Tutorials - Using Ki coordinates with "real" measurements

This tutorial will show how to use Ki coordinates in conjunction with other measurable assets to produce quality maps.

Setup. We've linked to an area that we wish to map. We immediately notice that there are square tiles on the ground covering a large area, and that we can use these squares as positioning/sizing information. But, first we have to prove that they are square.

So, we drop a firemarble on the ground near the square we want to measure; this helps us to ensure that we are measuring the same square as we go to our notebooks, paint programs, or whatever to write down our observations. Next, using the handy lubberline background, we align one edge of the square with the center of the view, and one edge with the bottom of the view (note to self; this should be more precise, amend the lubberline background with horizontal lines, as well).

Now, to measure the other side. We move ourselves around to the other edge of the square, line this side up like we did the other side, and take readings again.

Comfortable that this is indeed a square, we can pick a local origin for our map and begin mapping. I went over to a point that I think is a pretty good beginning for this floor pattern. Unfortunately, I couldn't get any closer to the wall, but I can see that I'm only about 1.5 squares from it. My Ki reads 60244/997/-71, and I make note of it.

Changing my view a little bit, I can count squares. Here, I've noted that my local origin is the red square, and the square I'm counting to is the blue square, which is at 60242/999/-71. The distance is 14 squares on one axis, and one square on the other. Note that I'm using the smaller squares, here (one light and one dark triangle), not the ones I previously measured (four dark triangles).

Wait a second... These are squares. I should be able to triangulate anything point in the grid created by these tiles!

Now for the real test. I've noted the green square's position as 60256/998/-71. Whipping out my trusty Ki converter, I put together a quick diagram using the coordinates I've already mined.

To get the toran lines for your grid, enter 1 toran to either side of each ki reading and draw the line through the points. For example, the green square I would get grid points for 60256 torans at 997, 998, and 999 torans.

Verifying this becomes a little more difficult. In the triangulation picture above, we know that the shortest possible distance of the hypotenuse of the triangle is the secant of the arc with endpoints at both torans (60242 and 60256) at the shortest distance (999 spans) from Rezeero. Put more simply, this is a line which is pretty close to perpendicular to both torans. So, I draw the smallest possible representation of my mapped area on the graph.

The next step is to simultaneously scale and rotate my map so that the red square is on 60242 torans and the blue one is on 60244 torans. There is a mathematical formula to identify the scaling, but I haven't codified it all, yet. Instead, I took the "map" layer of my work and slid it into place, making sure that my green point stayed in the same place, and my blue and red points ended up on their respective torans. This is roughly analgous to using the green point and the toran lines as LOPs.

At this point, my map has the right attitude and is scaled correctly. I can use planar geometry to find the actual size (in spans, or feet, or whatever) of the tiles, or to calculate any other distances that I'm able to map. Furthermore, I can use these fixed points as LOPs to other objects.

This map also shows me how inaccurate the spans reported by ki can be.

Summary of steps to get an accurate map:

1. Decide your unit square - that one piece of tile or wall hanging or cone that you can easily repeat squares with. This could also be paces in perpendicular directions.
2. Take reference Ki readings - go to a few locations where the unit square repeats and get ki readings.
3. Get toran lines and approximate grid from conversion calculator - enter in the readings you took for each point, and include multiple points on each toran to get your lines.
4. Plot the approximate grid and toran lines.
5. Plot your paced or counted map fragment.
6. Rotate and scale your fragment to line up on the torans it should.
7. Repeat, if necessary. Use this and other methods to complete your map.

Here's another example, this time in the Bevin classroom. I've left out all the steps and provided two simple renders from the Blender 3D environment. The main image is the overhead, ortho view which shows the bright-red tiles, the bluish triangle via Ki coordinates, and you can faintly see the toran lines with markers at the bottom. The purple inset shows a perpsective view from a point somewhere above and west of the classroom. Apologies for the poor color scheme!