- #1

- 56

- 0

## Main Question or Discussion Point

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?

- Thread starter JanClaesen
- Start date

- #1

- 56

- 0

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?

- #2

- 240

- 2

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.

- #3

- 56

- 0

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?

- #4

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 956

- #5

- 56

- 0

- #6

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 956

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

- #7

- 56

- 0

No no I think you misunderstood me.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 [itex]x^2+ y^2= R^2[/itex] in Cartesian coordinates has circumference [itex]2\pi R[/itex]. That same circle would be given, in polar coordinates, by r= R and it still has circumference [itex]2\pi R[/itex].

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.

- #8

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 956

Then this has nothing to do with polar coordinates.

- Last Post

- Replies
- 7

- Views
- 1K

- Last Post

- Replies
- 4

- Views
- 2K

- Last Post

- Replies
- 4

- Views
- 13K

- Replies
- 4

- Views
- 3K

- Replies
- 8

- Views
- 3K

- Last Post

- Replies
- 4

- Views
- 6K

- Replies
- 1

- Views
- 10K

- Replies
- 2

- Views
- 5K

- Replies
- 7

- Views
- 1K

- Last Post

- Replies
- 5

- Views
- 5K