Free lesson · Technical analysis

Dynamic support, trend lines and Gann angles

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Start with the idea

A moving reference changes with time. State its numerical rule before drawing it on a chart.

Symbols, units & horizon

E_t: current EMA in price units
C_t: current completed close
α: smoothing coefficient in (0,1]
E_(t−1): prior EMA
P₁,P₂: confirmed anchor prices
t₁,t₂: distinct anchor times in bars or stated time units
m: price units per time unit
K(t): projected line level

When and why to use this

Use distances to causal moving references and trend lines as features or explicit trigger boundaries.

A moving average can act as a candidate dynamic support/resistance reference. A touch or bounce remains a hypothesis, not a property of the average.
An EMA updates the previous average toward the latest completed value. Its smoothing coefficient controls response speed.
A trend line joins two previously confirmed anchors. It becomes available when the second anchor is confirmed; projecting it earlier is hindsight.
A Gann 1×1 line means one chosen price unit per chosen time unit. It is not an invariant 45-degree angle when the chart is resized.
Specify linear versus logarithmic price axes. A straight line on a log-price axis represents a different price path.
Normalize distances by prior volatility and test against a constant-level or simple trend baseline. Reject an interpretation that depends only on visual aspect ratio.

E_{t} = α C_{t} + (1 - α) E_{t - 1}, m = \frac{P_{2} - P_{1}}{t_{2} - t_{1}}, K (t) = P_{1} + m (t - t_{1})

Specified chart rule · derivation and arithmetic

Dynamic support, trend lines and Gann angles

EMA with prior 100, new close 104 and α=.25 gives .25×104+.75×100=101.
Anchors (0,100) and (4,108) give slope (108−100)/(4−0)=2 price units/bar.
At bar 6, the line projects to 100+2×6=112. A 1×1 line in these units would instead have slope 1.

Work it by hand

The same 2-price-units/bar slope looks steeper or flatter when the screen axes are resized, but its numerical rule does not change.

Use the rule

Fix the definition, units and information timestamp.
Compute the example and inspect the graph.
Compare with a simple baseline on untouched periods after costs.

Before moving on

Write the exact rule, its availability time, an invalidation condition and a fair out-of-sample test.

Research context: review the related evidence checkpoint. The numerical convention here defines a candidate feature; that related research does not validate this exact rule.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def ema_step(close,previous,alpha):
    if not 0<alpha<=1: raise ValueError("alpha in (0,1] required")
    return alpha*close+(1-alpha)*previous

def projected_line(t,t1,p1,t2,p2):
    if t1==t2: raise ValueError("Distinct anchor times required")
    slope=(p2-p1)/(t2-t1)
    return p1+slope*(t-t1)

print(ema_step(104,100,.25),projected_line(6,0,100,4,108))

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