Mixing
Sin, Cos, then?
$begingroup$
I hope this random thought is on topic. If not please notify me before penalising me for it.
I have reflected that Sin and Cos provide answers for trigonometry problems with cartesian pairs on a 2D plane inside a unit circle.
Why stop there?
Is there a further similar function which adds the z axis so problems can be solved in 3D, inside a unit sphere? If there isn't, is there an explanation for why?
trigonometry
edited Jan 30 at 14:50
Mauro ALLEGRANZA
67.7k449117
asked Jan 30 at 14:45
Georgina DavenportGeorgina Davenport
48118
$endgroup$
add a comment |
$begingroup$
I hope this random thought is on topic. If not please notify me before penalising me for it.
I have reflected that Sin and Cos provide answers for trigonometry problems with cartesian pairs on a 2D plane inside a unit circle.
Why stop there?
Is there a further similar function which adds the z axis so problems can be solved in 3D, inside a unit sphere? If there isn't, is there an explanation for why?
trigonometry
edited Jan 30 at 14:50
Mauro ALLEGRANZA
67.7k449117
asked Jan 30 at 14:45
Georgina DavenportGeorgina Davenport
48118
$endgroup$
2
$begingroup$
See Spherical trigonometry.
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 14:49
1
$begingroup$
Check out "solid trigonometry": mathworld.wolfram.com/SolidAngle.html
$endgroup$
– user247327
Jan 30 at 14:53
1
$begingroup$
The history of the move from two dimensions to three is an interesting one and led Hamilton to discover the Quaternions. Also observe that rotations of a sphere are possible with an infinite number of axes and that the central involution (reflection in the centre) has determinant $-1$ in three dimensions. None of this is precisely on your point, but it shows some issues which arise in finding the most convenient way to work with a sphere in contrast to a circle.
$endgroup$
– Mark Bennet
Jan 30 at 15:07
add a comment |
$begingroup$
I hope this random thought is on topic. If not please notify me before penalising me for it.
I have reflected that Sin and Cos provide answers for trigonometry problems with cartesian pairs on a 2D plane inside a unit circle.
Why stop there?
Is there a further similar function which adds the z axis so problems can be solved in 3D, inside a unit sphere? If there isn't, is there an explanation for why?
trigonometry
edited Jan 30 at 14:50
Mauro ALLEGRANZA
67.7k449117
asked Jan 30 at 14:45
Georgina DavenportGeorgina Davenport
48118
$endgroup$
I hope this random thought is on topic. If not please notify me before penalising me for it.
I have reflected that Sin and Cos provide answers for trigonometry problems with cartesian pairs on a 2D plane inside a unit circle.
Why stop there?
Is there a further similar function which adds the z axis so problems can be solved in 3D, inside a unit sphere? If there isn't, is there an explanation for why?
trigonometry
trigonometry
edited Jan 30 at 14:50
Mauro ALLEGRANZA
67.7k449117
asked Jan 30 at 14:45
Georgina DavenportGeorgina Davenport
48118
edited Jan 30 at 14:50
Mauro ALLEGRANZA
67.7k449117
asked Jan 30 at 14:45
Georgina DavenportGeorgina Davenport
48118
edited Jan 30 at 14:50
Mauro ALLEGRANZA
67.7k449117
edited Jan 30 at 14:50
Mauro ALLEGRANZA
67.7k449117
edited Jan 30 at 14:50
Mauro ALLEGRANZA
67.7k449117
67.7k449117
asked Jan 30 at 14:45
Georgina DavenportGeorgina Davenport
48118
asked Jan 30 at 14:45
Georgina DavenportGeorgina Davenport
48118
asked Jan 30 at 14:45
Georgina DavenportGeorgina Davenport
48118
48118
2
$begingroup$
See Spherical trigonometry.
