Tulip
Monday, January 13, 2014
Sunday, January 5, 2014
Paper transformations
Paper transformations - Making 3d shapes out of paper by cutting,folding,pasting etc.
Here we not only want to make shapes, but also we want to know how to make a 3d shape of given parameters. And also the dimensions of the paper that would be required to create the 3d shape.
Simplest 3d shape that can be formed of paper is a cylinder. You just take rectangular paper and paste two opposite sides together. Height of the cylinder will be breadth of the rectangle and circumference of the cylinder base will be height of the rectangle or vice-versa. We can thus calculate the height and breadth of the rectangle required to form the cylinder of desired height and base
Now if we have to form a simple geometric cone out of paper, what shape do we have to take and paste? To figure this out, lets take an imaginary cone object made of paper. If we flatten it what shape would it form? Equilateral Triangle? No. It would be a sector of a circle. If we take a sector of a circle and join and paste the lengths, we get a cone . Like below:
sector==> cone
The following figure explains how the sector of a circle and the cone formed out of it are related.
The other details can be easily worked out. Like the height of cone will be square root of sum of the squares of L and R using Pythagoras formula. And the angle theta of the sector of the first circle is length of the arc divided by radius L.
By this reverse mapping, we can create a cone with desired parameters.
Up Next : Creating inclined cones.
Here we not only want to make shapes, but also we want to know how to make a 3d shape of given parameters. And also the dimensions of the paper that would be required to create the 3d shape.
Simplest 3d shape that can be formed of paper is a cylinder. You just take rectangular paper and paste two opposite sides together. Height of the cylinder will be breadth of the rectangle and circumference of the cylinder base will be height of the rectangle or vice-versa. We can thus calculate the height and breadth of the rectangle required to form the cylinder of desired height and base
Now if we have to form a simple geometric cone out of paper, what shape do we have to take and paste? To figure this out, lets take an imaginary cone object made of paper. If we flatten it what shape would it form? Equilateral Triangle? No. It would be a sector of a circle. If we take a sector of a circle and join and paste the lengths, we get a cone . Like below:
sector==> cone
The following figure explains how the sector of a circle and the cone formed out of it are related.
The other details can be easily worked out. Like the height of cone will be square root of sum of the squares of L and R using Pythagoras formula. And the angle theta of the sector of the first circle is length of the arc divided by radius L.
By this reverse mapping, we can create a cone with desired parameters.
Up Next : Creating inclined cones.
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