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October 20, 2004
Nodal Point Pan and Tile: Part 2
A good way to set up a nodal point pan and tile is to camera project a series of images onto the inner surface of a sphere from a projection point in the exact center of that sphere.
I'm starting with a photograph I took which is full of straight lines going off in various directions. It's a photo of the inside of a parking garage. I've enhanced some of the garage's natural lines in red, and I've added a border in blue around the edges of the image.
Next I place a camera in the center of a sphere of arbitrary radius. Notice how when viewed from the center of the sphere, the longitude lines of the sphere become perfectly straight vertical lines. This is because longitude lines are great circles, and when we are in the center of the sphere, we are also in the center of each of those longitude circles, so they aim edge-on directly at us and therefore appear to be lines.
When I use camera projection to place the image of the parking garage onto the inner surface of the sphere, the image appears curved. The red lines appear curved, and the blue border lines also appear curved.
The only reason they appear curved, however, is because I am viewing the image from a place other than the center of the sphere.
If I were to go to the center of the spere, all the lines would appear straight.
Here is an animation of the curved frame wandering around on the curved sphere surface. Notice that from this vantage point in the center of the sphere, it looks like a simple rotated rectangle.
Here's the exact same animation as seen from a point other than the center of the sphere. From here you can see what is really going on.
Why do the curved red and blue lines appear straight when viewed from the center of the circle? Because when projected, they become great circles.
I lined up the edges of some purple circles along a couple of the red lines, just to show how the lines are, in fact, great circles.
Remember, the idea of a great circle is that the 2D center of the circle is in the same place as the 3D center of the sphere.
UPDATE 12/27/2004:
Here's a guy named
Dick Termes who goes out of his way to make sure the viewer is not at the center of the sphere:
http://www.boingboing.net/2004/12/27/spherical_paintings.html
The idea that points in space can be represented as if viewed from the center of a sphere has formed the basis of Astronomical observation since at least the time of Aristotle.
Posted by digital artform at October 20, 2004 07:42 PM