Indeed, you hit the nail on the head. Or more to the point, on the infinitely thin edges between the nailheads.
The first time I saw this explained clearly was in the original Inside Macintosh, describing the QuickDraw coordinate plane. Some excerpts (with emphasis added to a few key concepts):
# # #
The Coordinate Plane
All information about location or movement is given to QuickDraw in terms of coordinates on a plane. The coordinate plane is a two-dimensional grid, as illustrated in Figure 2.
Note the following features of the QuickDraw coordinate plane:
• All grid coordinates are integers (in the range -32767 to 32767).
• All grid lines are infinitely thin.
These concepts are important. First, they mean that the QuickDraw plane is finite, not infinite (although it's very large). Second, they mean that all elements represented on the coordinate plane are mathematically pure. Mathematical calculations using integer arithmetic will produce intuitively correct results. If you keep in mind that grid lines are infinitely thin, you'll never have "endpoint paranoia"—the confusion that results from not knowing whether that last dot is included in the line.
Points
There are 4,294,836,224 unique points on the coordinate plane. Each point is at the intersection of a horizontal grid line and a vertical grid line. As the grid lines are infinitely thin, so a point is infinitely small.
Figure 3 shows the relationship between points, grid lines, and. pixels, the physical dots on the screen. (Pixels correspond to bits in memory, as described in the next section.)
Rectangles
Any two points can define the top left and bottom right corners of a rectangle. As these points are infinitely small, the borders of the rectangle are infinitely thin (see Figure 4).
# # #
The full PDF is available here (or via a search for "Inside Macintosh"), and it's better with the illustrations:
The description of the QuickDraw coordinate plane is on pages 148-151 (I-138 to I-141).
Figure 3 is especially good. It shows how grid lines and points are infinitely thin/small, but pixels occupy the space between the gridlines.
Disclaimer and shameless plug: My friend Caroline Rose wrote Inside Macintosh, and she still writes. If you want someone who understands technical concepts and can explain them clearly, look her up.
Awesome reference! In fact, I was going to write that the interpretation with positions is "geometric" or "spatial", as opposed to "ordinal", but then I thought the analogy might be a little far-fetched so I left it out.
Pretty soon, I figured out that if Caroline had trouble
understanding something, it probably meant that the
design was flawed. On a number of occasions, I told her
to come back tomorrow after she asked a penetrating
question, and revised the API to fix the flaw that she
had pointed out.
It should be a reminder that to the engineer who wrote the API everything is obvious in retrospect, even the inconsistent or poorly thought details. A fresh set of (smart) eyes is essential to bring some perspective.
Inside Macintosh was an absolutely excellent series of books. I read volume I from front to back in middle school, and it taught me much of my English.
The first time I saw this explained clearly was in the original Inside Macintosh, describing the QuickDraw coordinate plane. Some excerpts (with emphasis added to a few key concepts):
# # #
The Coordinate Plane
All information about location or movement is given to QuickDraw in terms of coordinates on a plane. The coordinate plane is a two-dimensional grid, as illustrated in Figure 2.
Note the following features of the QuickDraw coordinate plane:
• All grid coordinates are integers (in the range -32767 to 32767).
• All grid lines are infinitely thin.
These concepts are important. First, they mean that the QuickDraw plane is finite, not infinite (although it's very large). Second, they mean that all elements represented on the coordinate plane are mathematically pure. Mathematical calculations using integer arithmetic will produce intuitively correct results. If you keep in mind that grid lines are infinitely thin, you'll never have "endpoint paranoia"—the confusion that results from not knowing whether that last dot is included in the line.
Points
There are 4,294,836,224 unique points on the coordinate plane. Each point is at the intersection of a horizontal grid line and a vertical grid line. As the grid lines are infinitely thin, so a point is infinitely small.
Figure 3 shows the relationship between points, grid lines, and. pixels, the physical dots on the screen. (Pixels correspond to bits in memory, as described in the next section.)
Rectangles
Any two points can define the top left and bottom right corners of a rectangle. As these points are infinitely small, the borders of the rectangle are infinitely thin (see Figure 4).
# # #
The full PDF is available here (or via a search for "Inside Macintosh"), and it's better with the illustrations:
http://www.pagetable.com/?p=50
The description of the QuickDraw coordinate plane is on pages 148-151 (I-138 to I-141).
Figure 3 is especially good. It shows how grid lines and points are infinitely thin/small, but pixels occupy the space between the gridlines.
Disclaimer and shameless plug: My friend Caroline Rose wrote Inside Macintosh, and she still writes. If you want someone who understands technical concepts and can explain them clearly, look her up.