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The localization characters of the first-order rogue wave (RW) solution $u$of the Kundu-Eckhaus equation is studied in this paper.  We discover a full process of the evolution for the contour line with height$c^2+d$ along the orthogonal direction of the ($t,x$)-plane for a first-orderRW $|u|^2$: A point at height $9c^2$ generates a convex curve for $3c^2\leqd<8c^2$, whereas it becomes a concave curve for $0<d<3c^2$, next it reduces toa hyperbola on asymptotic plane (i.e. equivalently $d=0$), and the two branchesof the hyperbola become two separate convex curves when $-c^2<d<0$, and finallythey reduce to two separate points at $d=-c^2$. Using the contour line method,the length, width, and area of the RW at height $c^2+d (0<d<8c^2)$ , i.e. abovethe asymptotic plane, are defined. We study the evolutions of threeabove-mentioned localization characters on $d$ through analytical and visualmethods. The phase difference between the Kundu-Eckhaus and the nonlinearSchrodinger equation is also given by an explicit formula.

On the evolution of a rogue wave along the orthogonal direction of the ($t,x$)-plane