Spherical Slope Field

How do you portray an infinite plane using a finite object? Stereographic projection, that's how. Stereographic projection makes use of a one-to-one and onto function with an infinite plane as its domain and a finite sphere (minus the top point) as its codomain. The infinite plane gets mapped onto a sphere by taking the point on the sphere that is furthest away from the plane and connecting it to any point on the plane with a line. The point on the sphere that the line intersects is the corresponding point to the point on the plane. Because the plane gets infinitely shrinked as it reaches the top of the sphere, the entire plane gets mapped to the sphere. If there is something graphed on the plane and you wanted it to be mapped onto the sphere, all you would need to do is connect every point on the graph to the top of the sphere and display the points of intersection on the sphere. Because the function is one-to-one and onto, the inverse exists, too. Take the top point of the sphere and any other point on the sphere, connect a line between them and follow the line until it hits the plane. That point on the plane is corresponding to the point on the sphere. If there were a pattern on the plane that was mapped onto the sphere, the pattern would get smaller and more warped on the sphere as you moved upwards. And conversely, a pattern on the sphere mapped onto the plane gets larger and more warped as you move away from the sphere on the plane. I used this aspect of stereographic projection as a basis for my model.
  I decided to map a slope field onto my sphere. A slope field is a visualization of solutions to a first order differential equation. The solutions to this slope field all have parallel asymptotes as they make parabola like shapes.

The slope field includes 121 arrows and you can see three of the asymptotes. Because the infinite plane can be mapped onto a sphere, this slope field can also be mapped onto a sphere. The solutions still have asymptotes, but they are now curves on the sphere and the parabola like shapes are also curves.

Some of the challenges with making this model was spacing the arrows and choosing a point to connect the arrows to in order to make enough space between the arrows on the sphere. I wanted enough arrows to show multiple asymptotes, but too many arrows would cause the arrows on the sphere to merge near the sides. Also, because the arrows are not convex shapes, a hull was needed for each triangle and rectangle separately making up each arrow. This design demonstrates how the visualization of solutions to a first order differential equation could be portrayed even on a sphere. It also shows that when mapping a square graph onto a sphere, the sides of the square become curved and no longer look like a square. Interestingly enough, if one were to shine a light from the top of the hemisphere, the slope field with unwarped arrows of equal size would be projected onto whatever surface the light was aimed at.