[R] Question about banking to 45 degrees.

[Charilaos Skiadas]

On May 20, 2008, at 5:59 PM, Deepayan Sarkar wrote:

> On 5/20/08, Charilaos Skiadas <cskiadas at gmail.com> wrote:
>
>>  Here is how I see it. Let me define a "visual y-unit" as the  
>> height of a
>> unit of data in the y-direction, and similarly for a visual x-unit.
>>  Then the aspect ratio is the quotient of the visual y-unit over  
>> the visual
>> x-unit. So the aspect ratio is the number of visual x-units that  
>> have the
>> same length as one visual y-unit.
>
> [Not that it matters, but it is not clear what you mean here. Let's
> say we have a 100cm x 100cm plot, with data ranges xlim=c(0, 100) and
> ylim=c(0, 200). Then, the aspect ratio is 1, your "visual y-unit" is
> 0.5cm, and "visual x-unit" is 1cm (so their ratio is 0.5).]

So in this case, the slope of the line y=x, which is 1, appears as  
0.5. I effectively wanted to combine the two effects, of the sizes of  
the two scales and of the sizes of the window. They both have an  
effect on how a line of slope 1 is seen. But perhaps I am missing  
something here?

>>  If a line has real (data) slope r, and the aspect ratio is b,  
>> then the line
>> appears with slope rb.
>
> Agreed.
>
>>  Now, there are two things one can compute (for simplicity I  
>> assume all
>> slopes are positive, insert absolute values as necessary):
>>  1. The value of the aspect ratio, that makes the median of the  
>> visual
>> slopes be 1. This would be obtained by requiring the median of all  
>> the rb to
>> be 1, which means that the aspect ratio would be 1/median(slopes).
>>  2. The median of the aspect ratios, that make each individual  
>> line have
>> slope 1. So for each line with slope r, we consider the aspect  
>> ratio 1/r,
>> and then take the median of that. So this would be median(1/slopes).
>
> I agree with your analysis, but would claim that both calculations are
> "right", since the median of 2 numbers is formally any number in
> between.

That's a very good point, I never thought of it that way (though I  
have to say, I haven't seen anything but the arithmetic average used  
in getting "THE median" before).

> I think it is unlikely that the difference in calculations
> leads to any difference in the perceptual benefits.

Agreed.

> Of course, the current calculation has the advantage of doing one less
> division! :-)

For me, that's reason enough to keep it as is ;)

> -Deepayan

Haris Skiadas
Department of Mathematics and Computer Science
Hanover College