Algorithm for "nice" grid line intervals on a graph

Question

I need a reasonably smart algorithm to come up with "nice" grid lines for a graph (chart).

For example, assume a bar chart with values of 10, 30, 72 and 60. You know:

Min value: 10 Max value: 72 Range: 62

The first question is: what do you start from? In this case, 0 would be the intuitive value but this won't hold up on other data sets so I'm guessing:

Grid min value should be either 0 or a "nice" value lower than the min value of the data in range. Alternatively, it can be specified.

Grid max value should be a "nice" value above the max value in the range. Alternatively, it can be specified (eg you might want 0 to 100 if you're showing percentages, irrespective of the actual values).

The number of grid lines (ticks) in the range should be either specified or a number within a given range (eg 3-8) such that the values are "nice" (ie round numbers) and you maximise use of the chart area. In our example, 80 would be a sensible max as that would use 90% of the chart height (72/80) whereas 100 would create more wasted space.

Anyone know of a good algorithm for this? Language is irrelevant as I'll implement it in what I need to.

Answer 1

I've done this with kind of a brute force method. First, figure out the maximum number of tick marks you can fit into the space. Divide the total range of values by the number of ticks; this is the minimum spacing of the tick. Now calculate the floor of the logarithm base 10 to get the magnitude of the tick, and divide by this value. You should end up with something in the range of 1 to 10. Simply choose the round number greater than or equal to the value and multiply it by the logarithm calculated earlier. This is your final tick spacing.

Example in Python:

import math

def BestTick(largest, mostticks):
    minimum = largest / mostticks
    magnitude = 10 ** math.floor(math.log(minimum) / math.log(10))
    residual = minimum / magnitude
    if residual > 5:
        tick = 10 * magnitude
    elif residual > 2:
        tick = 5 * magnitude
    elif residual > 1:
        tick = 2 * magnitude
    else:
        tick = magnitude
    return tick

Answer 2

There are 2 pieces to the problem:

1. Determine the order of magnitude involved, and
2. Round to something convenient.

You can handle the first part by using logarithms:

range = max - min;  
exponent = int(log(range));       // See comment below.
magnitude = pow(10, exponent);

So, for example, if your range is from 50 - 1200, the exponent is 3 and the magnitude is 1000.

Then deal with the second part by deciding how many subdivisions you want in your grid:

value_per_division = magnitude / subdivisions;

This is a rough calculation because the exponent has been truncated to an integer. You may want to tweak the exponent calculation to handle boundary conditions better, e.g. by rounding instead of taking the int() if you end up with too many subdivisions.

Answer 3

CPAN provides an implementation here (see source link)

See also Tickmark algorithm for a graph axis

FYI, with your sample data:

• Maple: Min=8, Max=74, Labels=10,20,..,60,70, Ticks=10,12,14,..70,72
• MATLAB: Min=10, Max=80, Labels=10,20,,..,60,80

Answer 4

Another idea is to have the range of the axis be the range of the values, but put the tick marks at the appropriate position.. i.e. for 7 to 22 do:

[- - - | - - - - | - - - - | - - ]
       10        15        20

As for selecting the tick spacing, I would suggest any number of the form 10^x * i / n, where i < n, and 0 < n < 10. Generate this list, and sort them, and you can find the largest number smaller than value_per_division (as in adam_liss) using a binary search.

Answer 5

See this related question about a function to find the min, max, and interval size for grid lines when given inputs related to the dataset.

Answer 6

I use the following algorithm. It's similar to others posted here but it's the first example in C#.

public static class AxisUtil
{
    public static float CalcStepSize(float range, float targetSteps)
    {
        // calculate an initial guess at step size
        float tempStep = range/targetSteps;

        // get the magnitude of the step size
        float mag = (float)Math.Floor(Math.Log10(tempStep));
        float magPow = (float)Math.Pow(10, mag);

        // calculate most significant digit of the new step size
        float magMsd = (int)(tempStep/magPow + 0.5);

        // promote the MSD to either 1, 2, or 5
        if (magMsd > 5.0)
            magMsd = 10.0f;
        else if (magMsd > 2.0)
            magMsd = 5.0f;
        else if (magMsd > 1.0)
            magMsd = 2.0f;

        return magMsd*magPow;
    }
}