Mixing
trapezoid approximation of sinus function
$begingroup$
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 Jan 3 at 15:58
trapickitrapicki
99
$endgroup$
add a comment |
$begingroup$
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 Jan 3 at 15:58
trapickitrapicki
99
$endgroup$
$begingroup$
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.
$endgroup$
– Paul Sinclair
Jan 4 at 0:31
$begingroup$
@PaulSinclair. These are the French names of these functions.
$endgroup$
– Claude Leibovici
Jan 4 at 10:45
add a comment |
$begingroup$
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 Jan 3 at 15:58
trapickitrapicki
99
$endgroup$
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 Jan 3 at 15:58
trapickitrapicki
99
asked Jan 3 at 15:58
trapickitrapicki
99
asked Jan 3 at 15:58
trapickitrapicki
99
asked Jan 3 at 15:58
trapickitrapicki
99
asked Jan 3 at 15:58
trapickitrapicki
99
99
$begingroup$
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.
$endgroup$
– Paul Sinclair
Jan 4 at 0:31
$begingroup$
@PaulSinclair. These are the French names of these functions.
$endgroup$
– Claude Leibovici
Jan 4 at 10:45
add a comment |
$begingroup$
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.
$endgroup$
– Paul Sinclair
Jan 4 at 0:31
$begingroup$
@PaulSinclair. These are the French names of these functions.
$endgroup$
– Claude Leibovici
Jan 4 at 10:45
$begingroup$
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.
$endgroup$
– Paul Sinclair
Jan 4 at 0:31
$begingroup$
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.
$endgroup$
– Paul Sinclair
Jan 4 at 0:31
$begingroup$
@PaulSinclair. These are the French names of these functions.
$endgroup$
– Claude Leibovici
Jan 4 at 10:45
$begingroup$
@PaulSinclair. These are the French names of these functions.
$endgroup$
– Claude Leibovici
Jan 4 at 10:45
add a comment |
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$begingroup$
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.
$endgroup$
– Paul Sinclair
Jan 4 at 0:31
$begingroup$
@PaulSinclair. These are the French names of these functions.
$endgroup$
– Claude Leibovici
Jan 4 at 10:45