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Angle of Repose

Angle of Repose – Part 1

Many traders discuss the virtues of plotting 45 degree lines onto their chart.  Generally, an introduction to 45 degree lines is gained from reading something about the works of W. D. Gann and how he plotted 45 degree lines on his charts.

Plotting a line on your computer generated charts physically at a 45 degree angle is worthless taking into account scaling issues.  To illustrate the problem with a 45 degree line, compare these two charts.

 

The blue line is plotted at a downward 45 degree angle in both charts, but as can be seen, the line passes through the chart bars in different places.  The line which looks very useful as an indicator of a trend in the left-hand chart suddenly looks useless in the right-hand chart.  So what happened?  The vertical spacing of the chart scale changed!

Computer generated charts typically use a scale range that covers the highest high and the lowest low of the data set that is being plotted.  This scale is mapped to the physical size of the chart window, which may range from a few centimetres to the full size of the monitor display.  Not only can the range be dynamic, but the bar spacing is also dynamic.  The following example uses the same range as the 1st chart, but with a narrower spacing between the bars.  The position of the 45 degree line appears quite differently again.

 

Since 45 degree lines are so arbitrary in their relationship to the bars, what then was W. D. Gann doing in plotting 45 degree lines on his charts?  Gann referred to the 45 degree lines as 1x1 lines (one by one lines).  The line was being plotted on his charts with a mathematical slope of one unit of price per one unit of time.  Gann would manually construct his charts using graph paper with a square grid.  The vertical price grid would be labelled with a price interval such as 2 cents.  Thus, the price unit is the grid interval of 2 cents.  The bars would be plotted on the horizontal grid, such as a daily bar on every grid interval.  Thus, the time unit would be one day.

A graph constructed in this manner would give his 1x1 line the following slope definition:  2 cents per day.  A line with this slope could be easily drawn using a 45 degree triangle because of the way the graph paper was laid out.  So, a 45 degree line and a 1x1 line would be one and the same thing only when a specific graph paper grid is used.

When you use computer charts with dynamic scale ranges and dynamic bar spacing, you must draw 1x1 lines according to a slope definition.  The plotted location of the 1x1 line may or may not (usually not) be at a 45 degree angle.

When you see a reference to a 45 degree line, always observe the price grid interval, and the time interval so you know the 1x1 definition for the slope.  The slope will be one unit of price for one unit of time.  Once the slope is known, the same line can be drawn on computer generated charts.

Angle of Repose – Part 2

It can be argued that finding a useful slope for the 1x1 Gann line is what Gann analysts refer to as 'squaring time and price.'  My current understanding of what is meant is that it is a literal relationship expressed mathematically as:

Price = Time squared  or

P = t ^ 2

This relationship gives us the needed mathematics for automatically calculating the slope for the 1x1 Gann angle.

To calculate the slope of the 1x1 line, two prices are needed, and a time interval.  The first price P1 will be the price on the chart where the 1x1 line (or Gann Fan) is anchored.  Usually this is the top or bottom price of a significant trend.  The time interval is calculated from P1 by normalizing P1 to fall in the range of 100 to 999.  If P1 is below 100, multiply it by 10 as many times as needed until it is in the range of 100 to 999.  If P1 is at or above 1000, repeatedly divide it by 10 until it is in the range of 100 to 999.  Then the time interval t is found by taking the square root of P1.

Gann's Square of Nine is used to determine the 2nd price P2.  P2 is related to P1 by some degree of rotation around the Square of Nine.  The commonly used degrees of rotation are 360, 180, 90, and 45 degrees.  P2 can be calculated using this formula:

P2 = (t + degrees of rotation / 180) ^ 2

Remember, the time interval t was determined by taking the square root of the normalized price P1.  Example:  If the trend top or bottom price is $144.00, then the time interval is 12 bars.  To find the price that is 180 degrees around the Square of Nine, P2 would be (12 + 180/180) ^ 2, which equals 13 squared or $169.00.

The slope of the 1x1 line is calculated using this formula:

slope = ( P2 - P1 ) / t

Continuing the example, slope = ($169.00 - $144.00) / 12 bars, which equals $2.08 per bar.  If the 1x1 line determined in this manner is too steep to be useful on the chart, then it is appropriate to use a smaller degree of rotation around the Square of Nine, such as 90, 45, 22.5, or 11.25 degrees, etc.  If the 1x1 line is too flat to be useful on the chart, then it is appropriate to use a higher degree of rotation such as 360 or 720 degrees.

