I consider 2 triangles $T_1$ and $T_2$ in a plane (as surfaces and not circumferences) What is the geometrical shape of the set S of points P=(w,z) such that there exists 2 points : $P_1=(x_1,y_1) \in T_1$ and $P_2=(x_2,y_2) \in T_2$ such that $w=x_1x_2$ and $z=y_1y_2$ ?
I don't know how to deal with this problem. I think the products of the vertices of the triangles should play the role of some kind of vertices in the searched set. But for the line segment, there is an infinity of way to chose the points. So I am a little bit puzzled
First of all, your transformation does not correspond to any particularly well behaved geometric transform, as it is not invariant to rotation. Component-wise product is not a vector operation.
Consider a "product" of a line segment with a point. The point here represents a stretch in both directions by some factor. So edge × point=edge. So consider edge × edge. Multiplying a vertex of either edge with the other edge gives you one of 4 straight edges that share vertices, so edge × edge = quadrilateral.
A triangle is a convex set with straight edges. A product of two edges we have seen is also a convex set with straight edges. As a triangle can be thought as a convex polygon delimited by 3 straight edges, so the product of two filled triangles will be a convex hull, delimited by the product of the edges. There are 9 combinations of vertex × vertex products. Compute all of them and find the convex hull of the result to obtain your final set. Notice that rotation of the coordinate frame changes the shape of the result. As some vertices may be collinear, or lie in the bulk of the shape, the number od vertices of the final set may vary.