Locus is path described by the point under given condition. Loci is plural form of the word locus.
For example for defining circle, we say a point is moving in such a way that always remains equidistance from a fixed point. Here fixed point is known as center and the path described by the point is circumference of the circle.
Let P(x,y) is moving point. C(x1,y1) is fixed point .
This is locus of moving point P.
Symmetry you can say image (congruence ) of the object . For example You left palm and right palm are symmetrical.
Symmetry we apply to siplify /visualise the problem and to avoid repitition.
No there are no specific formulas for these. Either for locus or for symmetry conditions prescribed in advance.