This technology is built into the Gann Fan tool in ExperCharts.  The Gann Fan is placed on the chart by selecting the point for the vertex.  The 1x1 line can be located manually by selecting a 2nd point, or let ExperCharts determine the 1x1 slope automatically using the mathematics described above.  The following charts show the Gann Fan with the slope of the 1x1 line determined automatically from the P1 anchor price at the fan's vertex.

Angle of Repose – Part 3 (Time and Price)

Gann's geometric angles are trend lines drawn from prominent tops or bottoms at certain angles.  The most important angle is 45 degrees, which means the line's slope is one unit of price per unit of time.  (Note:  Depending of the chart scale used, the line may or may not appear to be plotted at a 45 degree angle.).  One interpretation can be expressed as follows:

Price = Time squared or P = t ^ 2

The above relationship between price and time can be plotted on a chart as shown in the illustration below.  The time values of 10, 20, and 30 are marked by the three arrows.

 

For the sake of illustration, let's suppose a prominent top or bottom occurs at a price of 400.  The theory is that this significant point has a mathematical counterpart.  Start a new time curve at this point in time, and it will give us an expectation for a future top or bottom to occur on this curve.  This principle can be stated as 'When price meets time, a change is imminent.'  This 'price meets time' relationship is shown in the following chart.

 

 

With the prominent top or bottom at P, if price meets the curve at point A it will do so in 18 bars.  The time to A is the square root of the price at A.  Price at A is 324.  The square root of 324 is 18.

If price meets the curve at point B, it will do so in 20 bars.  The time to B is the square root of the price at B.  Price at B is 400, therefore the time to B is 20 bars.

If price meets the curve at point C, it will do so in 22 bars.  This is a very interesting concept!

Remember that price and time are related by the formula: P = t ^ 2 or t = sqrt( P )

Let us take this concept one step further to obtain the Trend Line Slope using the figure shown below.

 

 

From the previous discussion, we recognise the next time curve will be t bars away for a given price P.  At a time t+1 price would meet the curve at price P1.  Now, let’s solve for the slope of the trend line shown in blue which connects P and P1.

P = t ^ 2

P1 = (t + 1) ^ 2 = t ^ 2 + 2 t + 1 = P + 2 t + 1

Slope = (Change in price) / (Change in time)

Change in price = P1 - P = P + 2 t + 1 - P = 2 t + 1 = 2 t + 2 - 1 = 2 [ t + 1] -1

Change in time = t + 1

Therefore, slope of P to P1 is = (2 [ t + 1] - 1) / (t+1) = 2 - 1 / (t+1) = 2 - 1 / sqrt( P1 )

If we normalize all prices to consider three significant digits, then all prices will fall in the range of [100 ... 1000]. By substituting the price boundaries into the slope formula, we can get a range of slopes as follows.

For a P1 of 100, the slope of the up trend line to 100 = 2 - 1 / 10 = 1.9

For a P1 of 1000, the slope of the up trend line to 1000 = 2 - 1 / 100 = 1.99

The slope of the up trend line at the midpoint of this price range is 2 - 1 / sqrt(500) = 1.96

Let's call this trend line a 45 degree line because we developed the slope using one unit of price change from P to P1 with one unit of time t.  For this 45 degree line, the slope is basically 2.  I think this is strong justification as to why Gann used 2 cents as the price grid interval of his daily grain charts.  Such a scale layout would naturally give Gann 45 degree angles with a slope of 2 cents per daily bar.  I have shown that 2 is the slope of the upward 45 degree trend line that develops from the price and time relationship given by the formula:  P = t ^ 2.

One can solve for the slope of the downward trend line from P1 to P to obtain this result:

Slope of P1 to P = (-2 t - 1) / (t-1) = (-2 [t - 1] - 3 ) / (t-1) = -2 - 3 / (t-1) =-2 - 3 / (sqrt( P ) - 1)

For a P of 100, the slope of the down trend line to 100 = -2 - 3/9 = -2.33

For a P of 1000, the slope of the down trend line to 1000 = -2 - 3/99 = -2.03

Again, the slope of the down trend line approaches a value of -2. Therefore, -2 is a good approximation for the slope of a downward 45 degree trend line.

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Page last modified: May 08, 2008
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