This week, we made our last step away from Calculus topics, to something more in the realm of pure maths. Our topic for this week is "Stereo-graphic Projection", which if you do not know what that even means, it is okay! Despite not knowing exactly what this means, yet, you know many examples of this! The major one is what you would see on the globe. This globe is a projection of the world, represented on a sphere. As you may know, the proportions on globes are not always true to relative sizes of countries/continents/oceans, as they easily get distorted with this projection.
A stereo-graphic projection, for a base definition, is a projection from a sphere onto a plane. This is done with some mapping, or function, that takes objects on this sphere onto a plane (and also vis-ver-sa since this map is bijective). Anyway, that this looks like is imagine you are standing at the north pole on the globe. Next, take a light that can shine through the earth, and this light goes to a point on a plane at the base of the south pole. If you turn around, this will create a bunch of different points on the plane, all looking like small little circles (assuming your light source is circular).
Here is a small example of the shining down of the flashlight! Where the blackline is the light source, thus making all of these small points along a line! This example is more of a circle and a line rather than a sphere and a plane, but the premise is the exact same!
Stereographic Projections have some pretty interesting uses in your everyday life, as well as pure maths. As discussed above, your very basic example is a map or globe. The globe is the one that shows how distorted your shapes can be when taking something in 2D space onto 3D space and going backwards. Besides just the globe, your GPS is also a form of this stereographic projection. The GPS is taking some selection of the 3D space of the Earth and wanting to project it onto a 2D-Space, like your phone's map. As technology progresses, the error becomes more minimal, but it is still present. Besides other factors affecting a satellite, this conversion of a 3D space to a 2D resemblance can cause objects given to be off from their natural sizes. In one of the newer IOS updates, the maps now have more depth to buildings, making them appear more '3-D like' on the phone. There is some error again, when you want to show 3D space in 2D space, but this helps some with the projections since while it is still from 3D space to 2D, buildings and objects do not have to be presented as solely 2D.
These sort of projections are what we are tasked to create this week. In order to accomplish our goal, I wanted to project something that resembles a common object, which I felt was fitting with this 'shining' of the light! Thus, I chose to construct a diamond! Since you are shining a light, it seems fitting with the title of this blog. In case you are unfamiliar with the shape of a diamond, below is an example of the shape I wanted to construct.
A Nice Diamond:
This shape is not the worst thing ever to create, but can still cause some problems! In order to create this shape, you need to capitalize on difference() functions with different cubes centered at different points. This allows for us to shave off some edges, getting these flat sides that point off in almost perpendicular directions of one-another. To begin, below is a screenshot of this projection we had discussed from above, basically showing what appears on the plane below the sphere!
Projection:
Next, here is what the actual sphere looks like with the diamond cut outs! You can imagine this sphere with the projection from above, by pictureing a sphere and using sort of 'lasers' to cut out these diamonds from the sphere, with the angle of cutting being the angle to the top of that sphere!
Spherical Diamond:
Please note the little rough parts of the edge is just do in part to the resolution and the lines in openscad making it appear as a rigided.
This example is a nice showcase of taking this higher dimension shape and projecting it onto a 2-D space instead, allowing us to actually observe it. Since maps are our usual examples of such projections, I did not want to just create a mapping of a map, since it was what I wanted to use to explain what these projections are. Instead, I wanted to focus more on finding a shape that is fun to create and look at, connecting back to my discussions of real-life examples of our modelings in past blog posts! Also, with more holidays coming up, you can show your loved one's what you may like by simply shining a light here and showing the diamond!
A stereo-graphic projection, for a base definition, is a projection from a sphere onto a plane. This is done with some mapping, or function, that takes objects on this sphere onto a plane (and also vis-ver-sa since this map is bijective). Anyway, that this looks like is imagine you are standing at the north pole on the globe. Next, take a light that can shine through the earth, and this light goes to a point on a plane at the base of the south pole. If you turn around, this will create a bunch of different points on the plane, all looking like small little circles (assuming your light source is circular).
Here is a small example of the shining down of the flashlight! Where the blackline is the light source, thus making all of these small points along a line! This example is more of a circle and a line rather than a sphere and a plane, but the premise is the exact same!
Stereographic Projections have some pretty interesting uses in your everyday life, as well as pure maths. As discussed above, your very basic example is a map or globe. The globe is the one that shows how distorted your shapes can be when taking something in 2D space onto 3D space and going backwards. Besides just the globe, your GPS is also a form of this stereographic projection. The GPS is taking some selection of the 3D space of the Earth and wanting to project it onto a 2D-Space, like your phone's map. As technology progresses, the error becomes more minimal, but it is still present. Besides other factors affecting a satellite, this conversion of a 3D space to a 2D resemblance can cause objects given to be off from their natural sizes. In one of the newer IOS updates, the maps now have more depth to buildings, making them appear more '3-D like' on the phone. There is some error again, when you want to show 3D space in 2D space, but this helps some with the projections since while it is still from 3D space to 2D, buildings and objects do not have to be presented as solely 2D.
These sort of projections are what we are tasked to create this week. In order to accomplish our goal, I wanted to project something that resembles a common object, which I felt was fitting with this 'shining' of the light! Thus, I chose to construct a diamond! Since you are shining a light, it seems fitting with the title of this blog. In case you are unfamiliar with the shape of a diamond, below is an example of the shape I wanted to construct.
A Nice Diamond:
This shape is not the worst thing ever to create, but can still cause some problems! In order to create this shape, you need to capitalize on difference() functions with different cubes centered at different points. This allows for us to shave off some edges, getting these flat sides that point off in almost perpendicular directions of one-another. To begin, below is a screenshot of this projection we had discussed from above, basically showing what appears on the plane below the sphere!
Projection:
Next, here is what the actual sphere looks like with the diamond cut outs! You can imagine this sphere with the projection from above, by pictureing a sphere and using sort of 'lasers' to cut out these diamonds from the sphere, with the angle of cutting being the angle to the top of that sphere!
Spherical Diamond:
Please note the little rough parts of the edge is just do in part to the resolution and the lines in openscad making it appear as a rigided.
This example is a nice showcase of taking this higher dimension shape and projecting it onto a 2-D space instead, allowing us to actually observe it. Since maps are our usual examples of such projections, I did not want to just create a mapping of a map, since it was what I wanted to use to explain what these projections are. Instead, I wanted to focus more on finding a shape that is fun to create and look at, connecting back to my discussions of real-life examples of our modelings in past blog posts! Also, with more holidays coming up, you can show your loved one's what you may like by simply shining a light here and showing the diamond!
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