A brake disc for a rope brake for rope climbing, the brake disc comprising: a first portion for supporting a rope on the brake disc, wherein the first portion is a first arc of a first circle having a first radius; and a second portion for supporting the rope on the brake disc, wherein the second portion is a second arc of a second circle having a second radius, and wherein the second radius is larger than the first radius, wherein the second portion is arranged adjacent to the first portion, wherein the second portion is a portion of the surface of the brake disc and comprises a rope channel having an undulating contour, wherein the rope channel comprises one or more recesses (21a, 21b, 21c) on the surface of the brake disc, wherein each of the recesses (21a, 21b, 21c) is delimited by two narrowings (22a, 22b, 22c, 22d), wherein the one or more recesses (21a, 21b, 21c) comprise a first recess and a second recess (21b, 21c), wherein the second recess (21b, 21c) is arranged following the first recess on the surface of the brake disc when viewed from the slack strand of the rope to the load strand of the rope, and one of the narrowings (22a, 22b, 22c, 22d) is located between the first recess and the second recess (21b, 21c), which one of the narrowings (22a, 22b, 22c, 22d) delimits the first recess from the second recess (21b, 21c), characterized in that both the contour of the first recess and the contour (23b, 23c) of the second recess (21b, 21c) each have the shape of a surface portion of an ellipsoid, wherein a first angle (α1) is given between a main axis of the ellipsoid of the first recess and a tangent to the second arc in the region of the first recess, wherein a second angle (α2) is given between a main axis of the ellipsoid of the second recess and a tangent to the second arc in the region of the second recess, and wherein the first angle (α1) is larger than the second angle (α2). a first portion for supporting a rope on the brake disc, wherein the first portion is a first arc of a first circle having a first radius; and a second portion for supporting the rope on the brake disc, wherein the second portion is a second arc of a second circle having a second radius, and wherein the second radius is larger than the first radius, wherein the second portion is arranged adjacent to the first portion, wherein the second portion is a portion of the surface of the brake disc and comprises a rope channel having an undulating contour, wherein the rope channel comprises one or more recesses (21a, 21b, 21c) on the surface of the brake disc, wherein each of the recesses (21a, 21b, 21c) is delimited by two narrowings (22a, 22b, 22c, 22d), wherein the one or more recesses (21a, 21b, 21c) comprise a first recess and a second recess (21b, 21c), wherein the second recess (21b, 21c) is arranged following the first recess on the surface of the brake disc when viewed from the slack strand of the rope to the load strand of the rope, and one of the narrowings (22a, 22b, 22c, 22d) is located between the first recess and the second recess (21b, 21c), which one of the narrowings (22a, 22b, 22c, 22d) delimits the first recess from the second recess (21b, 21c), characterized in that both the contour of the first recess and the contour (23b, 23c) of the second recess (21b, 21c) each have the shape of a surface portion of an ellipsoid, wherein a first angle (α1) is given between a main axis of the ellipsoid of the first recess and a tangent to the second arc in the region of the first recess, wherein a second angle (α2) is given between a main axis of the ellipsoid of the second recess and a tangent to the second arc in the region of the second recess, and wherein the first angle (α1) is larger than the second angle (α2).