What is stereographic projection?
Stereographic projection is a really interesting mathematical concept. With it, you can essentially "map" or project a spere onto a plane. For our purposes, we will be using this concept to project a shape on a 3-D printed spehere onto a wall with light. The way the light passes through the steadily increasing shapes as they spread out from the light source at their respective angles results in a plane with equally projected proportions.
(from: https://mathworld.wolfram.com/StereographicProjection.html)
This picture may be confusing but it is essentially showing where each point on a circle will end up on a flat plane, from 3-D on the sphere to 2-D on the plane/surface. Looking at the top left diagram, the top point N follows a straight line all the way down though O to the very center of the plane at point S. For the rest of the points we will look at the example illistrated from point N to point P'. This shows the point following the line from N to P on the sphere being projected to point P' on the plane. You can see how if you go all the way around the sphere you will be able to map all the points of the sphere onto the plane in the same manner.
You can also imagine for illistrative purposes the sphere melting down from top to bottom and the resulting melted plate is the projection onto the plane. This helps identify how the outer region of the sphere forms the perimiter on the plane, but it may get a little messy when you look at other points.
This cocnept works with the way we map Earth. The Earth's continents are put onto a circular plane to be made into a proportional sphere and vice versa, it can be put onto a sphere and unraveled to form a flat circle. Angles and shapes are maintained although directions can be skewed when doing this.
(from: https://desktop.arcgis.com/en/arcmap/latest/map/projections/stereographic.htm )
When we form a 3-D printed sphere with a design on it and shine a light through it onto a wall, the wall acts as a plane, and the light literally illuminates the concept of how the points go from the top of the sphere through the outer point on the sphere and onto the plane. The resulting image will be a proportional display on the wall with the light and shadows forming equally sized images even though they are different sizes on the sphere.
The design I chose for my plane was a deconstructed golden snitch from the Harry Potter series. I chose this because I was playing with the line possibilities and found that you can curve lines and put lines next to one another to form something similar to mask. When I tried to find a mask to compare it to however I couldn't find one so it just looked like arbitrary lines and curved lines on a ball. When I made some changes I found something that looked like a golden snitch and I tinkered with the lines for a few hours until I got something that looked somwhat like it because I really like the design and concept surrounding the golden snitch visually and its role in the series. The open slots in the sphere for light to pass through form the top left and right open areas of the snitch, and the curving lines are a simplified version of the curving lines along either side of the snitch (their gradient), and the line down the middle increasign in length represents the hourglass center.
(from: https://www.pbteen.com/products/harry-potter-golden-snitch-clock/)
My only limits were that openSCAD kept crashing when I worked with a code that had a light source at the top, and I couldn't figure out how to add reversed lines that mirrored the lines I already had, or rings going around the sphere that didn't intersect with the lines I already had. If I were to add the mirrored lines they would have to follow the stereographic projection form of going from small to big as it diverged from the light source and the result would not have looked like a golden snitch. Similarly, the rings going around the circle would have to be increasing in size instead of being the same size so those were forgoed as well. I removed and reinstalled openSCAD and decreased the number of faces of the sphere from 20 to 10 so it could preview faster, but it only helped temporarily. I was able to open the file and change it from 10 to 20 just in time before it froze after reinstalling openSCAD and will be using a smoother spehere form my print than the one below. The proportions are a sphere of diameter 10 units, but I will scale it in Prusa slicer to be 1.5in.
Stereographic projection is a really interesting mathematical concept. With it, you can essentially "map" or project a spere onto a plane. For our purposes, we will be using this concept to project a shape on a 3-D printed spehere onto a wall with light. The way the light passes through the steadily increasing shapes as they spread out from the light source at their respective angles results in a plane with equally projected proportions.
This picture may be confusing but it is essentially showing where each point on a circle will end up on a flat plane, from 3-D on the sphere to 2-D on the plane/surface. Looking at the top left diagram, the top point N follows a straight line all the way down though O to the very center of the plane at point S. For the rest of the points we will look at the example illistrated from point N to point P'. This shows the point following the line from N to P on the sphere being projected to point P' on the plane. You can see how if you go all the way around the sphere you will be able to map all the points of the sphere onto the plane in the same manner.
You can also imagine for illistrative purposes the sphere melting down from top to bottom and the resulting melted plate is the projection onto the plane. This helps identify how the outer region of the sphere forms the perimiter on the plane, but it may get a little messy when you look at other points.
This cocnept works with the way we map Earth. The Earth's continents are put onto a circular plane to be made into a proportional sphere and vice versa, it can be put onto a sphere and unraveled to form a flat circle. Angles and shapes are maintained although directions can be skewed when doing this.
When we form a 3-D printed sphere with a design on it and shine a light through it onto a wall, the wall acts as a plane, and the light literally illuminates the concept of how the points go from the top of the sphere through the outer point on the sphere and onto the plane. The resulting image will be a proportional display on the wall with the light and shadows forming equally sized images even though they are different sizes on the sphere.
The design I chose for my plane was a deconstructed golden snitch from the Harry Potter series. I chose this because I was playing with the line possibilities and found that you can curve lines and put lines next to one another to form something similar to mask. When I tried to find a mask to compare it to however I couldn't find one so it just looked like arbitrary lines and curved lines on a ball. When I made some changes I found something that looked like a golden snitch and I tinkered with the lines for a few hours until I got something that looked somwhat like it because I really like the design and concept surrounding the golden snitch visually and its role in the series. The open slots in the sphere for light to pass through form the top left and right open areas of the snitch, and the curving lines are a simplified version of the curving lines along either side of the snitch (their gradient), and the line down the middle increasign in length represents the hourglass center.
My only limits were that openSCAD kept crashing when I worked with a code that had a light source at the top, and I couldn't figure out how to add reversed lines that mirrored the lines I already had, or rings going around the sphere that didn't intersect with the lines I already had. If I were to add the mirrored lines they would have to follow the stereographic projection form of going from small to big as it diverged from the light source and the result would not have looked like a golden snitch. Similarly, the rings going around the circle would have to be increasing in size instead of being the same size so those were forgoed as well. I removed and reinstalled openSCAD and decreased the number of faces of the sphere from 20 to 10 so it could preview faster, but it only helped temporarily. I was able to open the file and change it from 10 to 20 just in time before it froze after reinstalling openSCAD and will be using a smoother spehere form my print than the one below. The proportions are a sphere of diameter 10 units, but I will scale it in Prusa slicer to be 1.5in.
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