Cutting a round tube so it wraps perfectly around another round tube is one of the least intuitive jobs in fabrication. The joint looks trivial. The geometry is not. Where two cylinders intersect, the line of contact is a saddle curve, known in the trade as a fishmouth or cope. Only one case stays simple: equal diameters meeting at any angle produce a true ellipse, which is why an elliptical template works on same-size tube and fails on everything else. Change the diameter ratio and the curve becomes transcendental, with no tidy closed-form flat pattern. Unroll it and you get an asymmetric wave plotted point by point. Handrail work lives inside this problem. UK guidance converges on circular handrail of roughly 32-50mm outside diameter, and a continuous rail demands a joint at every post, return and transition. Every one of those is a cope. Machines attack it four ways: - Hole saw notchers clamp the tube in a swivelling vice and drive a saw of the mating diameter straight through. ...