In this animation you can see a moving blue curve and a fixed red curve. Moreover, the blue curve projects, while moving, its shadow on the ground and on the right wall. These two shadows should help you in visualizing the position of the curve in the laboratory. The blue curve describes the surface we were considering. The red, fixed curve (and its fixed shadow on the ground) describes the intersection of our laboratory with the plane of the four dimensional space given by the points whose coordinates satisfy the equations: u=0, v=0. Our laboratory is travelling along the z axis, therefore its intersectionwith the plane of equation u=0, v=0 is a fixed line. If you carefully look at the animation, you see that the blue line meets the red line twice (the first time at the beginning of its motion, the other at the end of the motion). Which is the meaning of these two intersections? First of all, they are the points of our surface that meet the plane u=0, v=0, but recalling that y was defined as u + i v, they are the two (complex) zeros of the equation x² - 3 x + 10 = 0. In other words, now we can see the two solutions
of this equation!
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