# Questions tagged [plane-curves]

Plane curves are continuous (or smooth) functions $\gamma\colon I\to\mathbb R^2$ from a real interval to the plane. Sometimes also the image $\gamma(I)$ is called curve.

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### Showing that a circle is an osculating circle of a unit-speed curve

Let $\alpha : I\to\mathbb{R}^2$ be a smooth plane curve parametrized by arc length, and assume that $0\in I$. A circle with radius $r$ centred at $p$ is called the osculating circle of $\alpha$ at $0$ ...
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### Stereographic Projection of circle of Complex plane to Sphere

Source : Complex Variables by Ablowitz and Fokas Q : Show that a circle in the z- plane corresponds to a circle on the sphere. Sphere : $X^2+Y^2+(Z-1)^2=1$ Attempt : x,y are in z-plane (complex) ...
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### Moving a point along an ellipse given initial point $P$ and arc-length $d$

Fix a reference frame in $\mathbb{R}^2$. Suppose you have an ellipse $\mathcal{E}$, a point $P \in \mathcal{E}$ and a real number $d$, where it is implicitly understood that $d>0$ means movement ...
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### convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
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### Formula for the osculating conic of a plane curve

A follow up to this question. Presumably similar curves have similar osculating conics, which in turn have identical eccentricities. Thus, the 'local eccentricity' of a plane curve at a point is the ...
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### Finding the tangent line(s) to a curve in 3D parallel to a plane

Given the curve $$r(t) = (1 − 2t)i + (t^2)j + (t^3/2)k, ~~ t > 0~.$$ Find a point on the curve at which the tangent line is parallel to the plane $$5x + y + z − 3 = 0~.$$ I have done everything ...
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### Derivation of the formula for the vertex of a parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the ...
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### A curve is a circle or a line

Let $\gamma(t):\mathbb{R}\to\mathbb{R}^2$ be a continuous curve in the plane such that for every $t_1,t_2\in\mathbb{R}$ the euclidean distance $d(\gamma(t_1),\gamma(t_2))$ depends only on $|t_1-t_2|$. ...
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### Fourier transform for a 2D curve

I am stuck on the following problem about a Fourier transform of a 2D curve: I have to calculate the Fourier transform (using 1D complex FT) (and the opposite of it) for a 2D curve z(t). The curve is ...
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### What is the general formula for NURBS curves?

Give me the general mathematical formula for NURBS curves, with special cases (B-spline and Bézier curves)
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### Formula to create a Reuleaux polygon

The Wikipedia articles for Reuleaux triangle and curve of constant width do a good job of describing the properties of a Reuleaux polygon, but they don't give a straightforward formula for computing ...