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$\mathbb N = \{1,2,3,4\dots\}$

Let us suppose we are starting at a point with coordinates $(0,0)$. Now draw a line from $(0,0)$ to $(1,0)$ and from $(1,0)$ to $(1,2)$. Now by the Pythagorean theorem, I will draw the hypotenuse i.e. a line from $(0,0)$ to $(1,2)$ of length $\sqrt{5}$. Now I will draw a line of length $3$ from the last point $(1,2)$ and perpendicular to the last hypotenuse we got. It will give another point, suppose $(x_1,y_1)$. Now again I will draw the hypotenuse and again draw a perpendicular line of length $4$ from this hypotenuse and so on. The values of length of the lines are coming from the set $\mathbb N$ of natural numbers.

Now the question is: How can I find out the equation of the curve satisfying the points $(0,0), (1,0), (1,2) , (x_1,y_1) , (x_2,y_2), \dots$ ?

It seems that the curve will be a spiral, but again I don't know how to find out the equation.

A similar figure can be -

enter image description here

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    Probably I missed a point here. Do you want to calculate a smooth interpolation of your spiral? I know, I merely showed the calculation of the single points...2012-08-20
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    Exactly, I think you got the point what I wanted.2012-08-20
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    For future reference, rather than saying "curves satisfy the points", I think you should be saying "points satisfy the equation (or curve)". I know these relationships can get turned around across language barriers sometimes :)2012-08-20
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    @rchwieb : thanks for the suggestion, surely ... It was my mistake2012-08-20
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    Surely the construction looks more like [this](http://i.stack.imgur.com/rFueE.png) than the figure in your question?2012-08-20

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