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JanClaesen
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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".JanClaesen said: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 said: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].
Polar coordinates use an angle and a distance from the origin to represent a point, while cartesian coordinates use x and y coordinates on a grid.
To convert from polar to cartesian coordinates, you can use the equations x = r*cos(theta) and y = r*sin(theta), where r is the distance and theta is the angle. To convert from cartesian to polar coordinates, you can use the equations r = sqrt(x^2 + y^2) and theta = arctan(y/x).
To find the angle of intersection between two lines in polar coordinates, you can use the formula theta = arctan((m1-m2)/(1+m1m2)), where m1 and m2 are the slopes of the two lines.
Yes, you can graph polar and cartesian coordinates on the same grid by converting between the two coordinate systems and plotting the points accordingly.
Polar coordinates are commonly used in physics and engineering, especially for representing circular and rotational motion. Cartesian coordinates are often used in geometry and algebra, and are useful for graphing and solving equations.