# Questions tagged [osculating-circle]

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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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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: $... 1answer 60 views ### Do all functions have an osculating circle? Radius of curvature is defined as the radius of a circle that has a section that follows/approximates a function/curve over some interval. Now, it's easy to Google pictures of curves that have ... 1answer 120 views ### Limacon curve and its osculating circle Consider the Limacon:$\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)$... 0answers 43 views ### Rational-radii circles packed along the x-axis Q0. Can all rationals in$(0,1)$be realized at$x$-coordinates of tangent circles in the arrangement below? I think the answer to Q0 is Yes. ... 1answer 112 views ### Deciding if$\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 ... 1answer 1k views ### 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 ... 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? 1answer 13k views ### 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 ... 1answer 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\...
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 ...
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 ...