Functional Differential Geometry
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a. The rectangular coordinate equation for the Lemniscate of Bernoulli is
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Find the expression in polar coordinates.
b. Describe a helix space curve in both rectangular and cylindrical coordinates.
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(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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2022.12.10 Saturday (c) All rights reserved by ACHK
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