What is the angle of the stairway?

Specifically, what is the angle of the stairway shown below? In particular, is the angle up the same as the angle down?

To use Digitizer 1.0, we use the relationship that for two given pairs of values and the tangent of the angle between them is

Our procedure will be this. Select an origin on the picture, and select points going down and then points going up. After obtaining our coordinates along the "suspected" circle we compute the values

for each However, we will allow the first point to be the origin.

Here's the detail:

• First import the image into Digitizer 1.0. The result will be something this:
  
  

• Now input an origin (red cross) and and x-axis value (blue cross). We set the scale to be 12.
  
  

• Now digitize points incline (decline) and show the points.
  
  

• The next step is to copy the data from the Digitizer data table into your spreadsheet and parse the data into columns. We show the parsed data. (In Microsoft Excel the raw data is parsed using the Data +Text_to_Columns keys. Be sure to select the "delimited" option.)
  
  

• The final step is the computation of the ( ) for each of the data points. We will reinsert the origin between the incline set of data and decline set.
  
  It is evident from the picture that the positive incline is about 34 and the decline is about -35 Note that we have used successive pairs of data for the angular computation. We could just have well always used the origin as the base value. You might consider the differences in the outcome. Though theoretically the same there are some issues of sampling error to discuss.

Discussion points:

1. The mathematical principal of finding the angle.

2. The procedure for sampling.

3. The formulas for determining the angle(s).

4. The possibility of using least squares to approximate the slope.

5. The sampling error and its effect on the differences of the results.

In summary, we required mathematics to use a proper model to test our hypothesis, the digitizer to make coordinates, and a spreadsheet to analyze the data.