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Questions tagged [osculating-circle]

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

An osculating circle is the circle tangent to a given curve at a given point and at infinitely close adjacent points. Consequently it is the largest circle tangent to the point on the inside of the curve. Given three mutually tangent circles, Descartes Theorem gives a formula to find the two circles which are tangent to each of the given three. The evolute of a curve is another curve traced by the center of an osculating circle as it travels along the given curve.
This tag should be used for questions relating to finding the osculating circle or evolute for a given curve or set of curves.

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On the number of ways to draw kissing circles

So I was watching Numberphile with Neil Sloane of OEIS fame on the number of ways to make circles intersect. During the intro, he explicitly forbid kissing (touching, tangent) circles. This made me think about the number of ways to draw circles that…
Jens
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Does $\frac{\sin(tx)}{\sin(x)}$ have a name?

Does the following function have a name? $$\operatorname{boxySine}(t,x) = \begin{cases} \frac{\sin(tx)}{\sin(x)} & x \neq 0 \\ t & \text{otherwise} \end{cases} $$ It appears in describing the position of a point on a circular segment in…
Cirdec
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How can i understand the graphical interpretation of Torsion of a curve?

I understand the graphical interpretation of the curvature of a curve in $\mathbb{R}^3$. Could you help me to understand the graphical meaning of the torsion of a curve? I know that if torsion is positive, the curve goes through the osculating…
Mika Ike
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Center and radius of the osculating circle - The limiting position of a circle trough three points

I am stucked on problem 1.7.2.b of Differential Geometry of Curves and Surfaces by Manfredo do Carmo. The problem is similar as this topic, but here the exercise defines the osculator circle, ie, this circle of exercise is whose we call osculator…
Quiet_waters
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Issue in my proof that the limit of circles through three points on a curve is the osculating circle

I'm going through some differential geometry exercises (from Kristopher Tapp's Differential Geometry of Curves and Surfaces) I worked on a while ago, and realised I missed a detail in Part (2) of the following: Let $\boldsymbol\gamma: I \to \mathbb…
themathandlanguagetutor
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Defining curvature via osculating circles

I am trying to figure out a geometrically accessible definition for the curvature of a smooth plane curve $c:I \to \mathbb{R}^2$ where $I$ is an interval and $c' \neq 0$ everywhere. My plan is to define the curvature via the osculating circle so…
russoo
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Limit definition of the osculating circle

Let $I$ be an interval and $c:I \to \mathbb{R}^2$ a smooth curve with $c' \neq 0$ everywhere. I am currently trying to figure out how to define the osculating circle at a point $c(t_0)$, $t_0 \in I$, without assuming that the curvature has…
russoo
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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: $\vec r_c(t)$ is the vector that identifies the…
user515933
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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 osculating circles drawn in and it seems obvious that…
S. Rotos
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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 tried to use the expression…
checkmath
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Triangle inscribed and circumscribed gap-filling radii sequences distinct?

Staring with an equilateral triangle $\Delta$, inscribe a circle, then in the gaps, inscribe other circles, ad infinitum. Similarly, inside the circumscribed circle but outside $\Delta$, continue to fill in every gap with a touching…
Joseph O'Rourke
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What is the Benice equation?

I keep seeing lots & lots of pictures generated by the Benice equation (usually spirograph type things or fractal like things) but nowhere have I seen a reference or an explicit explaination of what the Benice equation actually is. So I'm asking…
DrBwts
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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 center of the osculator sphere and $r(s_0)$ is the…
Relure
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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 equation of the osculating circle for the parabola at…
matryoshka
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Deriving the equation of osculating circle starting from the definition using order of contact

the question is 9.f ) section 5.4.6 from Mathematical Analysis I by Zorich page 263: Choose the constants a, b, and R so that the circle $(x − a)^2 + (y − b)^2 = R^2$ has the highest possible order of contact with the given parametrically defined…
zaknenou
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