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 14:49
1
$begingroup$
Check out "solid trigonometry": mathworld.wolfram.com/SolidAngle.html
$endgroup$
– user247327
Jan 30 at 14:53
1
$begingroup$
The history of the move from two dimensions to three is an interesting one and led Hamilton to discover the Quaternions. Also observe that rotations of a sphere are possible with an infinite number of axes and that the central involution (reflection in the centre) has determinant $-1$ in three dimensions. None of this is precisely on your point, but it shows some issues which arise in finding the most convenient way to work with a sphere in contrast to a circle.
$endgroup$
– Mark Bennet
Jan 30 at 15:07
add a comment |
2
$begingroup$
See Spherical trigonometry.
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 14:49
1
$begingroup$
Check out "solid trigonometry": mathworld.wolfram.com/SolidAngle.html
$endgroup$
– user247327
Jan 30 at 14:53
1
$begingroup$
The history of the move from two dimensions to three is an interesting one and led Hamilton to discover the Quaternions. Also observe that rotations of a sphere are possible with an infinite number of axes and that the central involution (reflection in the centre) has determinant $-1$ in three dimensions. None of this is precisely on your point, but it shows some issues which arise in finding the most convenient way to work with a sphere in contrast to a circle.
$endgroup$
– Mark Bennet
Jan 30 at 15:07
2
2
$begingroup$
See Spherical trigonometry.
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 14:49
$begingroup$
See Spherical trigonometry.
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 14:49
1
1
$begingroup$
Check out "solid trigonometry": mathworld.wolfram.com/SolidAngle.html
$endgroup$
– user247327
Jan 30 at 14:53
$begingroup$
Check out "solid trigonometry": mathworld.wolfram.com/SolidAngle.html
$endgroup$
– user247327
Jan 30 at 14:53
1
1
$begingroup$
The history of the move from two dimensions to three is an interesting one and led Hamilton to discover the Quaternions. Also observe that rotations of a sphere are possible with an infinite number of axes and that the central involution (reflection in the centre) has determinant $-1$ in three dimensions. None of this is precisely on your point, but it shows some issues which arise in finding the most convenient way to work with a sphere in contrast to a circle.
$endgroup$
– Mark Bennet
Jan 30 at 15:07
$begingroup$
The history of the move from two dimensions to three is an interesting one and led Hamilton to discover the Quaternions. Also observe that rotations of a sphere are possible with an infinite number of axes and that the central involution (reflection in the centre) has determinant $-1$ in three dimensions. None of this is precisely on your point, but it shows some issues which arise in finding the most convenient way to work with a sphere in contrast to a circle.
$endgroup$
– Mark Bennet
Jan 30 at 15:07
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
In spherical co-ordinates each point in $mathbb{R}^3$ has a distance from the origin $r$ and two angles $theta$ and $phi$. This is analogous to polar co-ordinates in $mathbb{R}^2$ except there are two angles instead of one.
Converting to Cartesian co-ordinates, the $x,y,z$ co-ordinates of a point are functions of $r$ and the two angles:
$(x,y,z) = left(rf(theta, phi), space rg(theta, phi), space rh(theta, phi) right)$
but, as it happens, the functions $f,g,h$ can be expressed simply in terms of $sin$ and $cos$:
$f(theta, phi)=sin (theta) cos (phi) \ g(theta, phi)=sin (theta) sin (phi)\ h(theta, phi)=cos (theta)$
As far as I know the functions $f, g, h$ have never been given specific names.
answered Jan 30 at 15:10
gandalf61gandalf61
9,219825
$endgroup$
add a comment |
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$begingroup$
In spherical co-ordinates each point in $mathbb{R}^3$ has a distance from the origin $r$ and two angles $theta$ and $phi$. This is analogous to polar co-ordinates in $mathbb{R}^2$ except there are two angles instead of one.
