mathmo y axis bounds about apps1

[gmp26]

A few related problems here which are due to the fact that we have a pretty dumb sampling routine at the moment. Ideally we want an adaptive sampler.

For an xy plot, Mathematica does this:

1. subdivide domain equally into (PlotPoints - 1) intervals
2. look at each 3 successive points ABC
3. If angle ABC > MaxBend, also plot midpoints of AB and BC
4. Terminate the subdivision when interval length is less than 1/(PlotPoints * PlotDivision)

Probably best to use differences between Math.atan2 over AB and BC to find angle ABC. atan2 values can be cached for use in the next triple.

atan2 has a branch at dy/dx == 0, so near horizontal line segments must test for near equalty or a diff of near ±2π.

An atan2 diff near ±π indicates a singular point. A null data value should be inserted at the singularity so d3 can chop the line and so avoid drawing a vertical stroke.

Refs:

1. Mathematica book Plot, PlotPoints, PlotDivision, MaxBend
2. http://demonstrations.wolfram.com/AdaptivePlotting/
3. http://scicomp.stackexchange.com/questions/2377/algorithms-for-adaptive-function-plotting

[gmp26]

Need a similar algorithm for a parametric plot.

[gmp26]

NB, this algorithm could generate lists of interesting plot features - like assymptotes, and turning values. These could
be indicated visually so student is prompted to label them.