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I am very interested in a certain problem and I am wondering what methods currently exist to solve it. Given a curve defined by a function which maps any given arc length, s, from an arbitrarily chosen original point on the curve (so s designates the arc length from the original point to another point on the curve), to a certain tangential angle, theta, what is the area enclosed by this curve if said curve is a Jordan curve?

Edit: More specifically, if a curve, A, is defined by the aforementioned function, and this curve is closed (it returns to its original point after some finite arc length), what is the area within the entire curve. For instance, with this construction, we have a Jordan curve. As the Jordan Curve Theorem states that a Jordan curve divides the plane into two sections, an inner section and an outer section, what is the area of the inner section define by the previous curve?

Edit 2: I understand that you can use Green's Theorem for this solution, but as for many different curves described in the aforementioned way, this would be a difficult, messy, and not very elegant way to solve the problem. Is there a simpler solution?

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    What's the significance of the title "Research Advice"?2011-11-12
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    @Srivatsan I am researching this problem and I need guidance.2011-11-12
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    This strikes me as something one could compute using line integrals and standard multivariable calculus techniques. However, I am unclear as to your exact meaning.2011-11-12
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    Do you mean to measure the area bounded by a portion of a curve and a secant line (a line from your initial point to a point $s$ arc length units down the curve)?2011-11-12
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    Could you give an example?2011-11-12
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    @BillCook More specifically, if a curve, A, is defined by the aforementioned function, and this curve is closed (it returns to its original point after some finite arc length), what is the area within the entire curve. For instance, with this construction, we have a Jordan curve. As the Jordan Curve Theorem states that a Jordan curve divides the plane into two sections, an inner section and an outer section, what is the area of the inner section define by the previous curve?2011-11-12
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    You can use [Green's Theorem](http://en.wikipedia.org/wiki/Green%27s_theorem). Towards the bottom of the wiki page there is a formula for computing the area enclosed by a curve using the curve.2011-11-12
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    @BillCook Thank you. But what if the curve is not a function of x or y and is simply described by the set of tangent angles at different arc lengths (so that the curve could be rotationally or translationally transformed and it would be the same) and it did not necessarily have continuous derivatives (and if this curve was not defined by x or y, then the derivative with respect to x or y would not say anything anyway). An example of this particular phenomena would be that of a triangle, but I am not predominantly interested in this specific problem (of non-differentiable curves).2011-11-12
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    So you are starting with knowledge of the [curvature](http://en.wikipedia.org/wiki/Curvature) of a curve (which determines a planar curve up to a rotation and translation). This should leave you with differential equations whose solution is a parametrization of your curve. This then could be plugged into the area formula from Green's. Maybe an expert in differential geometry knows a more direct route.2011-11-12
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    Also, continuity of the derivative is not necessary for applying Green's. I'm not sure what the minimal criterion are, but I think piecewise $C^1$ works.2011-11-12
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    @BillCook Thank you. I understand that it is possible to do it this way, but it seems that this would be a rather messy solution that would not be elegant. I wonder if there is a method which uses solely the arc length/tangential angle definition of the curve to find the enclosed area as opposed to converting it through a rather long, tedious process so that another method acting on a different definition of the curve can be used.2011-11-12

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