The equation of the curve of intersection of a torus with a plane perpendicular to both the midplane of the torus and to the plane x = 0. (The general intersection of a torus with a plane is called a toric section). Let the tube of a torus have radius a, let its midplane lie in the z = 0 plane, and let the center of the tube lie at a distance c from the origin. Now cut the torus with the plane y = r. The equation of the torus with y = r gives the equation (c - sqrt(x^2 + r^2))^2 + z^2 = a^2 c^2 - a^2 + x^2 + z^2 + r^2 = 2c sqrt(x^2 + r^2) (r^2 - a^2 + c^2 + x^2 + z^2)^2 = 4c^2(r^2 + x^2).

toric section | torus