I What is the term for a set of loci that are the same?

[swampwiz]

I'm thinking the term is congruent.

For example, let's say there are 2 parametric lines:

s₀( t ) = < 2 , 1 > t + < 0 , - 3 >

s₁( t ) = < -18 , -9 > t + < -6 , -6 >

The loci for both are the same line, just that one proceeds in the opposite as the other (and at a high rate) per change in parameter, and such that

s₁( - t ) = s₀( 9 t + 3 )

Surely, there must be a term for this.

 

[BvU]

'The loci are identical' is the way I would express it.

 

[Mark44]

For example, let's say there are 2 parametric lines
s₀( t ) = < 2 , 1 > t + < 0 , - 3 >

s₁( t ) = < -18 , -9 > t + < -6 , -6 >

I would say that the equations are equivalent, as both parametric equations describe the same set of points.

For your parametric line, there are an infinite number of parametric representations. For example, the direction along your line is given by the vector <2, 1>. Any scalar multiple of this vector will also have the same direction. In your second equation the direction vector is <-18, -9>, which is -9 * <2, 1>.
Also, the equation could be written using any point on the line.

In slope intercept form, the equation of the line is y = (1/2) x - 3, which could also be written as 2y - x + 6 = 0, and in many other ways as well.