# Questions tagged [bezier-curve]

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

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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 ...
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### Rational Beziér Surface isocurves

In Variational "Design of Rational Bezier Curves and Surfaces" Georges-Pierre Bonneau's PhD thesis it's stated that isocurves (which are curves obtained fixing one parameter of a ...
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### Conic section and Bézier curves

I'd like to prove or disprove the following: given a compact arc of a conic section $\mathcal{T} \subset \mathbb{R}^2$ such that for any $P \neq Q \in \mathcal{T}$ the tangent vectors $\tau_P, \tau_Q$ ...
1 vote
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### Why Bezier Curve is not undefined at t=0 and t = 1? [duplicate]

Sorry,this might be a dumb question, but I couldn't really understand it. If we define Bezier Curves as: $B(t) = \displaystyle\sum_{i = 0}^{n} P_i\binom{n}{i}t^i(1-t)^{n-i}$ when t and i are zero it ...
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### Rational Bézier Curves are projectively invariant

I want to prove that a Rational Bézier Curve is not only affine invariant but also a projective invariant. By affine invariance i mean that applying an affine map to the curve is the same as applying ...
1 vote
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### Variation diminishing property

in the book "CAGD" by Farin he gives a proof of the variation diminishing property for piecewise linear interpolation. Given a continuous curve $c$ in $\mathbb{R}^3$ we define a piecewise ...
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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 ...
37 views

### Equation to find the Principal Unit Normal of a Bezier curve at $t$ ($0\leq t\leq1$)?

I have recently been working on Bezier curves and have come across a very interesting snippet of code (don't worry its almost entirely mathematical) relating to finding the principal normal of a ...
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### Tensor product surface

I'm studying Bézier surfaces and they are a special case of a more general construction, tensor product surfaces. I'd really like to know why this name. I am familiar only with the abstract (algebraic)...
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### corners of a rectangular bended sheet in 3d

I write a mesher for special geometric forms to train Computer Aided Engineering with them. I need to display a rectangular with rounded corners in 3d. It is a Stadium Shape/Obolong/Capsule Shape, ...
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### Scaling Bezier curve first and second derivative?

I'm trying to use a cubic Bezier curve for ease-in-ease-out movement in 1 dimension. The output is position, first derivative is velocity, and second derivative is acceleration to control a motor. But ...
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### Shape invariant of a rational Bézier curve

I'd like to prove that $c_1=\frac{w_0w_2}{w_1^2}\,$ is a shape invariant for a quadratic Bézier curve, i.e. if the weights $w_i$ are modified but $c_1$ is kept constant, then the shape of the curve ...
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### Slope of bézier curve, with t not between 0 and 1

I work with so called "animation curves" in Unity which are basically bézier curves. For an algorithm I need to know the slope of the curve at point t. I already found some solutions on the ...
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### Measure for the number of curves in a shape in $\mathbb{R}^2$

I am looking for a measure of the following form: Say we have some geometric ribbon-like "shape/curve' in $\mathbb{R}^2$. Example: How can we model the number of "curves" (twists, ...
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### How to quantify the error of explicit methods of merging Bezier curves

I came across this paper here describing a method of creating an n_th order Bezier curve that approximates multiple other Bezier curves that are connected. The method is based on minimzing the error ...
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### When to choose Bezier curve over B-Spline curve?

I am reading about Bézier curves and B-Spline curves. I have understood both mathematically and intuitively that the big difference between these two kind of curves is that when dealing with Bézier ...
1 vote
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### Joining a straight line to a Bezier curve

Suppose that we have a Bezier curve of order n represented in the matrix notation as $P(t) = T(t) \Lambda P_c$ (where P(t) is a point on the Bezier curve, $\Lambda$ is a lower triangular matrix ...
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### Join two bezier o that the radius of curvature is smooth

I need to join more Bezier curves in order to obtain a smooth trajectory for a vehicle simulation. Let' say the curve 1 has the points P0, P1, P2, P3, and the curve 2 has the points Q0, Q1, Q2, Q3. I ...
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1 vote
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### Minimum (or mean?) radius of a cubic bezier curve

I'm trying to calculate the minimum radius in a cubic Bezier curve (in C#). I know this question is around on StackExchange, and have thoroughly browsed the answers and tried different implementations....
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### How do you move multiple points with a set distance (linear) between them on a non-linear spline?

I'm doing some things related to spline code, and I've run into a problem. Please take a quick look at the picture below. Black = spline (bezier), red circles = points on spline Imagine that the ...