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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?
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".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?
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].