Maths Question
Discussion
Not sure if anyone can work this out?
Just had a friend from a mate asking if i could, and i'm struggling. I've absolutely no idea why he wants to know what he does, but he's gone to the trouble of sending me a 3D image to illustrate, so it must be important!
Basically, after a formula to work out the height of the line, marked by the arrows, e.g if you were to wrap a piece a string half way around the base of a cup, rising at an angle perpendicular to that of the edge.
Just had a friend from a mate asking if i could, and i'm struggling. I've absolutely no idea why he wants to know what he does, but he's gone to the trouble of sending me a 3D image to illustrate, so it must be important!
Basically, after a formula to work out the height of the line, marked by the arrows, e.g if you were to wrap a piece a string half way around the base of a cup, rising at an angle perpendicular to that of the edge.
I may well be getting the question wrong, but wouldn't the formula be ∏ x R x tan X where R=the radius of the cup and X=the angle the string makes to the base?
ETA — ignore this; I was assuming the sides of the cup were vertical, which they're obviously not, and probably even then got it wrong.
ETA — ignore this; I was assuming the sides of the cup were vertical, which they're obviously not, and probably even then got it wrong.
Edited by samwilliams on Wednesday 17th November 01:50
t load more complicated than thatJQ said:
H squared = a squared + b squared
where H is the longest diagonal line and a and b are the 2 perpendicular lines.
The fact it's on a cup confuses the situation. Unravel the cup sides and they are all straight line - like on a globe. And as such the above equation works.
This. where H is the longest diagonal line and a and b are the 2 perpendicular lines.
The fact it's on a cup confuses the situation. Unravel the cup sides and they are all straight line - like on a globe. And as such the above equation works.
Ignore that it's a cup, ignore that the red line curves, just measure between three points. Done.
MiniMan64 said:
JQ said:
H squared = a squared + b squared
where H is the longest diagonal line and a and b are the 2 perpendicular lines.
The fact it's on a cup confuses the situation. Unravel the cup sides and they are all straight line - like on a globe. And as such the above equation works.
This. where H is the longest diagonal line and a and b are the 2 perpendicular lines.
The fact it's on a cup confuses the situation. Unravel the cup sides and they are all straight line - like on a globe. And as such the above equation works.
Ignore that it's a cup, ignore that the red line curves, just measure between three points. Done.
Diagram not to scale
Sonic said:
MiniMan64 said:
JQ said:
H squared = a squared + b squared
where H is the longest diagonal line and a and b are the 2 perpendicular lines.
The fact it's on a cup confuses the situation. Unravel the cup sides and they are all straight line - like on a globe. And as such the above equation works.
This. where H is the longest diagonal line and a and b are the 2 perpendicular lines.
The fact it's on a cup confuses the situation. Unravel the cup sides and they are all straight line - like on a globe. And as such the above equation works.
Ignore that it's a cup, ignore that the red line curves, just measure between three points. Done.
Diagram not to scale
maser_spyder said:
But the cup isn't 2D, so i don't think that works. In the same way a rhumb line is quicker than a direct line, etc.
Make the cup 2d (flatten it out) and the bottom would be curved...
Flatten out the cup (with the bottom removed) and it would be a trapezium (ie all straight lines)Make the cup 2d (flatten it out) and the bottom would be curved...
Stevenj214 said:
maser_spyder said:
But the cup isn't 2D, so i don't think that works. In the same way a rhumb line is quicker than a direct line, etc.
Make the cup 2d (flatten it out) and the bottom would be curved...
Flatten out the cup (with the bottom removed) and it would be a trapezium (ie all straight lines)Make the cup 2d (flatten it out) and the bottom would be curved...
Take a tapered paper cup like the one shown, and open it up, the top and bottom are curves, not straight.
Takes a while to get your head around it, but it's true.
Just to confirm, your picture there is totally wrong because the top and bottom are straight lines in your calculation.
