I would try to visualize placing a 2x4 board directly across a log, and then trying to balance on the log, while standing with your feet at each end of the 2x4 board. If the 2x4 straddles the log at a 90-degree angle, you will have a tendency to rock sideways as you attempt to balance the board across the log. If the 2x4 is placed along the length of the log, however, you will have a tendency to rock forwards and backwards. Although the log is now supporting the length of the 2x4, you will feel your ankles bending back and forth as you try to stabilize the width of the board on the rounded surface of the log.
Now, what happens when the 2x4 is placed at an angle across the log, no longer directly across it or directly along it? Now, you will have a tendency to to rock both sideways
as well as forwards and backwards. The 2x4 will actually rotate slightly as you distribute your weight to either leg, causing your body to twist or contort slightly. This log represents a really big cylinder, and the tendency to "twist" is the torsional component of the cylinder. You can demonstrate this effect for yourself with a soda can and a popsicle stick or some other flat object, like a credit card, by placing your fingers at each end of the stick or card and then rocking it at various angles around the can.
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In terms of differential calculus, a surface can be described by a function z = f(x,y). The curvatures of this surface at the center of the coordinate system are described by the "second partial derivatives" of the function. The horizontal curvature of this surface is equal to the second partial derivative with respect to x,
This is analogous to the sine-squared component of the formulas described earlier. And the vertical curvature is equal to the second partial derivative with respect to y,
This is analogous to the cosine-squared component of the formulas described earlier. Finally, the torsional-like curvature of the surface is given by the second mixed partial derivative (the derivative with respect to x, then y, or vice versa),
This is analogous to the sine-cosine component of the formulas described earlier, and is also associated with the amount of cylinder power or astigmatism at axis 045.
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