# Radius of curvature formula derivation

• chandran
T}In summary, the radius of curvature for a curve defined by y=f(x) is given by [f''(x)/(1+f''(x))]^(3/2). To derive this formula, one can simplify the vector function for curvature and use the fact that curvature is equal to the derivative of the unit tangent vector. This can be further simplified to find the desired formula. A more detailed explanation can be found at DGC.
chandran
for a curve defined by y=f(x) the radius of curvature is defined as
[f""(x)/(1+f"(x))] power 3/2. I need a good neat & understandable derivation for that. can anybody show a web.

Differential Geometry of Curves

I don't like the format of the forum reply, so click on the following link to view your derivative: DGC.

Well first understand that curvature for a vector function is given by:

$$\kappa=\frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|^3}$$

Now, let r = xi + f(x)j and simplify. To prove the first formula $\kappa$, use the following fact and compute r' x r''. The answer should be clear from there.

$$\kappa=\frac{d\mathbf{T}}{ds}$$

$$\mathbf{r}'=\frac{ds}{dt}\mathbf{T}$$

## 1. What is the formula for the radius of curvature?

The formula for the radius of curvature is given by R = (1 + (dy/dx)^2)^(3/2) / (|d^2y/dx^2|), where y is the function and x is the independent variable.

## 2. How is the radius of curvature calculated?

The radius of curvature can be calculated by taking the second derivative of the function and plugging it into the formula R = (1 + (dy/dx)^2)^(3/2) / (|d^2y/dx^2|).

## 3. What is the significance of the radius of curvature in mathematics?

The radius of curvature is an important concept in differential calculus and is used to measure the curvature of a curve at a specific point. It helps in understanding the shape and behavior of a curve and is often used in optimization problems.

## 4. Can the radius of curvature be negative?

No, the radius of curvature cannot be negative. It is always a positive value, as it represents the distance between the center of curvature and the curve at a given point.

## 5. How is the radius of curvature related to the curvature of a curve?

The radius of curvature and the curvature of a curve are inversely proportional to each other. This means that as the radius of curvature decreases, the curvature of the curve increases and vice versa. A small radius of curvature indicates a highly curved curve, while a large radius of curvature indicates a less curved curve.

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