To analyze the intersections of the Archimedean spiral \( r = \theta \) with a ray emanating from the origin, we start by parameterizing the ray. A ray in polar coordinates can be expressed with an angle \( \phi \): thus, any point on this ray can be represented in polar coordinates as \( (r, \phi) \).
The point on the ray corresponding to a distance \( r \) from the origin is given by:
\[ (r, \phi) = (r, \phi) \]
To find the intersection points of the Archimedean spiral \( r = \theta \) with this ray, we set the polar equations equal:
\[ r = \theta = k \phi \]
where \( k \) is some constant that represents a specific distance along the ray and varies as \(\theta\) increases. We can denote the points of intersection according to \( n \) as follows:
\[ \theta_n = k \phi_n \]
This relationship gives us the intersection points of the spiral with the ray. To find the specific points where these intersections occur, we can denote the \( n \)-th intersection point occurring at:
\[ \theta_n = n \cdot k \]
Then \( r \) at this intersection will be:
\[ r_n = \theta_n = n \cdot k \]
The positions along the ray are given by the polar coordinates:
\[ (r_n, \phi) = (n \cdot k, \phi) \]
Now let us consider the distance between two consecutive intersection points, namely, \( (r_n, \phi) \) and \( (r_{n+1}, \phi) \):
- The \( n \)-th intersection point is: \((n \cdot k, \phi)\)
- The \((n+1)\)-th intersection point is: \(((n+1) \cdot k, \phi)\)
To find the distance \( d \) between these points in Cartesian coordinates, we can convert from polar to Cartesian:
\[ (r, \theta) \rightarrow (r \cos \theta, r \sin \theta) \]
Calculating the Cartesian coordinates for both points gives:
- For \( (r_n, \phi) = (n \cdot k, \phi) \):
\[ (x_n, y_n) = (n \cdot k \cos \phi, n \cdot k \sin \phi) \]
- For \( (r_{n+1}, \phi) = ((n+1) \cdot k, \phi) \):
\[ (x_{n+1}, y_{n+1}) = ((n + 1) \cdot k \cos \phi, (n + 1) \cdot k \sin \phi) \]
The distance \( d \) between these two points is given by:
\[ d = \sqrt{(x_{n+1} - x_n)^2 + (y_{n+1} - y_n)^2} \]
Substituting the Cartesian coordinates, we get:
\[ d = \sqrt{(((n + 1) \cdot k \cos \phi) - (n \cdot k \cos \phi))^2 + (((n + 1) \cdot k \sin \phi) - (n \cdot k \sin \phi))^2} \]
This simplifies to:
\[ d = \sqrt{(k \cos \phi)^2 + (k \sin \phi)^2} = \sqrt{k^2 (\cos^2 \phi + \sin^2 \phi)} = \sqrt{k^2} = k \]
Thus, the distance \( d \) between two consecutive intersection points of the Archimedean spiral with the ray is constant and equal to \( k \). Therefore, we conclude:
The consecutive intersection points of the Archimedean spiral with any ray emanating from the origin are equidistant from each other, and that distance is \( k \).