Angle of intersection: polar versus cartesian

[JanClaesen]

Is it correct that the angle of intersection of two curves is the same in x,y coordinates as in r,theta coordinates? If so, why is this?

 

[elibj123]

Is it correct the a point represented in x,y coordinates is the same point we represent r,theta coordinates?
Either I don't get your question, or you don't understand that these are just representations, they cannot change the value of a property.

 

[JanClaesen]

My question is:

for example:
the straight line
x = y (1.1)
or
theta = 1/sqrt(2) (1.2)

and

the circle
x² + y² = 1 (2.1)
or
r = 1 (2.2)

In this case 1.1 and 2.1 have the same angle of intersection (so the angle described by the tangents of the straight line and the circle at their point of intersection) as do 1.2 and 2.2, but why is this true in the general case, what am I misunderstanding?

 

[HallsofIvy]

Yes, and elibj123's point is that things like length of a line segment, measure of an angle, area of a plane figure, etc. are geometric properties and are not dependent upon the coordinate system you choose to impose on them. Most people, after all, learn how to measure the length of a line segment, or to measure an angle, or to calculate area long before they learn anything about coordinate systems.

 

[JanClaesen]

Okay, but, for example, the arclength of a circle in cartesian coordinates is finite, while the same circle in the polar coordinate system, a straight line (y = the radius) has an infinite arclength, which is too a geometrical property, so appearently not all geometrical properties are preserved, so why is the angle of intersection preserved nonetheless?

 

[HallsofIvy]

Okay, but, for example, the arclength of a circle in cartesian coordinates is finite, while the same circle in the polar coordinate system, a straight line (y = the radius) has an infinite arclength, which is too a geometrical property, so appearently not all geometrical properties are preserved, so why is the angle of intersection preserved nonetheless?

What? A circle is a circle, no matter what the coordinate system! I have no idea what you mean by saying "the same circle in the polar coordinate system, a straight line (y = the radius) has an infinite arclength".

The circle given by x^2+ y^2= R^2 in Cartesian coordinates has circumference 2\pi R. That same circle would be given, in polar coordinates, by r= R and it still has circumference 2\pi R.

 

[JanClaesen]

What? A circle is a circle, no matter what the coordinate system! I have no idea what you mean by saying "the same circle in the polar coordinate system, a straight line (y = the radius) has an infinite arclength".

The circle given by x^2+ y^2= R^2 in Cartesian coordinates has circumference 2\pi R. That same circle would be given, in polar coordinates, by r= R and it still has circumference 2\pi R.

No no I think you misunderstood me.

The radius is on the y-axis and is, for a circle, a constant, theta is on the x-axis and may take any value. This is a circle, represented in this particular coordinate system by a straight line: y = the radius.
In rectangular coordinates the graph is a circle given by x^2 + y^2 = R^2.
Both represent the same entity, however, if I measure the length of the straight line in the first coordinate system it's infinite, in the second coordinate system I measure the circumference of a circle, 2*pi*R. Now if a geometrical property like this is not preserved, in the transformation of one coordinate system to the other, why would another one, like the angle of intersection of two curves be preserved?

I'd like to understand this because the proof I read why the angle of intersection of a logarithmic spiral and a radius through the origin is a constant uses this property.

 

[HallsofIvy]

Then this has nothing to do with polar coordinates.