Questions tagged [osculating-circle]
For questions about osculating-circles, Descartes Theorem, Radius of Curvature, and evolutes.
40 questions
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The set of centers from which some circle has $4$ intersections with $y=x^3-ax$ expands to cover the whole plane as $a → ∞$
For all $a\in\Bbb R$ let $S_a$ be the set of centers from which some circle has $4$ intersections with the graph of $y=x^3-ax$.
For example, in the image, $(1,0)$ is the center of a circle which has $...
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Maximum curvature of a Quintic Bezier curve
I am interested in solving the following problem.
Let's say we have a quintic bezier curve p(t) = {p0, p1, p2, p3, p4, p5}. I also have some curvature bound Kc. How can I find that the maximum ...
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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 ...
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How does the curvature relate to the osculating circle [duplicate]
In this image an osculating circle at point P is given. I understand this circle visually as a circle fitting the curve at point P most "snugly". But I want to understand it more clearly ...
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Osculating circle of ellipse
The osculating circle of ellipse at point $A$ interects the ellipse at $A,D$. I want to prove the tangent line $AB$ and the line $AD$ form equal angles with axis of the ellipse.
My attempt:
The ...
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Intersection of osculating circle with curve
Let $T$ be a planar curve that does not contain any circular arcs. Let $C$ be its osculating circle at the point $P$. Let $O$ be an intersection of $C$ with $T$ in a place other than $P,$ if it exists,...
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Two Definitions of Curvature for Plane Curves
Suppose we have a plane curve $C: I \rightarrow \mathbb{R}^2$ that is continuously differentiable ($C^1$) and parameterized by arc length. Tangent lines exists at each point on the curve, so an &...
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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 ...
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Why is $\arctan \left(\frac{\cos(\frac{\alpha}{2}t)-\cos(\frac{\alpha}{2})}{\sin(\frac{\alpha}{2}t)-t*\sin(\frac{\alpha}{2})}\right)$ a line?
Why is this arctangent a line on the interval $t = (-1, 1)$
$$
\arctan \left(\frac{\cos(\frac{\alpha}{2}t)-\cos(\frac{\alpha}{2})}{\sin(\frac{\alpha}{2}t)-t*\sin(\frac{\alpha}{2})}\right)
$$
I don't ...
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Proving the curvature formula for an arbitrary planar curve using perpendicular bisectors
Given an arbitrary (i.e. not necessarily arc-length parameterised) planar parametric curve $C(t) = \Big(x(t), y(t)\Big)$, I'm looking to prove the formula for its (signed) curvature
$$\kappa = \frac{x'...
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How to prove that a spiral that I have is logarithmic or archimedean?
I am conducting a research on modelling a spiral.. I know that the shape of the spiral on my pencil shavings is logarithmic indeed, How do i prove that? How do I prove it is logarithmic and not ...
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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 ...
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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 ...
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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$, ...
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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 ...
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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 ...
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The Path of the center of a cutting tool that will cut out an ellipse (x/4)2+(y/2)2=1
I am currently stuck on this question and it's implementation on Mathematica.
You are the chief engineer at the Badger Steel Plate Company in Madison, Wisconsin. In comes an order for 750 square steel ...
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Queer Case of Circle
Excuse me.
Let in $\mathbb{R}^2$ there is a circle $\{(x,y)\in\mathbb{R}^2~|~x^2+y^2=r^2\}$.
Then, let the position of the point $A$ in this space is $\mathbf{a}=r(\mathbf{i}\cos\alpha+\mathbf{j}\sin\...
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Trouble finding osculating circle
I'm presented with finding the equation of the osculating circle at the local minimum of $\mathbf f(x) = 3x^3-9x^2+5x-1 $.
Finding the local minimum wasn't that hard; I take the first derivative of $\...
user777589
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Osculating circle and its relation with level site
Find the osculating circle $C$ of the parabola $x^2+y=0$ at the origin $(0,0)$. Find a function $f(x,y)$ such that $C$ is a level curve of $f$.
What I have done so far was to find the quadratic ...
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Osculating circle with level curve of function
Find the osculating circle $C$ of the parabola $x^2+y=0$ at the origin $(0,0)$. Find a function $f(x,y)$ such that $C$ is a level curve of $f$.
Please, solve the problem completely with precise ...
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Finding an Osculating Circle
I am working on a problem with the following directions:
Find the osculating circle at the given points: $r(t)=<t,t^3> at $t=1$.
This image is my work so far. I cannot figure out how to find ...
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Parametrized curve along a sphere
If we parametrized a $C^{\infty}$ curve in $\mathbb{R^3}$ so that the curve lies on a sphere centred at an arbitrary point and the speed along that curve is never zero, how would we show that for any ...
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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 ...
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Nonzero Curvature implies local noncollinearity of any three points.
Let $\gamma:(a,b)\rightarrow\mathbb{R}^2$ be a regular smooth plane curve. Assume that the signed curvature of $\gamma$ at $t_0\in (a,b)$, i.e. $\kappa_*(t_0)$ is nonzero. Then there exists a ...
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Find circumradius of an equilateral triangle of side 7$\text{cm}$
I know that each length is 7 cm but how would I use that to work out the radius.
Thank you and your help is appreciated.
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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 ...
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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 ...
user596656
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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:
$\...
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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 ...
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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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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 $r(t)= <\cos(7t),\sin(7t),7t>$ at $t=0$
So I found the center of the osculating circle by calculating the radius of curvature and the normal ...
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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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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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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 ...
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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 ...
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