The 'flat' or 2D representation of a paper cup would have the top and bottom as curves, the only straight lines are the two vertical ones where the vertical parts meet.
Put it this way to prove it.
Take a tapered cup from your cupboard, and roll it on the table a full revolution. What shape does it make? The shape it makes in a full revolution is the 2d representation.
The top rolls further than the bottom, so has to be longer, hence the curved top and bottom of the template.
If the top and bottom were flat, the resulting paper cup would look great. It would be vertical at one side, but have a large angle on the other. Aesthetically pleasing, but useless for liquids.
A bit Iike this;
\ . . |
. \ . |
.. \_|
The 'flat' or 2D representation of a paper cup would have the top and bottom as curves, the only straight lines are the two vertical ones where the vertical parts meet.
Put it this way to prove it.
Take a tapered cup from your cupboard, and roll it on the table a full revolution. What shape does it make? The shape it makes in a full revolution is the 2d representation.
The top rolls further than the bottom, so has to be longer, hence the curved top and bottom of the template.
If the top and bottom were flat, the resulting paper cup would look great. It would be vertical at one side, but have a large angle on the other. Aesthetically pleasing, but useless for liquids.
A bit Iike this;
\ . . |
. \ . |
.. \_|
Edited by maser_spyder on Wednesday 17th November 00:58
Edited by maser_spyder on Wednesday 17th November 00:59
Having thought about this, it's relatively simple, but you lot have been barking up the wrong tree with the 'flat' shape of a cup.
Rolled out, the 'stripe' actually leaves the flat edge of the RH side of the cup at a right angle.
The bottom edge of the cup then increases in distance from the stripe in relation to the top/bottom sizes of the cup.
i.e. depending on the angle of the curves, depends on how far the stripe will be from the base at a given point.
You can work this out in terms of x, by using a fairly straightforward differentiation equation.
The paper cup, flattened out with the stripe, would actually look like this;
Hence, the distance of the stripe from the base increases in a non-linear way. You would have to use differentiation to work out the actual distance, at a given point (which is half-way across the diagram above).
It's too bloody late for me to do it, I'm sure somebody else will chip in tomorrow.
Rolled out, the 'stripe' actually leaves the flat edge of the RH side of the cup at a right angle.
The bottom edge of the cup then increases in distance from the stripe in relation to the top/bottom sizes of the cup.
i.e. depending on the angle of the curves, depends on how far the stripe will be from the base at a given point.
You can work this out in terms of x, by using a fairly straightforward differentiation equation.
The paper cup, flattened out with the stripe, would actually look like this;
Hence, the distance of the stripe from the base increases in a non-linear way. You would have to use differentiation to work out the actual distance, at a given point (which is half-way across the diagram above).
It's too bloody late for me to do it, I'm sure somebody else will chip in tomorrow.
maser_spyder said:
Hence, the distance of the stripe from the base increases in a non-linear way.
Are you sure about this? (I'm not certain either way) If you start by considering it wrapped around the base, then the line the stripe makes on that 2d diagram would be curved. As you increase the angle from 0, would it suddenly straighten out, or remain a curve where the distance from the base increases linearly?The OP asked if it is possible to determine the height of the line at a particular point with a formula. The answer to that is yes, but I don't think there is sufficient information available to determine which formula would be correct. The line forms part of a conical helix, but we don't know what type of helix it is. It could be Archimedean, logarithmic, or some other form.
maser_spyder said:
Stevenj214 said:
maser_spyder said:
But the cup isn't 2D, so i don't think that works. In the same way a rhumb line is quicker than a direct line, etc.
Make the cup 2d (flatten it out) and the bottom would be curved...
Flatten out the cup (with the bottom removed) and it would be a trapezium (ie all straight lines)Make the cup 2d (flatten it out) and the bottom would be curved...
Take a tapered paper cup like the one shown, and open it up, the top and bottom are curves, not straight.
Takes a while to get your head around it, but it's true.
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