Animations of traces

The following are animations of the x,y, and z traces being taken of various surfaces.

  • This is the sphere, defined by x^2+y^2+z^2=1. Notice that all three traces are families of circles. For the x traces, the radius of a circle in a given trace is \sqrt{1-x^2}. This explains why there is no trace when x^2>1, then grow from 0 to 1 along -1\leq x \leq 0 and then shrink from 1 back to 0 along 0\leq x \leq 1. By symmetry, identical properties hold when x is replaced by y or z.
    • sphere
  • This is the hyperboloid of one sheet, defined by x^2+y^2-z^2=1. Notice that in the x and y traces, one sees a family of hyperbolas whereas in the z traces, one sees a family of circles.
    • hloid1
  • This is the hyperboloid of two sheets, defined by x^2-y^2-z^2=1. By contrast to the hyperboloid of one sheet, the y and z traces are families of hyperbolas, whereas the x traces are a family of circles. Notice that there is a region of x values for which the trace does not intersect the hyperboloids, corresponding the the non-existence of a solution to y^2+z^2=k^2-1 when k is between 0 and 1.
    • hloid2