5 ms·

> their adjacent sides certainly aren't at right angles They certainly are. Parallels and meridians always meet at 90 degrees. https://www.geogebra.org/m/kNtN

> their adjacent sides certainly aren't at right angles They certainly are. Parallels and meridians always meet at 90 degrees. https://www.geogebra.org/m/kNtNZzsS https://www.geogebra.org/m/kNtNZzsS

doersino 6y ago
Right angles with regard to the latitude-longitude coordinate system, but not with regard to the surface of the earth. Recall that all meridians intersect at the poles.
- colanderman 6y ago
  @tantalor is technically correct; meridians and parallels intersect at right angles, even with respect to the (idealized) surface of the earth. Proof sketch: if the angles involved in a crossing were not all 90°, then either the crossing would not be symmetric about the meridian (it is), or one or both lines must have a sharp "kink" (i.e., have a non-smooth first derivative) at the intersection (neither does). However, parallels, while smooth, are not straight, they "curl" toward the nearest pole (i.e., non-zero second derivative, relative to the earth's surface). This accounts for the non-"rectangular" shape of quadrangles and can be replicated on a 2D plane.
- tantalor 6y ago
  Yes with regard to the surface of the earth. Take the lines tangent to the meridian & parallel at the point of intersection. Observe these lines are perpendicular. Hence the meridian & parallel meet at a right angle.
  - simonh 6y ago
    What about the ‘boxes’ where the top edge is, er, the point at one of the poles?
    - tantalor 6y ago
      Yeah not the pole of course, but you can cut them off at at ±89.999 degrees and it still works.
  - doersino 6y ago
    You’re right, I apologize for my smartass answer.
- kgwgk 6y ago
  > Recall that all meridians intersect at the poles. How is that relevant regarding “Parallels and meridians always meet at 90 degrees“?
roelschroeven 6y ago
I'm not 100% sure about that. Are angles between circles on a sphere defined by the angles between their tangent lines? If it were true, you could for each point/meridian combination construct an infinite amount of different circles that would all meet at right angles with that meridian (just vary the circle's radius). That doesn't feel quite right. According to Wikipedia [1], "in spherical geometry, angles are defined between great circles". Meridians are great circles, but parallels are not. Possibly the angle is simply not defined for circles that are not great circles? [1] https://en.wikipedia.org/wiki/Spherical_geometry https://en.wikipedia.org/wiki/Spherical_geometry
- tantalor 6y ago
  It only matters what is the angle between the lines tangent to the curves at the point of intersection. Here's an example: https://www.desmos.com/calculator/is7mush1ma https://www.desmos.com/calculator/is7mush1ma The red and blue circles are both perpendicular to the green circle at the origin.
  - roelschroeven 6y ago
    OK, thanks, today I learned something new.
caf 6y ago
Thankyou for the correction.

> their adjacent sides certainly aren't at right angles They certainly are. Parallels and meridians always meet at 90 degrees. https://www.geogebra.org/m/kNtNZzsS https://www.geogebra.org/m/kNtNZzsS

doersino 6y ago

Right angles with regard to the latitude-longitude coordinate system, but not with regard to the surface of the earth. Recall that all meridians intersect at the poles.

colanderman 6y ago
@tantalor is technically correct; meridians and parallels intersect at right angles, even with respect to the (idealized) surface of the earth. Proof sketch: if the angles involved in a crossing were not all 90°, then either the crossing would not be symmetric about the meridian (it is), or one or both lines must have a sharp "kink" (i.e., have a non-smooth first derivative) at the intersection (neither does). However, parallels, while smooth, are not straight, they "curl" toward the nearest pole (i.e., non-zero second derivative, relative to the earth's surface). This accounts for the non-"rectangular" shape of quadrangles and can be replicated on a 2D plane.
tantalor 6y ago
Yes with regard to the surface of the earth. Take the lines tangent to the meridian & parallel at the point of intersection. Observe these lines are perpendicular. Hence the meridian & parallel meet at a right angle.
- simonh 6y ago
  What about the ‘boxes’ where the top edge is, er, the point at one of the poles?
  - tantalor 6y ago
    Yeah not the pole of course, but you can cut them off at at ±89.999 degrees and it still works.
- doersino 6y ago
  You’re right, I apologize for my smartass answer.
kgwgk 6y ago
> Recall that all meridians intersect at the poles. How is that relevant regarding “Parallels and meridians always meet at 90 degrees“?

roelschroeven 6y ago

I'm not 100% sure about that. Are angles between circles on a sphere defined by the angles between their tangent lines? If it were true, you could for each point/meridian combination construct an infinite amount of different circles that would all meet at right angles with that meridian (just vary the circle's radius). That doesn't feel quite right. According to Wikipedia [1], "in spherical geometry, angles are defined between great circles". Meridians are great circles, but parallels are not. Possibly the angle is simply not defined for circles that are not great circles? [1] https://en.wikipedia.org/wiki/Spherical_geometry https://en.wikipedia.org/wiki/Spherical_geometry

tantalor 6y ago
It only matters what is the angle between the lines tangent to the curves at the point of intersection. Here's an example: https://www.desmos.com/calculator/is7mush1ma https://www.desmos.com/calculator/is7mush1ma The red and blue circles are both perpendicular to the green circle at the origin.
- roelschroeven 6y ago
  OK, thanks, today I learned something new.

caf 6y ago

Thankyou for the correction.