trapezoid approximation of sinus function
I want to find a reasonable good periodic trapezoid function approximation for the sinus (or cosinus) function.
My use case is it to divide the year in 4 epochs:
- visible light length stays roughly the same (around winter solistice),
- visible light lenghtens by roughly x minutes a day (around spring equinox),
- visible light length stays roughly the same (around summer solistice),
- visible light length shortens by roughly x minutes a day (around fall equinox).
I'd be interrested in 3 values:
- length of visible light during quasi-contant epochs
- steepness of change in the transit epoch
- days of transition between the epochs
If I did not make a mistake, this requires estimation or calculation of two parameters of the trapezoid function: amplitude and one of either lenght in high or low phase or steepness of the transition. Additionally of course some weighting function to establish a minimum distance would be required. If sensible/possible I'd go for sum of absolute difference.
I tried to establish a picewise formula for the difference between sinus and a periodic trapezoid function, to then minimize this, but it was over my head. Unfortunately I am not fluent enough in a mathematical program language to state that problem. Before I do a brute force numerical approximation I'd rather have a closed symbolic expression that yields a nice and exact solution.
optimization approximation periodic-functions
asked 2 days ago
trapicki
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I want to find a reasonable good periodic trapezoid function approximation for the sinus (or cosinus) function.
My use case is it to divide the year in 4 epochs:
- visible light length stays roughly the same (around winter solistice),
- visible light lenghtens by roughly x minutes a day (around spring equinox),
- visible light length stays roughly the same (around summer solistice),
- visible light length shortens by roughly x minutes a day (around fall equinox).
I'd be interrested in 3 values:
- length of visible light during quasi-contant epochs
- steepness of change in the transit epoch
- days of transition between the epochs
If I did not make a mistake, this requires estimation or calculation of two parameters of the trapezoid function: amplitude and one of either lenght in high or low phase or steepness of the transition. Additionally of course some weighting function to establish a minimum distance would be required. If sensible/possible I'd go for sum of absolute difference.
I tried to establish a picewise formula for the difference between sinus and a periodic trapezoid function, to then minimize this, but it was over my head. Unfortunately I am not fluent enough in a mathematical program language to state that problem. Before I do a brute force numerical approximation I'd rather have a closed symbolic expression that yields a nice and exact solution.
optimization approximation periodic-functions
asked 2 days ago
trapicki
99
New contributor
trapicki is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I assume you mean "sine" and "cosine", I've never heard them called "sinus" and "cosinus" before. The question that needs to be asked is what criterion you want to use to determine a "good" approximation. The two obvious choices are to either minimize the maximum difference between function values, or else to minimize the area between the two curves. But which would you prefer? Or is there some other criterion that would be better in this case? There is no need to "brute force" it. Either way, the problem is not hard to solve analytically.
– Paul Sinclair
yesterday
@PaulSinclair. These are the French names of these functions.
– Claude Leibovici
yesterday
add a comment |
I want to find a reasonable good periodic trapezoid function approximation for the sinus (or cosinus) function.
My use case is it to divide the year in 4 epochs:
- visible light length stays roughly the same (around winter solistice),
- visible light lenghtens by roughly x minutes a day (around spring equinox),
- visible light length stays roughly the same (around summer solistice),
- visible light length shortens by roughly x minutes a day (around fall equinox).
I'd be interrested in 3 values:
- length of visible light during quasi-contant epochs
- steepness of change in the transit epoch
- days of transition between the epochs
If I did not make a mistake, this requires estimation or calculation of two parameters of the trapezoid function: amplitude and one of either lenght in high or low phase or steepness of the transition. Additionally of course some weighting function to establish a minimum distance would be required. If sensible/possible I'd go for sum of absolute difference.
I tried to establish a picewise formula for the difference between sinus and a periodic trapezoid function, to then minimize this, but it was over my head. Unfortunately I am not fluent enough in a mathematical program language to state that problem. Before I do a brute force numerical approximation I'd rather have a closed symbolic expression that yields a nice and exact solution.
optimization approximation periodic-functions
asked 2 days ago
trapicki
99
New contributor
trapicki is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I want to find a reasonable good periodic trapezoid function approximation for the sinus (or cosinus) function.
