Equation of oscillating circle

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SUMMARY

The discussion focuses on finding the equation of the osculating circle for a given curve. The user has calculated the radius of curvature but is uncertain about deriving the equation of the circle. Key insights include the necessity of using the normal vector from the tangent line to determine the circle's center. The conversation highlights the importance of correcting differentiation errors to ensure accurate calculations.

PREREQUISITES
  • Understanding of calculus, specifically differentiation and curvature
  • Familiarity with the concept of osculating circles
  • Knowledge of vector mathematics, particularly normal vectors
  • Basic skills in analytical geometry
NEXT STEPS
  • Study the process of deriving the equation of an osculating circle
  • Learn about radius of curvature and its applications in geometry
  • Explore the use of normal vectors in curve analysis
  • Review differentiation techniques to avoid common errors
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are working on curve analysis and geometric applications, particularly those interested in osculating circles and curvature concepts.

sakkid95
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http://gyazo.com/bc7f6da4d4c4aca300bb5efb2410fc8d.png


The problem, and my solution are in the image. I need help on finding the equation of the osculating circle, I found radius but I don't know where to go from there. Also if you could just check if my math seems correct that would help me too, but my main problem is how I find the eqn for the circle... thanks in advance
 
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You made a mistake differentiating t3.
You have the radius of curvature; from the tangent you can find the normal vector.
 
Yea I noticed the error, I'm working on fixing it now. Can you elaborate more on how to find the radius? I'm not very good at this stuff..
 
sakkid95 said:
Yea I noticed the error, I'm working on fixing it now. Can you elaborate more on how to find the radius? I'm not very good at this stuff..

You said you had the radius. If so, all you need is a line the circle's centre must lie on. The normal to the curve is that line.
 

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