Visualizing Ellipse rotation and associated parametric equations

 


Diagonal matrix

  1. Ellipse axes are parallel to coordinate axes (no rotation).
  2. Zeros in off-diagonals means zero correlation.

Same values in diagonals

  1. Ellipse is rotated 45 degrees if correlation (off-diagonal) is positive regardless of its magnitude. 
  2. Similarly -45 degrees if correlation is negative.

Correlation matrix

  1. Correlation matrix has 1 in diagonals and values between -1 and 1 inclusive in off-diagonals.
  2. Ellipse size remains the same (always touches square of side 2 units). 
  3. Whether the correlation is positive or negative can be observed by the orientation of the ellipse.
  4. The amount of correlation can be interpreted by how thin the ellipse is.

How It Works

Below is a list of parametric equations starting from that of a general ellipse and modifying it step by step into a prediction ellipse, showing how different parts contribute at each step.








Reference

  1. How to Draw Ellipse of Covariance Matrix (cookierobotics.com)

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