Parametric equation for a space curve












5














With reference to the following image:



enter image description here



the blue curve has trivially a parametrization:



$$(x, y, z) = (costheta, , sintheta, , 0) ; ; ; text{with} ; theta in [0,,2pi)$$



I would like to determine the parametric equations of the red curve, very badly drawn in Paint, where I mean a sinusoidal curve along the blue circumference.



Although I thought about it a lot, I still could not figure out how to derive these parametric equation. Any ideas?






curves parametrization







share|cite|improve this question




















  • 1




    It looks like a curve of the form $$ r(theta) = r_0 + A cos{(omega theta )} $$ But it's not very clear what you mean with the hand-drawn curve. Is it supposed to come out of the $z=0$-plane?
    – Matti P.
    59 mins ago








  • 1




    @MattiP.: Yes, it must exit the $z = 0$ plane and have the circumference as the average line.
    – TeM
    38 mins ago


















5














With reference to the following image:



enter image description here



the blue curve has trivially a parametrization:



$$(x, y, z) = (costheta, , sintheta, , 0) ; ; ; text{with} ; theta in [0,,2pi)$$



I would like to determine the parametric equations of the red curve, very badly drawn in Paint, where I mean a sinusoidal curve along the blue circumference.



Although I thought about it a lot, I still could not figure out how to derive these parametric equation. Any ideas?






curves parametrization







share|cite|improve this question




















  • 1




    It looks like a curve of the form $$ r(theta) = r_0 + A cos{(omega theta )} $$ But it's not very clear what you mean with the hand-drawn curve. Is it supposed to come out of the $z=0$-plane?
    – Matti P.
    59 mins ago








  • 1




    @MattiP.: Yes, it must exit the $z = 0$ plane and have the circumference as the average line.
    – TeM
    38 mins ago
















5












5








5







With reference to the following image:



enter image description here



the blue curve has trivially a parametrization:



$$(x, y, z) = (costheta, , sintheta, , 0) ; ; ; text{with} ; theta in [0,,2pi)$$



I would like to determine the parametric equations of the red curve, very badly drawn in Paint, where I mean a sinusoidal curve along the blue circumference.



Although I thought about it a lot, I still could not figure out how to derive these parametric equation. Any ideas?






curves parametrization







share|cite|improve this question















With reference to the following image:



enter image description here



the blue curve has trivially a parametrization:



$$(x, y, z) = (costheta, , sintheta, , 0) ; ; ; text{with} ; theta in [0,,2pi)$$



I would like to determine the parametric equations of the red curve, very badly drawn in Paint, where I mean a sinusoidal curve along the blue circumference.



Although I thought about it a lot, I still could not figure out how to derive these parametric equation. Any ideas?






curves parametrization




curves parametrization






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 29 mins ago









dmtri

1,4181521




1,4181521










asked 1 hour ago









TeM

415315




415315








  • 1




    It looks like a curve of the form $$ r(theta) = r_0 + A cos{(omega theta )} $$ But it's not very clear what you mean with the hand-drawn curve. Is it supposed to come out of the $z=0$-plane?
    – Matti P.
    59 mins ago








  • 1




    @MattiP.: Yes, it must exit the $z = 0$ plane and have the circumference as the average line.
    – TeM
    38 mins ago
















  • 1




    It looks like a curve of the form $$ r(theta) = r_0 + A cos{(omega theta )} $$ But it's not very clear what you mean with the hand-drawn curve. Is it supposed to come out of the $z=0$-plane?
    – Matti P.
    59 mins ago








  • 1




    @MattiP.: Yes, it must exit the $z = 0$ plane and have the circumference as the average line.
    – TeM
    38 mins ago










1




1




It looks like a curve of the form $$ r(theta) = r_0 + A cos{(omega theta )} $$ But it's not very clear what you mean with the hand-drawn curve. Is it supposed to come out of the $z=0$-plane?
– Matti P.
59 mins ago






It looks like a curve of the form $$ r(theta) = r_0 + A cos{(omega theta )} $$ But it's not very clear what you mean with the hand-drawn curve. Is it supposed to come out of the $z=0$-plane?
– Matti P.
59 mins ago






1




1




@MattiP.: Yes, it must exit the $z = 0$ plane and have the circumference as the average line.
– TeM
38 mins ago






@MattiP.: Yes, it must exit the $z = 0$ plane and have the circumference as the average line.
– TeM
38 mins ago












1 Answer
1






active

oldest

votes


















6














You can try$$thetamapstoleft(costheta,sintheta,frac{cos(8theta)}8right),$$for instance.



enter image description here






share|cite|improve this answer























  • It's exactly what I wanted, thank you! Could you tell me how you managed to understand it?
    – TeM
    30 mins ago








  • 2




    The $z$ coordinate had to be a waving line again, and so I thought about $cos(8theta)$, but then the waves would go too high and too low. That's why I divided by $8$.
    – José Carlos Santos
    26 mins ago










  • Perfect! Before closing I also wanted to ask how to get the parametric equations of the red curve if it lay in the $z = 0$ plane. Is it better that I open a new question or change the request?
    – TeM
    17 mins ago