My use case is it to divide the year in 4 epochs:
- visible light length stays roughly the same (around winter solistice),
- visible light lenghtens by roughly x minutes a day (around spring equinox),
- visible light length stays roughly the same (around summer solistice),
- visible light length shortens by roughly x minutes a day (around fall equinox).
I'd be interrested in 3 values:
- length of visible light during quasi-contant epochs
- steepness of change in the transit epoch
- days of transition between the epochs
If I did not make a mistake, this requires estimation or calculation of two parameters of the trapezoid function: amplitude and one of either lenght in high or low phase or steepness of the transition. Additionally of course some weighting function to establish a minimum distance would be required. If sensible/possible I'd go for sum of absolute difference.
I tried to establish a picewise formula for the difference between sinus and a periodic trapezoid function, to then minimize this, but it was over my head. Unfortunately I am not fluent enough in a mathematical program language to state that problem. Before I do a brute force numerical approximation I'd rather have a closed symbolic expression that yields a nice and exact solution.
optimization approximation periodic-functions
optimization approximation periodic-functions
asked 2 days ago
trapicki
99
New contributor
trapicki is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
trapicki
99
New contributor
trapicki is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
trapicki
99
New contributor
trapicki is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
trapicki
99
asked 2 days ago
trapicki
99
99
New contributor
trapicki is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
trapicki is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
trapicki is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I assume you mean "sine" and "cosine", I've never heard them called "sinus" and "cosinus" before. The question that needs to be asked is what criterion you want to use to determine a "good" approximation. The two obvious choices are to either minimize the maximum difference between function values, or else to minimize the area between the two curves. But which would you prefer? Or is there some other criterion that would be better in this case? There is no need to "brute force" it. Either way, the problem is not hard to solve analytically.
– Paul Sinclair
yesterday
@PaulSinclair. These are the French names of these functions.
– Claude Leibovici
yesterday
add a comment |
I assume you mean "sine" and "cosine", I've never heard them called "sinus" and "cosinus" before. The question that needs to be asked is what criterion you want to use to determine a "good" approximation. The two obvious choices are to either minimize the maximum difference between function values, or else to minimize the area between the two curves. But which would you prefer? Or is there some other criterion that would be better in this case? There is no need to "brute force" it. Either way, the problem is not hard to solve analytically.
– Paul Sinclair
yesterday
@PaulSinclair. These are the French names of these functions.
– Claude Leibovici
yesterday
I assume you mean "sine" and "cosine", I've never heard them called "sinus" and "cosinus" before. The question that needs to be asked is what criterion you want to use to determine a "good" approximation. The two obvious choices are to either minimize the maximum difference between function values, or else to minimize the area between the two curves. But which would you prefer? Or is there some other criterion that would be better in this case? There is no need to "brute force" it. Either way, the problem is not hard to solve analytically.
– Paul Sinclair
yesterday
I assume you mean "sine" and "cosine", I've never heard them called "sinus" and "cosinus" before. The question that needs to be asked is what criterion you want to use to determine a "good" approximation. The two obvious choices are to either minimize the maximum difference between function values, or else to minimize the area between the two curves. But which would you prefer? Or is there some other criterion that would be better in this case? There is no need to "brute force" it. Either way, the problem is not hard to solve analytically.
– Paul Sinclair
yesterday
@PaulSinclair. These are the French names of these functions.
– Claude Leibovici
yesterday
@PaulSinclair. These are the French names of these functions.
– Claude Leibovici
yesterday
add a comment |
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I assume you mean "sine" and "cosine", I've never heard them called "sinus" and "cosinus" before. The question that needs to be asked is what criterion you want to use to determine a "good" approximation. The two obvious choices are to either minimize the maximum difference between function values, or else to minimize the area between the two curves. But which would you prefer? Or is there some other criterion that would be better in this case? There is no need to "brute force" it. Either way, the problem is not hard to solve analytically.
– Paul Sinclair
yesterday
@PaulSinclair. These are the French names of these functions.
– Claude Leibovici
yesterday