I have been trying to design a library to do some simple geometric computations in an Euclidean space regardless of its dimension. While it is easy to represent points, vectors, hyperspheres and hyperplanes in a generic fashion, I am still unable to find a generic way to represent a (infinite) line, even though lines share properties across dimensions.My best guess is that I could store some of the parameters of its parametric equation since it is easy to extend a parametric equation to a line in a space of any dimension:x = x0 + aty = y0 + btz = z0 + ct// can be extended to any dimensionBut even with this equation, I can't find what should be stored and what should not be in order to compare lines. With an ideal solution, two objects of type Line:would be programmatically equal (with operator==),would have equal representations in the memory.What should I store in order to achieve that?
How to represent a geometric line programmatically?
c++;math;implementations;geometry
I think you're on the right track with your parametric equation.What you have there is the vector form of the line equation.L = R + tVWhere R is [x0, y0, z0] and V is [a, b, c].You just need to normalize your equations. You would do that by finding the value of R such that |R| is at a minimum, which occurs when R is perpendicular to V, or R.V = 0.Also, since t can be scaled by any value, without changing the line, you should normalize V by dividing every coefficient by |V|