Curvature and Osculating circle

• bfr
In summary, the conversation was about finding a formula for the curvature of a given curve and writing an equation for the osculating circle at t=0. The formula for curvature was provided, but there was some confusion about its simplification. The final equation for the osculating circle was derived using the given formula for curvature and the coordinates of the curve at t=0.
bfr

Homework Statement

Find a formula for the curvature of the curve:

x=(e^t + e^(-t))/2
y=(e^t - e^(-t))/2

Write an equation of the osculating circle when t=0.

Homework Equations

curvature=|x'y'' - x''y'|/(x'^2 + y'^2)^(3/2)

The Attempt at a Solution

First, wouldn't the formula for curvature be:
sqrt(2)
---------
(e^(2t) + e^(-2t))/2 * sqrt(e^(2t) + e^(-2t))

? But for some reason, my teacher marked a big "X" across this and wrote question marks.

EDIT: Nevermind about the first part of the question - I just should have simplified it more.
Now, for the equation of the osculating circle when t=0, I'd get the center of the circle by doing: <x_center,y_center>=<x(0)-K*y'(0)/sqrt(x'(0)^2+y'(0)^2),y(0)+K*x'(0)/sqrt(x'(0)^2+y'(0)^2)>

where K=curvature?

Last edited:
Is this correct?EDIT 2:I think I got the equation of the osculating circle at t=0 as (x-3)^2+(y-1)^2=4.

1. What is curvature?

Curvature is a measure of how much a curve deviates from a straight line. It is defined as the rate of change of the tangent line to the curve at a given point.

2. How is curvature calculated?

Curvature can be calculated using the formula: k = |d𝛼/ds|, where k is the curvature, d𝛼 is the change in the angle of the tangent line, and ds is the corresponding change in the arc length of the curve.

3. What is an osculating circle?

An osculating circle is a circle that best approximates a curve at a given point. It has the same curvature as the curve at that point, and shares the same tangent line and direction of curvature.

4. How is the osculating circle related to curvature?

The osculating circle is closely related to curvature, as it is defined by the curvature of a curve at a specific point. It can be thought of as the circle that "kisses" the curve at that point, and provides a way to visualize the curvature of the curve at that point.

5. What are some applications of curvature and osculating circles?

Curvature and osculating circles have various applications in fields such as physics, engineering, and computer graphics. They are used to analyze the shapes of objects, design smooth and efficient curves for roads and roller coasters, and create realistic animations in video games and movies.

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