3D Least Squares Plane

by

If you have n data points (x[i], y[i], z[i]), compute the 3×3 symmetric matrix A whose entries are:

sum_i x[i]*x[i],    sum_i x[i]*y[i],    sum_i x[i]
sum_i x[i]*y[i],    sum_i y[i]*y[i],    sum_i y[i]
sum_i x[i],         sum_i y[i],         n

Also compute the 3 element vector b:

{sum_i x[i]*z[i],   sum_i y[i]*z[i],    sum_i z[i]}

Then solve Ax = b for the given A and b. The three components of the solution vector are the coefficients to the least-square fit plane {a,b,c}.

Note that this is the “ordinary least squares” fit, which is appropriate only when z is expected to be a linear function of x and y. If you are looking more generally for a “best fit plane” in 3-space, you may want to learn about “geometric” least squares.

Note also that this will fail if your points are in a line, as your example points are.

More Related Contents:

  1. Select k random elements from a list whose elements have weights
  2. Calculate distance between two latitude-longitude points? (Haversine formula)
  3. Unique (non-repeating) random numbers in O(1)?
  4. How to determine if a point is in a 2D triangle? [closed]
  5. How do I calculate a point on a circle’s circumference?
  6. Algorithm to find Largest prime factor of a number
  7. Fast permutation -> number -> permutation mapping algorithms
  8. Algorithm to detect intersection of two rectangles?
  9. How to make rounded percentages add up to 100%
  10. Writing your own square root function
  11. Sort points in clockwise order?
  12. How do you calculate the average of a set of circular data? [closed]
  13. Finding out nth fibonacci number for very large ‘n’
  14. Computing set intersection in linear time?
  15. Floating Point Divider Hardware Implementation Details
  16. “On-line” (iterator) algorithms for estimating statistical median, mode, skewness, kurtosis?
  17. Fastest primality test
  18. Circle-circle intersection points
  19. Efficient maths algorithm to calculate intersections
  20. How do I find a factorial? [closed]
  21. Calculating the position of points in a circle
  22. Strategies for simplifying math expressions
  23. How to express tetration function, for complex numbers
  24. An inverse Fibonacci algorithm?
  25. How to check if an integer is a power of 3?
  26. “Rounding” colour values to the nearest of a small set of colours
  27. How to generate a power set of a given set?
  28. determine if a point sits inside an arbitrary shape?
  29. How can I perform Collision Detection on rotated rectangles? [closed]
  30. Location of highest density on a sphere

Leave a Comment