 Taylor Series Approximations for Trigonometry:

1. The indicator starts by calculating sine and cosine values of the close price using Taylor Series approximations. These approximations use polynomial terms to estimate the values of these trigonometric functions.
Mathematical Component Formation:

2. The calculated sine and cosine values are then multiplied together. This gives us the primary mathematical component, termed as the 'trigComponent'.
Smoothing Process:

3. To ensure that our indicator is less susceptible to market noise and more reactive to genuine price movements, this 'trigComponent' undergoes a smoothing process using a simple moving average (SMA). The length of this SMA is defined by the user.
Logarithmic Transformation:

4. With our smoothed value, we apply a natural logarithm approximation. Again, this approximation is based on the Taylor expansion. This step ensures that all resultant values are positive and offers a different scale to interpret the smoothed component.
Dynamic Scaling:

5. To make our indicator more readable and comparable over different periods, the logarithmically transformed values are scaled between a range. This range is determined by the highest and lowest values of the transformed component over the user-defined 'lookback' period.
ROC (Rate of Change) Direction:

6. The direction of change in our scaled value is determined. This offers a quick insight into whether our mathematical component is increasing or decreasing compared to the previous value.
Visualization:

7. Finally, the indicator plots the dynamically scaled and smoothed mathematical component on the chart. The color of the plotted line depends on its direction (increasing or decreasing) and its boundary values.