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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How to call this Bezier curve?

With Anchor point inside and with two Handle lines that with different lengths and different angles (i.e. 90 degree between two Handle line). And Handle lines of two Anchors does not cross between each other? Thank you.
Javier Hernández
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Ray intersection with explicit (1 axis) Bezier triangle?

This question asks about how to intersect a ray with a bezier triangle: Intersect Ray (Line) vs Quadratic Bezier Triangle What would happen if we had a bezier triangle that had scalars for control points, so they only controlled the height of each…
Alan Wolfe
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How do I fill in points in an equation?

I'm doing research for Bézier curves for school, and I don't really understand how to fill in a point in an equation. I mean, I'd like to represent this on an x,y (or t,y) curve, but what do I need to fill in to calculate, say, B(1) or B(2)? I can't…
hansottowirtz
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what does "$t$" represent in De Casteljau's algorithm?

Hi everybody I need your help. My question is: what does "$t$" represent in De Casteljau's algorithm? We have the following formula to calculate the point $Q$: $Q=(1−t)P_1+tP_2,\;t\in[0,1]$ But what does $t$ mean here and why is it between $0$ and…
Mo Haidar
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Speed of Bezier curve

Let's assume, that we're working on 3rd degree Bezier curves only. $$ B(t) = p_1 \cdot (1-t)^3 + p_2 \cdot 3 \cdot (1-t)^2 \cdot t + p_3 \cdot 3 \cdot (1-t) \cdot t^2 + p_4 \cdot t^3 $$ One way to render Bezier curve on a screen is to convert it to…
Spook
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How do I find the optimal control points to fit a cubic Bézier curve to a known function?

I have the function $f(x) = \frac{x^3 − x^2 \sqrt{2 x^2 + 2} + 4 x} {x^2 + 2}$. I want to find the optimal control points $\{P_0, P_1, P_2, P_3\}$ for a cubic Bézier curve $B(t) = (1 − t)³ P_0 + 3(1 − t)² t P_1 + 3 (1 − t) t² P_2 + t³ P_3$ that most…
Lawton
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Algorithm for a Bezier Curve approximation on a Cartesian Grid

Given any bezier curve, I would like to find a set of lines such that: a) the lines are all connected in series b) the start point and end point of the series of lines are the start point and end point of the bezier curve c) the start and end point…
jaxoncreed
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That is the math behind interpolating circle using Bezier curve?

This is a basic circle build by a graphic editor using Bézier spline. The X here is 0.552125R: But how this value had been gotten? I mean reverse engineering, the mathematical equation which results to this value
Crantisz
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Cubic bezier speed curve solve for t for a given d

I am trying to currently achieve an animation as following 1)There are 4 elements in the canvas- 2) green square at (5,20) with a width 10 and height 10 3) 3 lines located respectively at (55,20), (80,20), (125,20). 4) the horizontal…
smpa01
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Number of Quadratic Bezier Curve-Ray Intersections

Given some quadratic bezier curve $B(t)$ and some ray $R$ is there an equation to calculate the number of intersections between the two. (For my application I only need to consider 2d space).
alexa griffin
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Find value Y on cubic Bézier given value X

Okay, I know that this has been asked here a lot, but I've read through ~10 other questions exactly like this one, and none of them have provided me with any useful information. In fact, 3 of the answers that I found were straight up…
OOPS Studio
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Maximum distance of Bezier curve

I have a Bezier curve given by four points: a start point ($P_0$), an end point ($P_3$), and two control points ($P_1$ and $P_2$). The points lie in a certain way. $P_1$ and $P_3$ lie on the x axis. $P_1$ lies on a vertical line with $P_0$ and $P_2$…
InsertNameHere
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Cusps on quadratic Bezier curve

A quadratic Bezier curve is given in parametric form by: $$C(t) = (1-t)^2P_0 + 2(1-t)tP_1 + t^2P_2.$$ My points are: $(1,1)$, $(2,2)$ and $(3,3)$. How do I show that this curve has cusps? Best regards, Sergey
user504068
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Calculating the two outer Bezier Curves of a thick Bezier curve

I'm creating a Bezier curve using 4 control points (left photo). If the same curve was 100px thick (right photo) is it possible to calculate the control points needed to create the 2 red curves using the original points? Where can I learn more about…
CrackerBarrelKid55
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Can Bézier curves be generalized to any number of dimensions or are they purely 2D?

Can you generalize a Bezier curve to a function $\acute e : [\mathbb R^n] \to C_n$ where $C_n$ represents a curve in $\mathbb R^n$ and $[x]$ is a list of $x$?
schuelermine
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