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I'd like to integrate a function over a closed circle-like contour around an arbitrary point on a torus and I assume I would use the expression:

$$ \int_{t_1}^{t_2} f(x,y,z) \sqrt{x'(t)^2+y'(t)^2+z'(t)^2}dt$$

And I cannot come up with an explicit parameterization of the variables in terms of t. Here's what I have:

The torus is given by ##\{(R+\rho \cos(\alpha))\cos(\phi),(R+\rho \cos(\alpha))\sin(\phi), \rho \sin(\phi)\}## or ##(x^2+y^2+z^2+b^2-a^2)^2=4b^2(x^2+y^2)##. Let's just use R=2 and ##\rho=1## and let me pick a point on the torus with ##\alpha=\pi/3## and ##\rho=\pi/4##. I now want to draw a small closed circle-like contour around that point on the torus so I'll just use a torus section consisting of a plane slicing through the torus and normal to the surface at the point and a little below it. That plane is given by: $$ n_x(x-a x_1)+n_y(y-a y_1)n_z(z-a z_1)=0$$ with ##a## being a scale factor say ##a=0.9##. The plot below is what that looks like. So where the plane intersects the torus is my blue contour. So that I need $$\int_{\text{blue}} f(x,y,z) \sqrt{x'(t)^2+y'(t)^2+z'(t)^2}dt$$

I can solve for the solution set numerically and then do some curve-fitting for ##x(t),y(t),z(t)##.

So I was wondering if there is a more elegant way to paramaterize x, y and z in this case or how to do the integration in general?

Thanks for reading.

$$ \int_{t_1}^{t_2} f(x,y,z) \sqrt{x'(t)^2+y'(t)^2+z'(t)^2}dt$$

And I cannot come up with an explicit parameterization of the variables in terms of t. Here's what I have:

The torus is given by ##\{(R+\rho \cos(\alpha))\cos(\phi),(R+\rho \cos(\alpha))\sin(\phi), \rho \sin(\phi)\}## or ##(x^2+y^2+z^2+b^2-a^2)^2=4b^2(x^2+y^2)##. Let's just use R=2 and ##\rho=1## and let me pick a point on the torus with ##\alpha=\pi/3## and ##\rho=\pi/4##. I now want to draw a small closed circle-like contour around that point on the torus so I'll just use a torus section consisting of a plane slicing through the torus and normal to the surface at the point and a little below it. That plane is given by: $$ n_x(x-a x_1)+n_y(y-a y_1)n_z(z-a z_1)=0$$ with ##a## being a scale factor say ##a=0.9##. The plot below is what that looks like. So where the plane intersects the torus is my blue contour. So that I need $$\int_{\text{blue}} f(x,y,z) \sqrt{x'(t)^2+y'(t)^2+z'(t)^2}dt$$

I can solve for the solution set numerically and then do some curve-fitting for ##x(t),y(t),z(t)##.

So I was wondering if there is a more elegant way to paramaterize x, y and z in this case or how to do the integration in general?

Thanks for reading.