Subject: intersection of perpendicular cylindrical surfaces I've encountered this situation in the course of normal architectural business affairs, and my atrophied math abilities no longer serve: Please consider two right circular cylinders, perpendicular one to the other, and of unlike radii in a 3 dimensional Cartesian space with mutually perpendicular x,y,z axes. If one cylinder is centered on the y axis with radius ra, and the other on the z axis with radius rb, then the expression for the first surface would be x^{2} + z^{2} = ra^{2}, y = any number. Likewise, the second cylinder's surface would be x^{2} + y^{2} = rb^{2}, z = any number. It is my intent to define the curve at the intersection of these two cylindrical surfaces. From sketching the conditions it appears that this intersection resembles an ellipse folded about its minor axis. My approach was to solve one of the two surface expressions for one variable and substitute that result into the other expression. So, taking the second expression and solving for x^{2} = rb^{2}  y^{2}, and then pitching that back into the first gives rb^{2}  y^{2} + z^{2} = ra^{2}. Rearranging, z^{2}  y^{2} = ra^{2}  rb^{2}, the right hand side of the equation being a constant. This seems to represent some sort of hyperbolic paraboloid, and not at all what I expected. Could you please render some guidance, and maybe recommend a good thorough text in which to find recipes for this sort of thing? Thank you for your time Hi Charlie,The curve is easily drawn, but it is not easily described in elementary terms  it is "twisted" in the sense that any plane intersects it in a finite number of isolated points; and it is "smooth" in the sense that you can find the direction of its tangent vector at any point. It comes in two disconnected pieces. You can get a feeling for what a piece looks like by setting a rubber band on a cylindrical object (or by running your car over a bicycle wheel). Your calculations show that by looking at the curve from a point far out on the xaxis (or equivalently, by projecting the picture into the yzplane), you see a hyperbola (namely z^{2}  y^{2} = ra^{2}  rb^{2}). Looking at it from a point on the yaxis you see a circle, and from the zaxis you see a circular arc. If you have a computer that draws curves in 3dimensions, have it draw  for t between ra and ra  When rb = ra the intersection of two cylinders is more familiar  the shape of the intersection (of the solid cylinders) is common in domes of public buildings. The boundary curve is no longer disconnected  it consists of two congruent ellipses in perpendicular planes, attached at the ends of their minor axes. Chris
