Decorating a Tree with a Ribbon
Decorating a Tree with a Ribbon
This Demonstration calculates the length of a ribbon wrapped around a tapered post or a tree (geometrically, a frustum, which is a truncated cone) in two ways:
1. approximately, by unfolding the cone and adding the lengths of all segments, where is the number of turns.
n
n
2. exactly, using the formula for arc length, ++dh, where is the height of frustum and derivatives are taken with respect to , the height, of the components of the parametric equation of the ribbon (as a curve):
H
∫
0
2
′
x
2
′
y
2
′
z
H
h
x=rcos(2πnh/H)
y=rsin(2πnh/H)
z=h
where is the radius of the frustum at height and slope .
r=(h-b)/m
h
m
In addition, the Demonstration calculates the surface area and the volume for different heights and slopes.