Convert equation from cartesian to spherical

In summary, converting the equation y=x to spherical coordinates results in the simplified equation sin(t) = cos(t), which can be further simplified to tan(t) = 1. The value of t is approximately 0.78 and can be expressed as the fraction pi/4.
  • #1
glog
17
0
This should be relitively simple:

y = x ... convert to spherical coords:

p*sin(r)*sin(t) = p*sin(r)*cos(t)

which reduces to...

sin(t) = cos(t)

tan(t) = 1 (is this right?)
t =~ 0.78... (Can i get a nice fraction for this?)

Any help is appreciated.

- glog
 
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  • #2
Looks good to me.

pi/4

you simply substitute and simplify.
 
  • #3
in other words, "y= x" in spherical coordinates reduces to the set of points where t= [itex]\theta= \pi/4[/itex], r= [itex]\phi[/itex] and p= [itex]\rho[/itex] can be anything. Do you see that that is, in fact, the same as the plane y= x?
 
  • #4
yep makes perfect sense...
only one angle is fixed :)
 

1. What is the formula for converting an equation from Cartesian to spherical coordinates?

The formula for converting an equation from Cartesian to spherical coordinates is:
x = r * sin(θ) * cos(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(θ)
Where r is the distance from the origin, θ is the polar angle, and φ is the azimuthal angle.

2. How do I know when to use Cartesian or spherical coordinates?

Cartesian coordinates are typically used when dealing with flat, 2D surfaces, while spherical coordinates are used for 3D surfaces that are symmetrical around a central point. It is also important to consider the complexity of the equation and which coordinate system would make it easier to solve.

3. Can I convert an equation from spherical to Cartesian coordinates?

Yes, you can convert an equation from spherical to Cartesian coordinates by using the inverse of the formula mentioned above:
x = r * sin(θ) * cos(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(θ)
Where r is the distance from the origin, θ is the polar angle, and φ is the azimuthal angle.

4. How do I plot an equation in spherical coordinates?

To plot an equation in spherical coordinates, you can use a graphing calculator or software that allows you to input equations in spherical coordinates. Alternatively, you can convert the equation to Cartesian coordinates and plot it on a 2D graph.

5. What are the advantages of using spherical coordinates over Cartesian coordinates?

Spherical coordinates are better suited for dealing with 3D surfaces that have spherical symmetry, such as spheres, cones, and cylinders. They also make it easier to visualize and solve equations involving polar angles and azimuthal angles. However, Cartesian coordinates are more commonly used in everyday applications and are easier to work with when dealing with flat surfaces.

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