# Tagged Questions

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

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### Which side of a 2d curve is a point on?

Given a point $Q$ and $2d$ Cubic Bezier Curve: $$P = A(1-t)^3 + 3Bt(1-t)^2 + 3Ct^2(1-t) + Dt^3$$ Is there a way to know which side of the curve the point lies on? I know that the term "side" is a ...
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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 ...
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### Do Bezier control points aproximate their curve?

I was just reading here about degree elevation in Bezier curves and I noticed that in the diagrams of the progressively higher degree curve, that the control points began to approximate the curve ...
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### Conditions that guarantee a composite Bezier curve in the cartesian plane represents a function?

Context I am allowing users of my application to control a curve connecting $(0,0)$ and $(1,1)$. There are a finite number of knots that are evenly spaced horizontally. The user can specify the ...
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### 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 $y=f(x)$ ...
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### What does the 2nd degree derivative of a cubic Bezier curve actually represent?

I have a $3D$ Bezier curve. Each co-ordinate along its path is defined by the equation: $$f(t) = t^3 \bigl(a_2+3(c_1-c_2)-a_1\bigr) + 3t^2 (a_1-2c_1+c_2) + 3t(c_1-a_1) + a_1$$ where $a_1, a_2$ are ...
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### Moving a control points on a bezier curve to best fit a moved end point

I have a bezier curve, which I am wanting to manipulate in a certain way. So that it is clear what part of the curve I am wanting to adjust, here is an illustration that labels the parts of the curve ...
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### Bezier extrapolation

The red dots are my data: I know that they are on a Bézier curve of order 5 (6 control points). There are extra restrictions on the 6 control points A,B,C,D,E & F: A & B are on a ...
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### Determining whether two 2D polynomial curves are everywhere close to each other

Let's say we have two curves $P(t), Q(t): [0, 1] \to \mathbb{R}^2$. $P_x(t), P_y(t), Q_x(t), Q_y(t)$ are all polynomials of some degree $n$. We can further restrict this to Bernstein basis polynomials ...
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### Determining whether images of two curves are close to each other

I am coding an implementation of Boolean operations on SVG paths, and need to solve the following problem: Given two sequences of curves, determine whether the distance between their images never ...
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### Approximating a cubic Bézier curves with a collection of quadratic ones

I need to approximate a cubic Bézier curve with a minimal collection of quadratic ones given a maximum acceptable error. Trying to read up on this problem, it seems like there are about as many ...
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### How to find XYZ Coordinates of the Major and Minor Axis end-points in an orbit?

To give some context to this problem, I'm attempting to convert an orbit into a Cubic Bezier Spline, by first plotting four points around the Orbits Ellipse and then computing the Control points of ...
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### I want to create a closed path around another path. (SVG)

I have an svg path, like this: ...
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### Find cubic Bézier control points given four points

What I need is to generate an SVG file while having a series of (x,y) ready. P0(x0,y0) P1(x1,y1) P2(x2,y2) P3(x3,y3) P4(x4,y4) P5(x5,y5) ... I need to make a ...
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### How to plot a smooth curve function from given points

I have several points, how can I plot a smooth curve that pass through those points? Is there any function that I can create or formula that I can use to get all points in the curve? I have read ...
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### Calculating control points of Cubic Bézier curve

I'm trying to draw different arcs with a Cubic Bezier curve and my problem is that after reading different blogs that explain it, I can draw only a 90degree arc using this article. Is it possible if I ...
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### maximum of a 5th order bezier curve with restrictions

Say you have a Bézier Curve of the 5th order with restrictions on the Control points: P0 & P1 are on a horizontal line P2 & P3 are on a horizontal line P4 & P5 are on a horizontal line ...
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### What's the technical term for “ternary interpolation”?

While researching how to render 2D bezier curves given the control points, I found a simple formula and the resource where I found this marked this iterative process as a ternary interpolation and ...
I have an cubic bezier curve, which is representing an arc by an approximation. The approximation was calculated with the kappa constant: $$\\k = \frac43*(\sqrt{2}-1)$$ This means, that the ...