# 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 ...
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### 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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### 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 ...
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### What are the conditions for the union of two Elastica curves to be an Elastica curve as well?

An Elastica curve is defined as one that minimises the bending energy, i.e. the curvature squared. Suppose I have two Elastica curves, and I decide to join them end to end. Then what are the ...
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1 vote