Does a line have 2 degrees of freedom or 1 degree of freedom?

geometry linear-algebra plane-geometry

Solution 1:

You are confused because these are different things.

  1. To pick a point from the set of points on a given line: one degree of freedom (by the argument you give)

  2. To pick a line from the set of lines in a plane: two degrees of freedom (e.g., its direction and its distance from the origin; or: pick two points in the plane = 4 degrees, but subtract 2 degrees because their position on the line is irrelevant; or ...)

  3. To pick a line from the set of lines in 3d space: four degrees of freedom (why?)

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