# Converting intrinsic equation to cartesian

In summary, the conversation discusses the intrinsic and cartesian equations of a curve, specifically the equation s = 12(sin φ)² and its corresponding cartesian equation (8-x)^(2/3)+y^(2/3)=4. The conversation also includes equations for dy/dx, dy/ds, and dx/ds, and the process of finding the cartesian equation from the intrinsic equation. The solution presented results in the equation y^(2/3)-x^(2/3)=4, which is close to the correct answer but lacks the constants of integration.

## Homework Statement

Intrinsic eqn of a curve is $s = 12(sin \varphi)^{2}$ where $s$ is length of arc from origin and $\varphi$ is angle of tangent at a point with x axis.

Show the cartesian eqn is $(8-x)^{\frac{2}{3}}+y^{\frac{2}{3}}=4$

## Homework Equations

^{}

$$\frac{dy}{dx}=tan\varphi$$
$$\frac{dy}{ds}=sin\varphi$$
$$\frac{dx}{ds}=cos\varphi$$

## The Attempt at a Solution

$$s=12(\frac{dy}{ds})^{2}$$

$$y=\frac{1}{2\sqrt{3}}\int(s^{\frac{1}{2}})ds$$

which comes out as:$$3y^{\frac{2}{3}}=s$$

now doing the same process for x:

$$s=12(1-(\frac{dx}{ds})^{2})$$

$$x=\int(1-\frac{s}{12})^{\frac{1}{2}}ds$$

$$x=-8(1-\frac{s}{12})^{\frac{3}{2}}$$

which comes out as:$$s=12+3x^{\frac{2}{3}}$$

so now equating the two equations for s i get
$$y^{\frac{2}{3}}-x^{\frac{2}{3}}=4$$

this obviously isn't right so where am i going wrong?!

Last edited by a moderator:
Seven years too late of course - still worth answering i suppose: I am no expert but I looked through and nothing you appear to have done is wrong. It occurs to me that the two results (correct answer and our answer) are very similar, one is simply shifted to the left of the other. I think you have to remember the constants of integration.

x infact equals -8(1 - s/12)^3/2 + c
so
(c - x) = 8(1 - s/12)^3/2
so
s = 12 - 3(c - x)^2/3
Equating y and x:
4 = (c - x)^2/3 + 3y^2/3(where c appears to be 8). It just so happens that the y constant of integration equals 0.

One intrinsic equation can have more than 1 cartesian equivalent.
Those are my thoughts at least - hope this helps.

## 1. What is an intrinsic equation?

An intrinsic equation is a mathematical representation of a curve or surface that does not depend on any external coordinate system. It describes the shape and properties of the object in its own intrinsic coordinates.

## 2. Why would you need to convert an intrinsic equation to cartesian?

Cartesian coordinates are the most commonly used system for representing mathematical equations and physical quantities, making them more intuitive and easier to work with. Converting an intrinsic equation to cartesian allows for easier visualization and analysis of the object.

## 3. How do you convert an intrinsic equation to cartesian?

To convert an intrinsic equation to cartesian, you first need to determine the relationship between the intrinsic coordinates and the cartesian coordinates. This can be done by using geometric or trigonometric properties of the object. Once the relationship is established, you can use it to rewrite the intrinsic equation in terms of cartesian coordinates.

## 4. Can any intrinsic equation be converted to cartesian?

Yes, any intrinsic equation can be converted to cartesian, as long as the necessary relationship between the coordinates can be determined. However, the resulting cartesian equation may not always be in a simple or familiar form.

## 5. Are there any challenges in converting an intrinsic equation to cartesian?

Yes, there can be challenges in converting an intrinsic equation to cartesian, especially if the relationship between the coordinates is complex or not well-defined. Additionally, the resulting cartesian equation may be difficult to interpret or manipulate, depending on the complexity of the original intrinsic equation.

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