Ex 2.1 Curves

Functional Differential Geometry

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a. The rectangular coordinate equation for the Lemniscate of Bernoulli is

\displaystyle{(x^2 + y^2)^2 = 2 a^2 (x^2 - y^2)}.

Find the expression in polar coordinates.

b. Describe a helix space curve in both rectangular and cylindrical coordinates.

~~~

(define-coordinates (up x y) R2-rect)

(define-coordinates (up r theta) R2-polar)

;

(define R2-rect-chi (chart R2-rect))

; R2-rect-chi
;     generates the rectangle coordinates of a point.

(define R2-rect-chi-inverse (point R2-rect))

; R2-rect-chi-inverse
;     gets the abstract representation.

(x (R2-rect-chi-inverse (up 'x0 'y0)))

; Function x
;     gets the x coordinate
;          of an (abstract-represented) point.

;

(define R2-polar-chi (chart R2-polar))

(define R2-polar-chi-inverse (point R2-polar))

(x (R2-polar-chi-inverse (up 'r0 'theta0)))

(r (R2-polar-chi-inverse (up 'r0 'theta0)))

(r (R2-rect-chi-inverse (up 'x0 'y0)))

(theta (R2-rect-chi-inverse (up 'x0 'y0)))

;

(define h (+ (* x (square r)) (cube y)))

(define R2-rect-point
  (R2-rect-chi-inverse (up 'x_0 'y_0)))

(show-expression
 (h R2-rect-point))

(show-expression
 (h (R2-polar-chi-inverse (up 'r_0 'theta_0))))

(show-expression
 ((- r (* 2 'a (+ 1 (cos theta))))
  ((point R2-rect) (up 'x 'y))))

; Ex 2.1 a

(show-expression
 ((- (square (+ (square x) (square y)))
     (* 2 (square 'a) (- (square x) (square y))))
  ((point R2-rect) (up 'x 'y))))

(show-expression
 ((- (square (+ (square x) (square y)))
     (* 2 (square 'a) (- (square x) (square y))))
  ((point R2-polar) (up 'r 'theta))))

; Ex 2.1 b

(define-coordinates (up r theta z) R3-cyl)

(define-coordinates (up x y z) R3-rect)

(show-expression
 ((- (up r z) (up 'R (* 'a theta)))
  ((point R3-cyl) (up 'r 'theta 'z))))

(show-expression
 ((- (up r z) (up 'R (* 'a theta)))
  ((point R3-rect) (up 'x 'y 'z))))

— Me@2022-12-10 10:29:59 AM

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