Coordinate Systems

A manifold is a space that locally looks like $\mathbb{R}^n$. To do calculus on a manifold, we need coordinate systems — smooth maps that identify regions of the manifold with regions of $\mathbb{R}^n$.

A coordinate system consists of two inverse functions:

• chart $\chi$: maps manifold points → coordinate tuples (e.g., $(x,y)$ or $(r,\theta)$)
• point $\chi^{-1}$: maps coordinate tuples → manifold points

The book introduces these as (chart coordsys) and (point coordsys) in Scheme. In Python, we represent a coordinate system as a pair of functions.

Example: Polar Coordinates on $\mathbb{R}^2$

A point $p = (x, y)$ in rectangular coordinates has polar coordinates:

$r = \sqrt{x^2 + y^2}, \quad \theta = \text{atan2}(y, x)$

Going back from polar to rectangular:

$x = r\cos\theta, \quad y = r\sin\theta$

Coordinate Independence

The key insight: a manifold point is a geometric object independent of coordinates. The point $(3, 4)$ in rectangular and the point $(5, \text{atan2}(4,3))$ in polar refer to the same point on the manifold. Both coordinate systems describe the same reality.

Your Task

Implement the polar coordinate system:

• polar_chart(p): convert rectangular point $(x, y)$ → polar coordinates $(r, \theta)$
• polar_point(c): convert polar coordinates $(r, \theta)$ → rectangular point $(x, y)$