I was recently presented this in differential geometry class which I cannot seem to solve involving curve of intersection of two surfaces in $ R^3 $ which reads as
Let us define two surfaces in $ R^3 $ as follows:
$ S_1 $ : $ x^{11} + y^{10} + z^9 + x ^ 3 y ^ 2 + x^2 y^2 + y^3 + z^2 + x^2 +y^2 + x +10y + 2z = 0 $
$ S_2 $ : $ x^4 -y^2 z^2 + x^3 -y^2 z + z = 0 $
We are asked to find, on the curve of intersection of these two surfaces at point $ O := (0,0,0) $ its curvature, its geodesic curvature relative to S1 and its normal curvature relative to S2
My trouble starts as the equations are too algebraically complicated so no hope of extracting anything directly, also I know the formulas for these quantities and how to use them but my problem is I cannot extract anything from this equations and that's where I am stuck and someone to show me how to get the desired quantities Thanks all
Uhm, if the surfaces are smooth and if at the origin you can express both of them as $z=S_i(x,y)$, then, using implicit differentiation, you can compute the tangent planes in the origin, and find the direction of the tangent line to the intersection line simply intersecting those planes. Furthermore, the gradients of the surfaces give you the basis in the tangent space, so that you should be able to build the first and second fundamental forms at the origin and proceed from there.