Notation Used for Points, Vectors, and Matrices

Basic Notation (Compare to Appendix C in Farin. His notation does not generally distinguish between points and vectors.)

Quantity	In print	Handwritten
point in an affine space	upper case italic P	upper case P
'geometric' vector in a vector space	lower case bold u	lower case with arrow on top
a unit length vector	lower case bold with 'hat'	lower case with 'hat'
a scalar (i.e., a floating point numeric constant)	lower case italic f	lower case f
a matrix	upper case bold M	upper case M

Allowed Vector Space Operations

Operation in print	Underlying coordinate-based interpretation
u + v	( u_x+v_x , u_y+v_y , u_z+v_z )
fu	( fu_x , fu_y , fu_z )

Allowed Affine Space Operations

Operation in print	Underlying coordinate-based interpretation
P - Q	( P_x-Q_x , P_y-Q_y , P_z-Q_z )
P + u	( P_x+u_x , P_y+u_y , P_z+u_z )

"Convenience" Operations

While not directly permitted by the definitions, the validity of the operations shown in the first column of the table below can be immediately established as indicated in the second column.

Operation in print	Expressed using valid operations	Underlying coordinate-based interpretation
u - v	u + (-1)*v	( u_x-v_x , u_y-v_y , u_z-v_z )
u / f	(1/f) * u	( u_x / f , u_y / f , u_z / f )
P - u	P + (-1)*u	( P_x-u_x , P_y-u_y , P_z-u_z )

An Issue

The following two expressions are algebraically equivalent. Therefore, they would both evaluate to the same triple of numbers, given values for the constituent points and vectors. One of these expressions uses only the operations directly permitted by our vector and affine space definitions. The other does not.

Should we be concerned about this? How should we handle this in general? What are the conditions under which some given expression -- which utilizes undefined operations -- is in fact a sensible, well-defined expression?

(P - Q) + 3(R - X) + 4u

3R - 3X - Q + P + 4u

J. R. Miller