Slope Distance Coordinates

Figure 1: The md coordinate system

The point m=2, d=4 shown on a graph

Coordinate system that locates a point by the slope of the line that would intersect it and the origin (m), and its distance from the origin (d). This is not much different from polar, just slope instead of angle. Function will be d as a function of m.

Figure 2: d = 10

First graph shows the md representation of d = 10, it winds up being the right half of a circle with radius 10, centered on the origin

Note the small space between what would be the y axis and the graph. The only place an md graph can touch this vertical axis is at zero. It can get arbitrarily close, of course, but this graph's domain was -20 < m < 15

Figure 3: d = m

Where Figure 2 was maybe trivial but still instructive, the graph of d = m is less obvious. The next few figure will illustrate what happens when we show classic x,y graphs on this m,d graph paper.

Figure 4: d = m - 7

Keeping it simple, d = m - 7. This is obviously a very cool graph, and I conjecture that it would make a good Formula 1 curve. The dotted line tracks with the value of m.

Figure 5: d = cos(m)

This graph shows how more and more information is displayed as m -> infinity. It also shows the solutions where d < 0 appear to the left of the m = infinity axis (I'd rather not call it the y-axis).

Figure 6: Trigonometric functions look better when pushed away from the origin, so here's d = sin(5m) + 5

Multiplied the argument by 5 to fit more spring activity between -1 and 1, where fully half of the graph gets drawn in md.

Figure 7: Polynomials make knots

Lots of different polynomials look pretty cool, I chose this cubic. Note it passes through the origin three times, at the appropriate slope value.

Figure 8: Drawing fake x,y graphs in m,d

Figure shows a vertical straight line passing through the m axis at d = 3

Undoing the transformation carefully yields some familiar looking graphs, starting with one that looks like y = 3. Important note: it never touches the 'y' axis, so it differs from y = 3 in that respect.

Figure 9: What about other lines?

Another straight line, this one meant to look like y = 5x + 3

The same idea of having a classic x,y graph appear on the m,d plane.

Figure 10: Adding md flavor to these 'x,y' classics

The vertical line again, but with bumps caused by the addition of an md sin curve. The bumps are arranged towards the origin, so as you look toward the top of the figure, they run together until the line just looks thicker

At this point just trying to make cool looking pictures.

Figure 11: Another spring looking shape

Shows a 'spring' affixed below the origin as in Figure 6, but this time the other end goes right then vertical like in Figure 10

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