# Homework Help: Finding The Coordinates of The Center Of Curvature

1. Jul 3, 2013

### Bashyboy

1. The problem statement, all variables and given/known data
Let C be a curve given by y = f(x). Let K be the curvature ($K \ne 0$) and let $z = \frac{1+ f'(x_0)^2}{f''(x_0)}$. Show that the coordinates $( \alpha , \beta )$ of the center of curvature at P are $( \alpha , \beta ) = (x_0 -f'(x_0)z , y_0 + z)$

2. Relevant equations

3. The attempt at a solution

I attached a picture of the solution. The portion of the solution with a half-box I would appreciate someone helping me with.

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Last edited: Jul 3, 2013
2. Jul 3, 2013

3. Jul 3, 2013

### Bashyboy

No, it is no longer that day in which I learn Linear Algebra. Today is devoted to reviewing multivariable calculus and electricity and magnetism. Tomorrow is when I resume Linear Algebra.

4. Jul 3, 2013

### LCKurtz

With respect to your question in this thread, if f'' > 0 the curve is concave up, so the circle is above the curve. At the point shown, f'' < 0 so the curve is concave down and the circle is below the curve. That determines whether $\beta$ is less or greater than $y_0$.

5. Jul 3, 2013

### Bashyboy

Ooh, you are very right. Thank you much for your help.