersections Between a Cubic Bézier Curve and a Line

If one end and a control point of a cubic Bézier curve is connected by a straight line, is there a simple way to find out whether this straight line intersects the Bezier curve? If it intersects then what will be the corresponding Bézier curve…
codeKhor
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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 $1$. The property is useful in a variety of ways,…
bubba
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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 their sharpness is minimized. I know that a…
Octopus
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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 determined by the following equation, from…
Lazlo
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Why it is impossible to draw a circle by Bezier Curve?

I have see a lot of people said it is not possible to draw a circle by Bezier Curve. However, I want to know why? I did see somebody explain, but I am not quite sure what they mean. Like, Bezier curve is parabola, so it is impossible to draw a…
Mars Lee
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