opengl basic transformation

Question

A vector can be rotated and scaled, since it has direction and scale. But does it mean by plotting a point. Point can only be translated. But wikipedia says "For example the matrix R = [ cos0,-sin0] [ sin0,cos0] rotates points in the xy-Cartesian plane counter-clockwise through an angle θ about the origin of the Cartesian coordinate system.

Also what does it mean by "since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system."? Does this mean I cannot perform rotation around any point other than the origin?

Answer 1

When describing transformations, Wikipedia and other sites often refer to the effect on "points"; however this implicitly applies to every point in the coordinate system (with explicit exceptions like rotation of the origin.) These transformations - typically rotating, translating, and scaling - apply to the entire frame of reference and any derivative frames of reference. Use of the word 'point' is in the mathematical sense, a choice of coordinates within a coordinate system, and doesn't imply anything about a point in the graphical sense, like a "plotted" or "drawn" point (although graphing a point is just a visualization of this concept, so the distinction is moot.)

While it's true that a rotation has no effect on the origin, you are free to translate the origin itself, or equivalently to translate your models relative to the origin. Once you have applied the rotation, you can reverse the translation to restore the original origin.

Answer 2

Absolutely, to rotate about another point than the origin, you have to create a matrix that translates your vertices from your rotation center to the origin, rotates, then translates back from the origin to your rotation center.