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# Advanced Math/Plotting the intersection with a plane onto a flattened cylindar

Question
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?

Volunteer

#### Scott A Wilson

##### Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

##### Experience

Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.