A torus in ℝ³ from two circles. Start with a ring in the x–y plane… — کانال تخصصی ریاضیات — TG.ME

Forwarded fromکاکانال تخصصی ریاضیات

🔻A torus in ℝ³ from two circles. Start with a ring in the x–y plane: (r cos u, r sin u), 0 ≤ u ≤ 2π, r ≥ 0. Now let the radius r and height z move around a circle in the r–z plane: r = a + d cos v, z = b + d sin v, 0 ≤ v ≤ 2π. Substitute r(v) into the ring: ((a + d cos v) cos u, (a + d cos v) sin u, b + d sin v) ∈ ℝ³. Restrictions: 0 ≤ u ≤ 2π, 0 ≤ v ≤ 2π, a, b ∈ ℝ, d ≥ 0, and a ≥ d. #geometry #هندسه 🔹کانال تخصصی ریاضیات ♾ @mathematics_learn