Skip to main content

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.

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
Post a Comment
Read more

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...
Post a Comment
Read more