Questions tagged [osculating-circle]

For questions about osculating-circles, Descartes Theorem, Radius of Curvature, and evolutes.

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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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Find the amplitude of the oscillation of the particle.

The displacement of a particle varies according to $x=3(\cos t +\sin t)$. Then find the amplitude of the oscillation of the particle. Can someone kindly explain the concept of amplitude and ...
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1answer
110 views

Curvature vector and osculating circle radius

I have found an incongruity into the evaluation of the osculating circle radius of the curve $\gamma(t) = R(cos(t),sin(t))$ using the formula: $$\vec r_c(t) = \vec \gamma(t) + \vec k(t)$$ Where: $...
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60 views

Do all functions have an osculating circle?

Radius of curvature is defined as the radius of a circle that has a section that follows/approximates a function/curve over some interval. Now, it's easy to Google pictures of curves that have ...
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1answer
120 views

Limacon curve and its osculating circle

Consider the Limacon: $\gamma(t)=((1+3cost)cost, (1+3cost)sint)$. (i) Compute $A(\gamma)=\frac{1}{2}\int_\gamma (x\frac{dy}{dt}-y\frac{dx}{dt})dt$. (ii) Determine the osculating circle $C$ at $(4,0)$...
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Rational-radii circles packed along the x-axis

Q0. Can all rationals in $(0,1)$ be realized at $x$-coordinates of tangent circles in the arrangement below? I think the answer to Q0 is Yes.                ...
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112 views

Deciding if $\gamma(s)$ cross the osculator sphere on $\gamma(s_0)$.

Let $\gamma(s)$ be a curve in $\mathbb{R}^3$ parametrized by its arc length, with curvature and torsion not $0$. Let $f(s)=\mid\mid \gamma(s) - C(s_0) \mid \mid ^2-r(s_0)^2$, where $C(s_0)$ is the ...
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Parametrization of the osculating circle to a space curve?

Find a parametrization of the osculating circle to r(t)= at t=0 So I found the center of the osculating circle by calculating the radius of curvature and the normal vector. I've also found the ...
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1answer
7k views

Three circles touch. What is the radius of smallest circle?

Three circles touch. The two biggest have radii of $2 \,\rm{cm}$ and $1 \,\rm{cm}$. What is the radius of smallest circle?
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How do I find the equation of an osculating circle when I'm given the parabola?

This is a question given out by my calculus professor, and I'm completely stumped as to how I need to go about solving it. Let the parabola $y=x^2$ be parameterized by $r(t)=ti+t^2j$. Find the ...
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726 views

Apollonian gasket

Okay , is there a way to find the radius of the nth circle in a apollonian gasket .. Something like this Its like simple case of apollonian gasket .. I found from descartes' theorem $R_n = 2\cdot\...
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How can I find a point where an osculating circle goes through a certain point?

Given a point $P = (x_P, y_P)$ and a function $f(x)$, how can I find the set of all points $Q\in f$ such that the periphery of the osculating circle to $f$ in $Q$ goes through $P$? Is there a curve ...
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1answer
195 views

Osculating circles intersecting a given point

Well, the problem is a question in Montiel's book. How to prove that a planar curve $\alpha$ such that all osculating circles intersects a given point is actually a circle (or a part of it)? I've ...