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Dear Scott,

I want to develop a method for mounting a flat card on the top of an angled cardboard cylinder like a toilet paper tube. Clearly no other angle of intersection besides a perpendicular one results in a straight line drawn around the cylinder.

Imagine that I want to print out paper forms that you can wrap around a standard sized cylinder and cut on the dotted line to achieve a mounting surface equivalent to the intersection of a plane cutting through a cylinder segment at one angle or another. So we're talking about (virtually# slicing the cylinder segment along its axis and flattening it out, resulting in a curve that describes the intersection of the chosen plane with the cylinder segment.

A complication: in 3d Cartesian space, a plane intersects a cylinder at two angles: assuming that the axis of the cylinder is parallel to the vertical z axis, there is the angle of displacement from the horizontal x axis, and angle of displacement from the horizontal y axis.

Is there a way #using, say, Excel formulas and graphing# to get from a table of displacement angles #x,y) to a set of graphs that could be printed out and used as templates to cut a uniform diameter paper tubes so as to create mounting surfaces for planes at various angles?

Forgive my inexpert use of terms and concepts here--I never took solid geometry.

Sincerely,

Webb Mealy

The equation of a circle is x²+y²=r² where (x,y) are the coordinates in a plane and

r is the radius of the circle.

Take this circle in the x-y plane and lift it so that only y lifts,

leaving x the same. Lift it by an angle A.

The height of the circle at any spot y is z = y*tanA.

The length of this circle in the sloping direction is then L = 2*r*secA.

The equation of drawing this would be a ellipse with width 2r and length 2*r*secA.

The equation in the x-y plane would then be (x/r)²+(y/(r*secA))² = 1.

This could be cut out and placed in the cone at angle A in the y direction with the x-y axis.

Is this what you're after?

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