  • No need for that. Just consider:$$thetamapstoleft(costheta+frac{cos(8theta)}8,sintheta,0right).$$
    – José Carlos Santos
    10 mins ago











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









6














You can try$$thetamapstoleft(costheta,sintheta,frac{cos(8theta)}8right),$$for instance.



enter image description here






share|cite|improve this answer























  • It's exactly what I wanted, thank you! Could you tell me how you managed to understand it?
    – TeM
    30 mins ago








  • 2




    The $z$ coordinate had to be a waving line again, and so I thought about $cos(8theta)$, but then the waves would go too high and too low. That's why I divided by $8$.
    – José Carlos Santos
    26 mins ago










  • Perfect! Before closing I also wanted to ask how to get the parametric equations of the red curve if it lay in the $z = 0$ plane. Is it better that I open a new question or change the request?
    – TeM
    17 mins ago












  • No need for that. Just consider:$$thetamapstoleft(costheta+frac{cos(8theta)}8,sintheta,0right).$$
    – José Carlos Santos
    10 mins ago
















6














You can try$$thetamapstoleft(costheta,sintheta,frac{cos(8theta)}8right),$$for instance.



enter image description here






share|cite|improve this answer























  • It's exactly what I wanted, thank you! Could you tell me how you managed to understand it?
    – TeM
    30 mins ago








  • 2




    The $z$ coordinate had to be a waving line again, and so I thought about $cos(8theta)$, but then the waves would go too high and too low. That's why I divided by $8$.
    – José Carlos Santos
    26 mins ago










  • Perfect! Before closing I also wanted to ask how to get the parametric equations of the red curve if it lay in the $z = 0$ plane. Is it better that I open a new question or change the request?
    – TeM
    17 mins ago












  • No need for that. Just consider:$$thetamapstoleft(costheta+frac{cos(8theta)}8,sintheta,0right).$$
    – José Carlos Santos
    10 mins ago














6












6








6






You can try$$thetamapstoleft(costheta,sintheta,frac{cos(8theta)}8right),$$for instance.



enter image description here






share|cite|improve this answer














You can try$$thetamapstoleft(costheta,sintheta,frac{cos(8theta)}8right),$$for instance.



enter image description here







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 34 mins ago

























answered 59 mins ago









José Carlos Santos

150k22121221




150k22121221












  • It's exactly what I wanted, thank you! Could you tell me how you managed to understand it?
    – TeM
    30 mins ago








  • 2




    The $z$ coordinate had to be a waving line again, and so I thought about $cos(8theta)$, but then the waves would go too high and too low. That's why I divided by $8$.
    – José Carlos Santos
    26 mins ago










  • Perfect! Before closing I also wanted to ask how to get the parametric equations of the red curve if it lay in the $z = 0$ plane. Is it better that I open a new question or change the request?
    – TeM
    17 mins ago












  • No need for that. Just consider:$$thetamapstoleft(costheta+frac{cos(8theta)}8,sintheta,0right).$$
    – José Carlos Santos
    10 mins ago


















  • It's exactly what I wanted, thank you! Could you tell me how you managed to understand it?
    – TeM
    30 mins ago








  • 2




    The $z$ coordinate had to be a waving line again, and so I thought about $cos(8theta)$, but then the waves would go too high and too low. That's why I divided by $8$.
    – José Carlos Santos
    26 mins ago










  • Perfect! Before closing I also wanted to ask how to get the parametric equations of the red curve if it lay in the $z = 0$ plane. Is it better that I open a new question or change the request?
    – TeM
    17 mins ago












  • No need for that. Just consider:$$thetamapstoleft(costheta+frac{cos(8theta)}8,sintheta,0right).$$
    – José Carlos Santos
    10 mins ago
















It's exactly what I wanted, thank you! Could you tell me how you managed to understand it?
– TeM
30 mins ago






It's exactly what I wanted, thank you! Could you tell me how you managed to understand it?
– TeM
30 mins ago






2




2




The $z$ coordinate had to be a waving line again, and so I thought about $cos(8theta)$, but then the waves would go too high and too low. That's why I divided by $8$.
– José Carlos Santos
26 mins ago




The $z$ coordinate had to be a waving line again, and so I thought about $cos(8theta)$, but then the waves would go too high and too low. That's why I divided by $8$.
– José Carlos Santos
26 mins ago












Perfect! Before closing I also wanted to ask how to get the parametric equations of the red curve if it lay in the $z = 0$ plane. Is it better that I open a new question or change the request?
– TeM
17 mins ago






Perfect! Before closing I also wanted to ask how to get the parametric equations of the red curve if it lay in the $z = 0$ plane. Is it better that I open a new question or change the request?
– TeM
17 mins ago














No need for that. Just consider:$$thetamapstoleft(costheta+frac{cos(8theta)}8,sintheta,0right).$$
– José Carlos Santos
10 mins ago




No need for that. Just consider:$$thetamapstoleft(costheta+frac{cos(8theta)}8,sintheta,0right).$$
– José Carlos Santos
10 mins ago


















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