Answer to Question #271916 in Differential Geometry | Topology for Angel Nodado

Question #271916

Find the curvature, the radius of curvature and the center of curvature having a parametric equations at the given point:



y=e^x , (0,1)


1
Expert's answer
2021-12-01T14:04:55-0500

Solution;

"y=e^x"

"P(x_0,y_0)=(0,1)"

"y'=e^x"

"y'(0)=1"

"y''=e^x"

"y''(0)=1"

Radius of curvature;

"R=\\frac{(1+y'(x_0)^2)^{\\frac32}}{y''(x_0)}"

By substitution ;

"R=\\frac{(1+1)^{\\frac32}}{1}=\\sqrt{2^3}=2\\sqrt2"

The curvature;

"k=\\frac1R=\\frac{1}{2\\sqrt2}"

The centre of curvature;

Let P be the center of curvature , given as (a,b)

"a=x_o-\\frac{\\frac{dy}{dx}[1+(\\frac{dy}{dx})^2}{\\frac{d^2y}{dx^2}}"

"a=0+\\frac{e^{2x}}{e^x}=e^0=1"

"b=y_o+\\frac{1+(\\frac{dy}{dx})^2}{\\frac{d^2y}{dx^2}}"

"b=1+\\frac{1+e^{2x}}{e^x}=1+2=3"

Therefore;

P=(1,3)


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