Converting to Cartesian co-ordinates, the $x,y,z$ co-ordinates of a point are functions of $r$ and the two angles:
$(x,y,z) = left(rf(theta, phi), space rg(theta, phi), space rh(theta, phi) right)$
but, as it happens, the functions $f,g,h$ can be expressed simply in terms of $sin$ and $cos$:
$f(theta, phi)=sin (theta) cos (phi) \ g(theta, phi)=sin (theta) sin (phi)\ h(theta, phi)=cos (theta)$
As far as I know the functions $f, g, h$ have never been given specific names.
answered Jan 30 at 15:10
gandalf61gandalf61
9,219825
$endgroup$
add a comment |
$begingroup$
In spherical co-ordinates each point in $mathbb{R}^3$ has a distance from the origin $r$ and two angles $theta$ and $phi$. This is analogous to polar co-ordinates in $mathbb{R}^2$ except there are two angles instead of one.
Converting to Cartesian co-ordinates, the $x,y,z$ co-ordinates of a point are functions of $r$ and the two angles:
$(x,y,z) = left(rf(theta, phi), space rg(theta, phi), space rh(theta, phi) right)$
but, as it happens, the functions $f,g,h$ can be expressed simply in terms of $sin$ and $cos$:
$f(theta, phi)=sin (theta) cos (phi) \ g(theta, phi)=sin (theta) sin (phi)\ h(theta, phi)=cos (theta)$
As far as I know the functions $f, g, h$ have never been given specific names.
answered Jan 30 at 15:10
gandalf61gandalf61
9,219825
$endgroup$
add a comment |
$begingroup$
In spherical co-ordinates each point in $mathbb{R}^3$ has a distance from the origin $r$ and two angles $theta$ and $phi$. This is analogous to polar co-ordinates in $mathbb{R}^2$ except there are two angles instead of one.
Converting to Cartesian co-ordinates, the $x,y,z$ co-ordinates of a point are functions of $r$ and the two angles:
$(x,y,z) = left(rf(theta, phi), space rg(theta, phi), space rh(theta, phi) right)$
but, as it happens, the functions $f,g,h$ can be expressed simply in terms of $sin$ and $cos$:
$f(theta, phi)=sin (theta) cos (phi) \ g(theta, phi)=sin (theta) sin (phi)\ h(theta, phi)=cos (theta)$
As far as I know the functions $f, g, h$ have never been given specific names.
answered Jan 30 at 15:10
gandalf61gandalf61
9,219825
$endgroup$
In spherical co-ordinates each point in $mathbb{R}^3$ has a distance from the origin $r$ and two angles $theta$ and $phi$. This is analogous to polar co-ordinates in $mathbb{R}^2$ except there are two angles instead of one.
Converting to Cartesian co-ordinates, the $x,y,z$ co-ordinates of a point are functions of $r$ and the two angles:
$(x,y,z) = left(rf(theta, phi), space rg(theta, phi), space rh(theta, phi) right)$
but, as it happens, the functions $f,g,h$ can be expressed simply in terms of $sin$ and $cos$:
$f(theta, phi)=sin (theta) cos (phi) \ g(theta, phi)=sin (theta) sin (phi)\ h(theta, phi)=cos (theta)$
As far as I know the functions $f, g, h$ have never been given specific names.
answered Jan 30 at 15:10
gandalf61gandalf61
9,219825
answered Jan 30 at 15:10
gandalf61gandalf61
9,219825
answered Jan 30 at 15:10
gandalf61gandalf61
9,219825
answered Jan 30 at 15:10
gandalf61gandalf61
9,219825
9,219825
add a comment |
add a comment |
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2
$begingroup$
See Spherical trigonometry.
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 14:49
1
$begingroup$
Check out "solid trigonometry": mathworld.wolfram.com/SolidAngle.html
$endgroup$
– user247327
Jan 30 at 14:53
1
$begingroup$
The history of the move from two dimensions to three is an interesting one and led Hamilton to discover the Quaternions. Also observe that rotations of a sphere are possible with an infinite number of axes and that the central involution (reflection in the centre) has determinant $-1$ in three dimensions. None of this is precisely on your point, but it shows some issues which arise in finding the most convenient way to work with a sphere in contrast to a circle.
$endgroup$
– Mark Bennet
Jan 30 at 15:07