Osculating Circle and Osculating Sphere





Osculating Circle

Let \( C:\vec{r}=\vec{r}( s )\) be a space curve with neighboring points \( P,Q,R\) on \( C\) , then osculating circle at \( P\) is defined as limiting position of circle through \( P,Q,R\) as \( Q,R\) approaches to\( P\) .
In mathematical notation
Osculating circle at \( P = \displaystyle \lim_{Q,R\to P} \)circlePQR

Note:
  1. Osculating circle lies in the osculating plane
  2. Osculating circle at \( P\) has in general three point contact with the curve at \( P\) .
  3. Tangent of osculating circle and space curve are same at point of contact

Osculating circle at P lies in the osculating plane at P, and osculating circle at P has three point contact with the curve at at P.




Center of Osculating Circle

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Properties on Osculating Circle

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Osculating Sphere

Let\( C:\vec{r}=\vec{r}( s )\)be a space curve. If \( P,Q,R,S\) are four neighboring points on\( C\), then osculating sphere at\( P\)is defined as limiting position of sphere through\( P,Q,R,S\) as\( Q,R,S\) approaches to\( P\).
\( \displaystyle \lim_{Q,R,S\to P} \)= spherePQRS= Osculating sphere at \( P\)

Osculating sphere at P has four point contact with the curve at at P.




Center of Osculating Sphere

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Properties on Osculating Sphere

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Theorems

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Relation of Center of Osculating Circle and Osculating Sphere




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