# How do i find the arc length of an implicit curve given by f[x,y]=0?

• okkvlt
In summary, the conversation discusses the use of the implicit function theorem and the arc length formula to find the derivative and integrate with respect to x. The topic of closed curves and the difficulty of finding a general formula or algorithm is also mentioned. The possibility of parametrizing any curve, regardless of complexity, is brought up by one of the speakers.
okkvlt
?
i used the implicit function theorem to find dy/dx, then applied that to the arc length formula, but i have to integrate with respect to x and there is the implicit function y[x] inside the radical.
also, if it matters, the curve is assumed to be closed.

That depends strongly on what the specific function is. What is f(x,y) for this problem?

So there's no general formula/algorithm? bummer. there is no information about the curve other than it cannot be parameterized(and trivially, cannot be put into the form y[x]

I am puzzled by this. Every curve can be given some parametric functions. And every curve can be written in terms of piecewise functions. Do you mean simply that you do not know what they are or that they would be very complicated?

in general they would range in complexity. I am writing a program that finds the arc length of a level curve of some function f[x,y]. do you mean every implicit curve has a parametric representation?

I second HallsofIvy assertion that any curve (at least any curve that is implicitly or explicitly defined with analytic functions) can be parametrized.

## What is an implicit curve?

An implicit curve is a mathematical curve that is defined by an implicit equation, where the coordinates of points on the curve satisfy the equation. In other words, the curve is not given explicitly as a function of one variable.

## How do I find the arc length of an implicit curve?

To find the arc length of an implicit curve, you can use the arc length formula: L = ∫√(1 + (dy/dx)^2) dx, where dy/dx is the derivative of the curve with respect to x. This formula can be derived from the Pythagorean theorem.

## What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

## What is the derivative of an implicit curve?

The derivative of an implicit curve can be found by implicitly differentiating the equation with respect to x. This means that you take the derivative of each term in the equation and use the chain rule when necessary.

## Can I use a computer to find the arc length of an implicit curve?

Yes, you can use a computer program or calculator to numerically approximate the arc length of an implicit curve. However, for more complex curves, it may be necessary to use numerical integration techniques to find an accurate approximation.

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