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(define R2 (make-manifold R^n 2))
(define U (patch 'origin R2))
; these are coordinate systems
(define R2-rect (coordinate-system 'rectangular U))
(define R2-polar (coordinate-system 'polar/cylindrical U))
; these are charts, and their inverses
(define R2-rect-chi (chart R2-rect))
(define R2-rect-chi-inverse (point R2-rect))
(define R2-polar-chi (chart R2-polar))
(define R2-polar-chi-inverse (point R2-polar))
((compose R2-polar-chi R2-rect-chi-inverse) (up 'x0 'y0))
; (up
; (sqrt (+ (expt x0 2) (expt y0 2))) ; radius
; (atan y0 x0)) ; angle (theta)
((compose R2-rect-chi R2-polar-chi-inverse) (up 'r0 'theta0))
; (up
; (* r0 (cos theta0)) ; x
; (* r0 (sin theta0))) ; y
((D (compose R2-rect-chi R2-polar-chi-inverse)) (up 'r0 'theta0))
; (down
; (up
; (cos theta0)
; (sin theta0))
; (up
; (* -1 r0 (sin theta0))
; (* r0 (cos theta0))))
(define R2->R (-> (UP Real Real) Real))
(define f (compose (literal-function 'f-rect R2->R) R2-rect-chi))
(define R2-rect-point (R2-rect-chi-inverse (up 'x0 'y0)))
(define corresponding-polar-point
(R2-polar-chi-inverse
(up (sqrt (+ (square 'x0) (square 'y0)))
(atan 'y0 'x0))))
(f R2-rect-point)
; (f-rect (up x0 y0))
(f corresponding-polar-point)
; (f-rect (up x0 y0))
; confirms that the CAS simplifies to the same point
(define-coordinates (up x y) R2-rect)
(define-coordinates (up r theta) R2-polar)
(x (R2-rect-chi-inverse (up 'x0 'y0)))
; x0
; expect r0 * cos(theta0)
(x (R2-polar-chi-inverse (up 'r0 'theta0)))
; (* r0 (cos theta0))
; expect r0 * sin(theta0)
(y (R2-polar-chi-inverse (up 'r0 'theta0)))
; (* r0 (sin theta0))
;(r (R2-polar-chi-inverse (up 'r0 'theta0)))
; r0
; expect sqrt(x0^2 + y0^2)
(r (R2-rect-chi-inverse (up 'x0 'y0)))
; (sqrt (+ (expt x0 2) (expt y0 2)))
; expect atan(y0, x0)
(theta (R2-rect-chi-inverse (up 'x0 'y0)))
; (atan y0 x0)
; h: x * r^2 + y^3
(define h (+ (* x (square r)) (cube y)))
(h R2-rect-point)
; (+ (expt x0 3) (* x0 (expt y0 2)) (expt y0 3))
; aka: x0^3 + x0 y0^2 + y0^3
; x0 (x0^2 + y0^2) + y0^3
(h (R2-polar-chi-inverse (up 'r0 'theta0)))
; (+ (* (expt r0 3) (expt (sin theta0) 3)) (* (expt r0 3) (cos theta0)))
;;; Exersize 2.1a
((- r (* 2 'a (+ 1 (cos theta))))
((point R2-rect) (up 'x 'y)))
; (/
; (+ (* -2 a x)
; (* -2 a (sqrt (+ (expt x 2) (expt y 2))))
; (expt x 2) (expt y 2))
; (sqrt (+ (expt x 2) (expt y 2))))
; "Lemniscate of Bernoulli"
; (x^2 + y^2)^2 = 2 a^2 (x^2 - y^2)
((-
(square (+ (square x) (square y)))
(* 2
(square 'a)
(- (square x) (square y))))
((point R2-polar) (up 'r0 'theta0)))
; (+ (* -4 (expt a 2) (expt r0 2) (expt (cos theta0) 2))
; (* 2 (expt a 2) (expt r0 2))
; (expt r0 4))
;
; r^2 / a^2 + 2 = 4 cos^2 (theta)
; => r^2 = 2 a^2 cos(2 theta)
; this matches Wikipedia, with substitution of a^2 = 2 c^2 (in the rectangular version)
;;; Exersize 2.1b
(define R3-rect-chi (chart R3-rect))
(define R3-rect-chi-inverse (point R3-rect))
(define R3-cyl-chi (chart R3-cyl))
(define R3-cyl-chi-inverse (point R3-cyl))
(define-coordinates (up x y z) R3-rect)
(define-coordinates (up r theta z-cyl) R3-cyl)
; R->R3
(define (helix-rect t)
(up (* 'R (cos t))
(* 'R (sin t))
(* 's t)))
; R->R3
(define (helix-cyl t)
(up 'R
t
(* 's t)))
((compose R3-rect-chi R3-cyl-chi-inverse helix-cyl) 't)
; (up
; (* R (cos t))
; (* R (sin t))
; (* s t))
((compose R3-cyl-chi R3-rect-chi-inverse helix-rect) 't)
; (up R
; t
; (* s t))
((- helix-rect (compose R3-rect-chi R3-cyl-chi-inverse helix-cyl)) 't)
; (up 0 0 0)
;;; Exersize 2.2
; given polar coordinates of point on plane, get to spherical coordinates of point on sphere with:
((compose
(chart S2-spherical)
(point S2-Riemann)
(chart R2-rect)
(point R2-polar))
(up 'rho 'theta))
; (up (acos (/ (+ -1 (expt rho 2))
; (+ 1 (expt rho 2))))
; theta)
; given spherical coordinates of point on sphere, what are the polar
; coordinates of corresponding point on the plane?
((compose
(chart R2-polar)
(point R2-rect)
(chart S2-Riemann)
(point S2-spherical))
(up 'phi 'lambda))
; (up (/ (sin phi)
; (+ -1 (cos phi)))
; (atan (* -1 (sin lambda))
; (* -1 (cos lambda))))
; this is just the inverse of the above, right?
; huh, I would expect the second term to just be lambda.
; atan (-sin(lambda), -cos(lambda)) -> tan^-1(tan(lambda)) -> lambda
; seems like an identity, ok
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